Should make this a little more maintainable.
PS_TARGET = $(TARGET_ROOT).ps
GS = gs
+EPS2EPS = eps2eps
PS2PDF = ps2pdf
PS2PNG = $(GS) -dSAFER -dBATCH -dNOPAUSE -sDEVICE=png16m -r300
rm -f $(DVI_TARGET)
.SUFFIXES: .eps
+.SUFFIXES: .eps.in
.SUFFIXES: .pdf
.SUFFIXES: .png
+.eps.in.eps:
+ $(EPS2EPS) $(EPS2EPSFLAGS) $< $@.tmp && mv -f $@.tmp $@
+
.eps.pdf:
$(PS2PDF) -dEPSCrop -sOutputFile=$@.tmp $< && mv -f $@.tmp $@
fig/loglogistic.eps \
# end of FIGURES_EPS
+$(FIGURES_EPS): fig/common/ad.ps
+$(FIGURES_EPS): fig/common/math.ps
+$(FIGURES_EPS): fig/common/newton.ps
+$(FIGURES_EPS): fig/common/plot.ps
+$(FIGURES_EPS): fig/common/splinterp.ps
+
+EPS2EPSFLAGS = -Ifig
+
include $(top_srcdir)/make-common
--- /dev/null
+# The master copies are in .eps.in files.
+/*.eps
+/*.pdf
+/*.png
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
-
-% Interpolation parameters.
-/nspline 20 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 1 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 2 1 0 tick 2 1 yticklabel
- 0 1 1 0 tick
- 0 -1 1 0 tick
-grestore
-
-% Plot another one: expm1 condition number.
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax { % Tx => T(f(x))
- dup Texpm1 % Tx T(e^x-1)
- dup Tq 0.0 eq {
- Texp
- } {
- exch dup % T(e^x-1) Tx Tx
- Texp Tmul % T(e^x-1) T(x*e^x)
- exch Tdiv % T(x*e^x/(e^x-1))
- } ifelse
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
+
+% Interpolation parameters.
+/nspline 20 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 2 1 0 tick 2 1 yticklabel
+ 0 1 1 0 tick
+ 0 -1 1 0 tick
+grestore
+
+% expm1 condition number.
+gsave
+ setupconditionnumberplot
+ nspline axmin axmax { % Tx => T(f(x))
+ dup Texpm1 % Tx T(e^x-1)
+ dup Tq 0.0 eq {
+ Texp
+ } {
+ exch dup % T(e^x-1) Tx Tx
+ Texp Tmul % T(e^x-1) T(x*e^x)
+ exch Tdiv % T(x*e^x/(e^x-1))
+ } ifelse
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 0.5 def /aymin -2.8 def /aymax 0.5 def
-
-% Interpolation parameters.
-/nspline 75 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 1 1 0 tick
- 0 -1 1 0 tick -1 1 yticklabel
- 0 -2 1 0 tick -2 1 yticklabel
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- 5 dict begin
- /Tf {
- dup Texp % Tx T(e^x)
- dup Tneg Tln1p % Tx T(e^x) T(log(1-e^x))
- 2 index Texpm1 Tneg % Tx T(e^x) T(log(1-e^x)) T(1-e^x)
- Tmul % Tx T(e^x) T((1-e^x)*log(1-e^x))
- Tdiv Tmul Tneg % T((-x*e^x)/((1-e^x)*log(1-e^x)))
- } def
- /fxmax -0.0001 def
- /fxmin axmin def
- /n nspline def
- /fxwidth fxmax fxmin sub def
- newpath
- nspline axmin -0.0001 {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n axmin -0.0001 {Tf} cubicsplinterpostep curveto } for
- % Fabricate a final point at the origin with positive
- % derivative. (Too lazy to compute the limit...)
- 0.0 0.0 T % Tx0 Ty0 Tx3
- 0.0 1.0 T % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol
- 5 index Tq 5 index Tq
- curveto
- cubicsplinterpostop
- stroke
- end
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 0.5 def /aymin -2.8 def /aymax 0.5 def
+
+% Interpolation parameters.
+/nspline 75 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 1 1 0 tick
+ 0 -1 1 0 tick -1 1 yticklabel
+ 0 -2 1 0 tick -2 1 yticklabel
+grestore
+
+gsave
+ setupconditionnumberplot
+ 5 dict begin
+ /Tf {
+ dup Texp % Tx T(e^x)
+ dup Tneg Tln1p % Tx T(e^x) T(log(1-e^x))
+ 2 index Texpm1 Tneg % Tx T(e^x) T(log(1-e^x)) T(1-e^x)
+ Tmul % Tx T(e^x) T((1-e^x)*log(1-e^x))
+ Tdiv Tmul Tneg % T((-x*e^x)/((1-e^x)*log(1-e^x)))
+ } def
+ /fxmax -0.0001 def
+ /fxmin axmin def
+ /n nspline def
+ /fxwidth fxmax fxmin sub def
+ newpath
+ nspline axmin -0.0001 {Tf} cubicsplinterpostart moveto % Tx0 Ty0
+ 0 1 n 1 sub { n axmin -0.0001 {Tf} cubicsplinterpostep curveto } for
+ % Fabricate a final point at the origin with positive
+ % derivative. (Too lazy to compute the limit...)
+ 0.0 0.0 T % Tx0 Ty0 Tx3
+ 0.0 1.0 T % Tx0 Ty0 Tx3 Ty3
+ cubicsplinterpocontrol
+ 5 index Tq 5 index Tq
+ curveto
+ cubicsplinterpostop
+ stroke
+ end
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -1.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
-
-% Interpolation parameters.
-/nspline 20 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- -1 aymin a2bb moveto
- -1 aymax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick
-
- % y ticks
- 0 2 1 0 tick 2 1 yticklabel
- 0 1 1 0 tick
- 0 -1 1 0 tick
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline -0.9 axmax {
- dup dup % Tx Tx Tx
- 1.0 Tconst Tadd % Tx Tx T(1+x)
- Tdiv % Tx T(x/(1+x))
- exch Tln1p % T(x/(1+x)) T(log(1+x))
- Tdiv % T(x/[(1+x)*log(1+x)])
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -1.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
+
+% Interpolation parameters.
+/nspline 20 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ -1 aymin a2bb moveto
+ -1 aymax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick
+
+ % y ticks
+ 0 2 1 0 tick 2 1 yticklabel
+ 0 1 1 0 tick
+ 0 -1 1 0 tick
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline -0.9 axmax {
+ dup dup % Tx Tx Tx
+ 1.0 Tconst Tadd % Tx Tx T(1+x)
+ Tdiv % Tx T(x/(1+x))
+ exch Tln1p % T(x/(1+x)) T(log(1+x))
+ Tdiv % T(x/[(1+x)*log(1+x)])
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
-
-% Interpolation parameters.
-/nspline 20 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
- newpath
- axmin 1 a2bb moveto
- axmax 1 a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont % encoding => encoding
- {
- dup 45 /minus put % replace hyphen by minus
- } reencodefont
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 2 -1 0 tick 2 -1 yticklabel
- 0 1 -1 0 tick
- 0 -1 -1 0 tick -1 -1 yticklabel
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- nspline axmin axmax { % Tx => T(f(x))
- dup Texp % Tx T(e^x)
- dup Tln1p % Tx T(e^x) T(log(1+e^x))
- 1 index 1.0 Tconst Tadd % Tx T(e^x) T(log(1+e^x)) T(1+e^x)
- Tmul % Tx T(e^x) T((1+e^x)*log(1+e^x))
- Tdiv Tmul % T((x*e^x)/((1+e^x)*log(1+e^x)))
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
+
+% Interpolation parameters.
+/nspline 20 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+ newpath
+ axmin 1 a2bb moveto
+ axmax 1 a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 2 -1 0 tick 2 -1 yticklabel
+ 0 1 -1 0 tick
+ 0 -1 -1 0 tick -1 -1 yticklabel
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline axmin axmax { % Tx => T(f(x))
+ dup Texp % Tx T(e^x)
+ dup Tln1p % Tx T(e^x) T(log(1+e^x))
+ 1 index 1.0 Tconst Tadd % Tx T(e^x) T(log(1+e^x)) T(1+e^x)
+ Tmul % Tx T(e^x) T((1+e^x)*log(1+e^x))
+ Tdiv Tmul % T((x*e^x)/((1+e^x)*log(1+e^x)))
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -2.5 def /aymax 1.5 def
-
-% Interpolation parameters.
-/nspline 8 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 1 -1 0 tick 1 -1 yticklabel
- 0 0.5 -0.5 0 tick
- 0 -0.5 -0.5 0 tick
- 0 -1 -1 0 tick
- 0 -1.5 -0.5 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax { % Tx => T(f(x))
- dup Tneg Texp % Tx T(e^-x)
- dup 1.0 Tconst Tadd % Tx T(e^-x) T(1 + e^-x)
- Tdiv % Tx T(e^-x/(1 + e^-x))
- Tmul % T(x*e^-x/(1 + e^-x))
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -2.5 def /aymax 1.5 def
+
+% Interpolation parameters.
+/nspline 8 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 1 -1 0 tick 1 -1 yticklabel
+ 0 0.5 -0.5 0 tick
+ 0 -0.5 -0.5 0 tick
+ 0 -1 -1 0 tick
+ 0 -1.5 -0.5 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline axmin axmax { % Tx => T(f(x))
+ dup Tneg Texp % Tx T(e^-x)
+ dup 1.0 Tconst Tadd % Tx T(e^-x) T(1 + e^-x)
+ Tdiv % Tx T(e^-x/(1 + e^-x))
+ Tmul % T(x*e^-x/(1 + e^-x))
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -4.5 def /axmax 4.5 def /aymin -0.5 def /aymax 1.2 def
-
-% Interpolation parameters.
-/nspline 8 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin 1 a2bb moveto
- axmax 1 a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 4 0 0 -1 tick 4 -1 xticklabel
- 3 0 0 -1 tick
- 2 0 0 -1 tick 2 -1 xticklabel
- 1 0 0 -1 tick
- -1 0 0 -1 tick
- -2 0 0 -1 tick -2 -1 xticklabel
- -3 0 0 -1 tick
- -4 0 0 -1 tick -4 -1 xticklabel
-
- % y ticks
- 0 1 -1 0 tick 1 -1 yticklabel
- 0 0.75 -0.5 0 tick
- 0 0.5 -0.5 0 tick
- 0 0.25 -0.5 0 tick
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax { % Tx => T(f(x))
- dup Tneg Texp % Tx T(e^-x)
- dup dup Tmul % Tx T(e^-x) T(e^-2x)
- 1.0 Tconst exch Tsub % Tx T(e^-x) T(1-e^-2x)
- Tdiv % Tx T(e^-x/(1-e^-2x))
- 2.0 Tconst Tmul Tmul % T(2x*e^-x/(1-e^-2x))
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -4.5 def /axmax 4.5 def /aymin -0.5 def /aymax 1.2 def
+
+% Interpolation parameters.
+/nspline 8 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin 1 a2bb moveto
+ axmax 1 a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 4 0 0 -1 tick 4 -1 xticklabel
+ 3 0 0 -1 tick
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1 0 0 -1 tick
+ -1 0 0 -1 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+ -3 0 0 -1 tick
+ -4 0 0 -1 tick -4 -1 xticklabel
+
+ % y ticks
+ 0 1 -1 0 tick 1 -1 yticklabel
+ 0 0.75 -0.5 0 tick
+ 0 0.5 -0.5 0 tick
+ 0 0.25 -0.5 0 tick
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline axmin axmax { % Tx => T(f(x))
+ dup Tneg Texp % Tx T(e^-x)
+ dup dup Tmul % Tx T(e^-x) T(e^-2x)
+ 1.0 Tconst exch Tsub % Tx T(e^-x) T(1-e^-2x)
+ Tdiv % Tx T(e^-x/(1-e^-2x))
+ 2.0 Tconst Tmul Tmul % T(2x*e^-x/(1-e^-2x))
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -0.2 def /axmax 1.2 def /aymin -3.2 def /aymax 6.2 def
-
-% Interpolation parameters.
-/nspline 75 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-/newton % n x0 Tf => x
-{
- 0 1 5 -1 roll % x0 Tf start step n
- {
- pop % x0 Tf
- 1 index Tvar % x0 Tf Tx0
- 1 index % x0 Tf Tx0 Tf
- exec % x0 Tf Ty0
- Tqv % x0 Tf y0 dy0
- dup 0.0 eq { pop pop exit } if
- div % x0 Tf y0/dy0
- 2 index exch % x0 Tf x0 y0/dy0
- sub % x0 Tf x1
- dup 0.0 ne {
- dup 3 index sub % x0 Tf x1 x1-x0
- 1 index div abs % x0 Tf x1 |(x1-x0)/x1|
- eps le { pop exit } if
- } if
- exch % x0 x1 Tf
- 3 -1 roll % x1 Tf x0
- pop % x1 Tf
- } for
- pop % x
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rmoveto %show
- gsave
- dup % v v
- true charpath pathbbox % v llx lly urx ury
- 3 -1 roll % v llx urx lly ury
- 2 copy add 2 div % v llx urx lly ury avgy
- 3 1 roll % v llx urx avgy lly ury
- exch sub % v llx urx avgy dy
- 4 2 roll % v avgy dy llx urx
- 2 copy add 2 div % v avgy dy llx urx avgx
- 3 1 roll % v avgy dy avgx llx urx
- exch sub % v avgy dy avgx dx
- exch 4 1 roll % v avgx avgy dy dx
- min % v avgx avgy majaxis
- 0 360 arc % v
- closepath
- 1 setgray
- fill
- grestore
- show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-/f
-{
- dup Tlogit exch % T(log(p/(1-p))) Tp
- 1.0 Tconst exch Tsub % T(log(p/(1-p))) T(1-p)
- 1.0 Tconst exch Tdiv % T(log(p/(1-p))) T(1/(1-p))
- exch Tdiv % T((1/(1-p))/(log(p/(1-p))))
-} def
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- 0.5 aymin a2bb moveto
- 0.5 aymax a2bb lineto
- stroke
- newpath
- 1 aymin a2bb moveto
- 1 aymax a2bb lineto
- stroke
-
- % Find the minimum of the dip.
- 10 0.75 { % numerator of f'
- dup dup % Tp Tp Tp
- 1.0 Tconst exch Tsub % Tp Tp T(1-p)
- 1.0 Tconst exch Tdiv % Tp Tp T(1/(1-p))
- Tmul % Tp T(p/(1-p))
- Tln % Tp T(log(p/(1-p)))
- Tmul % T(p*log(p/(1-p)))
- 1.0 Tconst Tsub % T(p*log(p/(1-p))-1)
- } newton
- Tvar f Tq
- newpath
- axmin 1 index a2bb moveto
- axmax 1 index a2bb lineto
- stroke
- pop
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.875 0 0 -0.25 tick
- 0.75 0 0 -0.25 tick
- 0.625 0 0 -0.25 tick
- 0.5 0 0 -0.5 tick 0.5 -1 xticklabel
- 0.375 0 0 -0.25 tick
- 0.25 0 0 -0.25 tick
- 0.125 0 0 -0.25 tick
-
- % y ticks
- 0 5 -0.5 0 tick
- 0 4 -1 0 tick 4 -1 yticklabel
- 0 3 -0.5 0 tick
- 0 2 -1 0 tick 2 -1 yticklabel
- 0 1 -0.5 0 tick
- 0 -1 -0.5 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline 0.00001 0.499 {f} cubicsplinterpolate
- nspline 0.501 0.999 {f} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -0.2 def /axmax 1.2 def /aymin -3.2 def /aymax 6.2 def
+
+% Interpolation parameters.
+/nspline 75 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/newton.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+/f
+{
+ dup Tlogit exch % T(log(p/(1-p))) Tp
+ 1.0 Tconst exch Tsub % T(log(p/(1-p))) T(1-p)
+ 1.0 Tconst exch Tdiv % T(log(p/(1-p))) T(1/(1-p))
+ exch Tdiv % T((1/(1-p))/(log(p/(1-p))))
+} def
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+
+ newpath
+ 0.5 aymin a2bb moveto
+ 0.5 aymax a2bb lineto
+ stroke
+ newpath
+ 1 aymin a2bb moveto
+ 1 aymax a2bb lineto
+ stroke
+
+ % Find the minimum of the dip.
+ 10 0.75 { % numerator of f'
+ dup dup % Tp Tp Tp
+ 1.0 Tconst exch Tsub % Tp Tp T(1-p)
+ 1.0 Tconst exch Tdiv % Tp Tp T(1/(1-p))
+ Tmul % Tp T(p/(1-p))
+ Tln % Tp T(log(p/(1-p)))
+ Tmul % T(p*log(p/(1-p)))
+ 1.0 Tconst Tsub % T(p*log(p/(1-p))-1)
+ } newton
+ Tvar f Tq
+ newpath
+ axmin 1 index a2bb moveto
+ axmax 1 index a2bb lineto
+ stroke
+ pop
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.875 0 0 -0.25 tick
+ 0.75 0 0 -0.25 tick
+ 0.625 0 0 -0.25 tick
+ 0.5 0 0 -0.5 tick 0.5 -1 xticklabel
+ 0.375 0 0 -0.25 tick
+ 0.25 0 0 -0.25 tick
+ 0.125 0 0 -0.25 tick
+
+ % y ticks
+ 0 5 -0.5 0 tick
+ 0 4 -1 0 tick 4 -1 yticklabel
+ 0 3 -0.5 0 tick
+ 0 2 -1 0 tick 2 -1 yticklabel
+ 0 1 -0.5 0 tick
+ 0 -1 -0.5 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline 0.00001 0.499 {f} cubicsplinterpolate
+ nspline 0.501 0.999 {f} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -3.2 def /axmax 0.5 def /aymin -4.5 def /aymax 4.5 def
-
-% Interpolation parameters.
-/nspline 70 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin 1 a2bb moveto
- axmax 1 a2bb lineto
- stroke
- newpath
- 0.5 ln aymin a2bb moveto
- 0.5 ln aymax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
- -2.25 0 0 -0.5 tick
- -2.5 0 0 -0.5 tick
- -2.75 0 0 -0.5 tick
- -3 0 0 -1 tick -3 -1 xticklabel
-
- % y ticks
- 0 3 -0.5 0 tick
- 0 2 -1 0 tick 2 0.5 yticklabel
- 0 1 -0.5 0 tick
- 0 -1 -0.5 0 tick
- 0 -2 -1 0 tick -2 0.5 yticklabel
- 0 -3 -0.5 0 tick
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin -0.694 {
- dup Texpm1 Tneg % Tx T(1-e^x)
- 1 index Texp Tlogit % Tx T(1-e^x) T(log(e^x/(1-e^x)))
- Tmul Tdiv % T(x/((1-e^x)*log((e^x/(1-e^x)))))
- } cubicsplinterpolate
- nspline -0.693 -.0001 {
- dup Texpm1 Tneg % Tx T(1-e^x)
- 1 index Texp Tlogit % Tx T(1-e^x) T(log(e^x/(1-e^x)))
- Tmul Tdiv % T(x/((1-e^x)*log((e^x/(1-e^x)))))
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -3.2 def /axmax 0.5 def /aymin -4.5 def /aymax 4.5 def
+
+% Interpolation parameters.
+/nspline 70 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ axmin 1 a2bb moveto
+ axmax 1 a2bb lineto
+ stroke
+ newpath
+ 0.5 ln aymin a2bb moveto
+ 0.5 ln aymax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+ -2.25 0 0 -0.5 tick
+ -2.5 0 0 -0.5 tick
+ -2.75 0 0 -0.5 tick
+ -3 0 0 -1 tick -3 -1 xticklabel
+
+ % y ticks
+ 0 3 -0.5 0 tick
+ 0 2 -1 0 tick 2 0.5 yticklabel
+ 0 1 -0.5 0 tick
+ 0 -1 -0.5 0 tick
+ 0 -2 -1 0 tick -2 0.5 yticklabel
+ 0 -3 -0.5 0 tick
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline axmin -0.694 {
+ dup Texpm1 Tneg % Tx T(1-e^x)
+ 1 index Texp Tlogit % Tx T(1-e^x) T(log(e^x/(1-e^x)))
+ Tmul Tdiv % T(x/((1-e^x)*log((e^x/(1-e^x)))))
+ } cubicsplinterpolate
+ nspline -0.693 -.0001 {
+ dup Texpm1 Tneg % Tx T(1-e^x)
+ 1 index Texp Tlogit % Tx T(1-e^x) T(log(e^x/(1-e^x)))
+ Tmul Tdiv % T(x/((1-e^x)*log((e^x/(1-e^x)))))
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -0.75 def /axmax 0.75 def /aymin -3.2 def /aymax 6.2 def
-
-% Interpolation parameters.
-/nspline 90 def % must be even here to avoid zero
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- -0.5 aymin a2bb moveto
- -0.5 aymax a2bb lineto
- stroke
- newpath
- 0.5 aymin a2bb moveto
- 0.5 aymax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 0.5 0 0 -1 tick 0.5 -1 xticklabel
- 0.375 0 0 -0.25 tick
- 0.25 0 0 -0.5 tick
- 0.125 0 0 -0.25 tick
- -0.125 0 0 -0.25 tick
- -0.25 0 0 -0.5 tick
- -0.375 0 0 -0.25 tick
- -0.5 0 0 -1 tick -0.5 -1 xticklabel
-
- % y ticks
- 0 5 -0.5 0 tick
- 0 4 -1 0 tick 4 -1 yticklabel
- 0 3 -0.5 0 tick
- 0 2 -1 0 tick 2 -1 yticklabel
- 0 1 -0.5 0 tick
- 0 -1 -0.5 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline -.499 .499 {
- dup dup % Tx Tx Tx
- dup Tmul % Tx Tx T(x^2)
- 4.0 Tconst Tmul % Tx Tx T(4x^2)
- 1.0 Tconst exch Tsub % Tx Tx T(1-4x^2)
- Tdiv % Tx T(x/(1-4x^2))
- exch Tlogithalf % T(x/(1-4x^2)) T(log((1/2+x)/(1/2-x)))
- Tdiv
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -0.75 def /axmax 0.75 def /aymin -3.2 def /aymax 6.2 def
+
+% Interpolation parameters.
+/nspline 90 def % must be even here to avoid zero
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ -0.5 aymin a2bb moveto
+ -0.5 aymax a2bb lineto
+ stroke
+ newpath
+ 0.5 aymin a2bb moveto
+ 0.5 aymax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 0.5 0 0 -1 tick 0.5 -1 xticklabel
+ 0.375 0 0 -0.25 tick
+ 0.25 0 0 -0.5 tick
+ 0.125 0 0 -0.25 tick
+ -0.125 0 0 -0.25 tick
+ -0.25 0 0 -0.5 tick
+ -0.375 0 0 -0.25 tick
+ -0.5 0 0 -1 tick -0.5 -1 xticklabel
+
+ % y ticks
+ 0 5 -0.5 0 tick
+ 0 4 -1 0 tick 4 -1 yticklabel
+ 0 3 -0.5 0 tick
+ 0 2 -1 0 tick 2 -1 yticklabel
+ 0 1 -0.5 0 tick
+ 0 -1 -0.5 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline -.499 .499 {
+ dup dup % Tx Tx Tx
+ dup Tmul % Tx Tx T(x^2)
+ 4.0 Tconst Tmul % Tx Tx T(4x^2)
+ 1.0 Tconst exch Tsub % Tx Tx T(1-4x^2)
+ Tdiv % Tx T(x/(1-4x^2))
+ exch Tlogithalf % T(x/(1-4x^2)) T(log((1/2+x)/(1/2-x)))
+ Tdiv
+ } cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -3.5 def /aymax 1.5 def
-
-% Interpolation parameters.
-/nspline 40 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin 1 a2bb moveto
- axmax 1 a2bb lineto
- stroke
- newpath
- axmin 0 axmin sub a2bb moveto
- axmax 0 axmax sub a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 1 -1 0 tick 1 1 yticklabel
- 0 -1 -1 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
- 0 -3 -1 0 tick
-grestore
-
-gsave
- 1 0 0 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax {
- dup Tneg Texp % Tx T(e^-x)
- dup 1.0 Tconst Tadd % Tx T(e^-x) T(1+e^-x)
- 1 index Tln1p % Tx T(e^-x) T(1+e^-x) T(log(1+e^-x))
- Tmul % Tx T(e^-x) T((1+e^-x)*log(1+e^-x))
- Tdiv % Tx T(e^-x/((1+e^-x)*log(1+e^-x)))
- Tmul % T(x*e^-x/((1+e^-x)*log(1+e^-x)))
- Tneg % T(-x*e^-x/((1+e^-x)*log(1+e^-x)))
- } cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -3.5 def /aymax 1.5 def
+
+% Interpolation parameters.
+/nspline 40 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ axmin 1 a2bb moveto
+ axmax 1 a2bb lineto
+ stroke
+ newpath
+ axmin 0 axmin sub a2bb moveto
+ axmax 0 axmax sub a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 1 -1 0 tick 1 1 yticklabel
+ 0 -1 -1 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+ 0 -3 -1 0 tick
+grestore
+
+gsave
+ setupconditionnumberplot
+ nspline axmin axmax {
+ dup Tneg Texp % Tx T(e^-x)
+ dup 1.0 Tconst Tadd % Tx T(e^-x) T(1+e^-x)
+ 1 index Tln1p % Tx T(e^-x) T(1+e^-x) T(log(1+e^-x))
+ Tmul % Tx T(e^-x) T((1+e^-x)*log(1+e^-x))
+ Tdiv % Tx T(e^-x/((1+e^-x)*log(1+e^-x)))
+ Tmul % T(x*e^-x/((1+e^-x)*log(1+e^-x)))
+ Tneg % T(-x*e^-x/((1+e^-x)*log(1+e^-x)))
+ } cubicsplinterpolate
+grestore
+
+showpage
--- /dev/null
+% Automatic differentiation.
+
+/T { 2 array astore } def % x dx => [x dx]
+/Tq { 0 get } def % Tx => x (`position')
+/Tv { 1 get } def % Tx => dx (`velocity')
+/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
+/Tvar { 1.0 T } def % x => [x 1]
+/Tconst { 0.0 T } def % x => [x 0]
+
+/Tadd % Tx Ty => T(x+y)
+{
+ Tqv % Tx y dy
+ 3 -1 roll % y dy Tx
+ Tqv % y dy x dx
+ exch % y dy dx x
+ 4 -1 roll add % dy dx x+y
+ 3 1 roll add % x+y dx+dy
+ T % T(x+y)
+} def
+
+/Tmul % Tx Ty => T(x*y)
+{
+ Tqv % Tx y dy
+ 3 -1 roll % y dy Tx
+ Tqv % y dy x dx
+ 3 -1 roll % y x dx dy
+ 2 index mul % y x dx x*dy
+ exch % y x x*dy dx
+ 3 index mul % y x x*dy y*dx
+ add % y x y*dx+x*dy
+ 3 1 roll mul % y*dx+x*dy x*y
+ exch % x*y y*dx+x*dy
+ T % T(x*y)
+} def
+
+/Tneg % Tx => T(-x)
+{
+ Tqv % x dx
+ 0 exch sub % x -dx
+ exch % -dx x
+ 0 exch sub % -dx -x
+ exch % -x -dx
+ T % T(-x)
+} def
+
+/Trecip % Tx => T(1/x)
+{
+ Tqv exch % x dx
+ 1 exch div % dx 1/x
+ dup dup % dx 1/x 1/x 1/x
+ 4 1 roll % 1/x dx 1/x 1/x
+ 0 exch sub % 1/x dx 1/x -1/x
+ mul mul % 1/x -dx/x^2
+ T % T(1/x)
+} def
+
+/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
+/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
+
+/Tln % Tx => T(log x)
+{
+ Tqv % x dx
+ exch dup ln % dx x log(x)
+ 3 1 roll % log(x) dx x
+ div % log(x) dx/x
+ T % T(log x)
+} def
+
+/Tln1p % Tx => T(log(1+x))
+{
+ Tqv % x dx
+ exch dup % dx x x
+ ln1p % dx x log(1+x)
+ 3 1 roll % log(1+x) dx x
+ 1 add % log(1+x) dx 1+x
+ div % log(1+x) dx/(1+x)
+ T % T(log(1+x))
+} def
+
+/Texp % Tx => T(e^x)
+{
+ Tqv % x dx
+ exch 2.718281828 exch exp exch % e^x dx
+ 1 index mul % e^x e^x*dx
+ T % T(e^x)
+} def
+
+/Texpm1 % Tx => T(e^x-1)
+{
+ Tqv % x dx
+ exch dup expm1 % dx x e^x-1
+ 3 1 roll % e^x-1 dx x
+ 2.718281828 exch exp % e^x-1 dx e^x
+ mul % e^x-1 e^x*dx
+ T % T(e^x-1)
+} def
+
+/Tsin % Tx => T(sin(x))
+{
+ Tqv % x dx
+ exch dup % dx x x
+ sin exch cos % dx sin(x) cos(x)
+ 3 -1 roll % sin(x) cos(x) dx
+ 0.017453292519943295 mul
+ mul % sin(x) cos(x)*dx
+ T % T(sin(x))
+} def
+
+/Tcos % Tx => T(cos(x))
+{
+ Tqv % x dx
+ exch dup % dx x x
+ cos exch sin % dx cos(x) sin(x)
+ 3 -1 roll % cos(x) sin(x) dx
+ 0.017453292519943295 mul
+ mul 1 exch sub % cos(x) -sin(x)*dx
+ T % T(cos(x))
+} def
+
+/Tlogistic % Tx => T(1/(1+e^{-x}))
+{
+ 0.0 Tconst exch Tsub Texp % T(e^{-x})
+ 1.0 Tconst Tadd % T(1+e^{-x})
+ 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
+} def
+
+/Tlogit % Tp => T(log(p/(1-p)))
+{
+ dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
+ 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
+ or {
+ dup 1.0 Tconst exch Tsub % Tp T(1-p)
+ Tdiv Tln % T(log(p/(1-p)))
+ } {
+ dup 2.0 Tconst Tmul % Tp T2p
+ 1.0 Tconst exch Tsub % Tp T(1-2p)
+ exch Tdiv % T((1-2p)/p)
+ Tln1p % T(log1p((1-2p)/p))
+ 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
+ } ifelse
+} def
+
+/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
+{
+ dup Tq abs 0.5 1.0 3.718281828 div sub le {
+ dup 2.0 Tconst Tmul % Tp T(2p)
+ exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
+ Tdiv Tln1p % T(log(1+2p/(1/2-p)))
+ } {
+ dup 0.5 Tconst Tadd % T(p) T(1/2+p)
+ exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
+ Tdiv Tln % T(log((1/2+p)/(1/2-p)))
+ } ifelse
+} def
+
+/Tlerp % Tt fxmin fxmax => Tx
+{
+ 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
+ 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
+ exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
+} def
--- /dev/null
+
+/point % x y greyness => ...
+{
+ gsave
+ newpath
+ 3 1 roll % greyness x y
+ 1 0 360 arc
+ gsave setgray fill grestore
+ stroke
+ grestore
+} def
+
+/showcurveto % x1 y1 x2 y2 x3 y3 => ---
+{
+ gsave
+ .5 setlinewidth
+ .5 setgray
+ currentpoint
+ newpath moveto
+ 5 index 5 index
+ lineto
+ stroke
+ 2 copy
+ newpath
+ moveto
+ 3 index 3 index
+ lineto
+ stroke
+ grestore
+ gsave
+ .1 setlinewidth
+ 0 setgray
+ currentpoint 0 point
+ 5 index 5 index
+ 1 point
+ 3 index 3 index
+ 1 point
+ 2 copy
+ 0 point
+ grestore
+ curveto
+} def
--- /dev/null
+% Linear `spline' interpolation.
+
+/linearsplinterpolate % n fxmin fxmax Tf => ---
+{
+ 6 dict begin
+ /Tf exch def
+ /fxmax exch def
+ /fxmin exch def
+ /n exch def
+ /fxwidth fxmax fxmin sub def
+ newpath
+ 0 1 n {/i exch def
+ i n linearsplinterpoint Tconst
+ fxwidth Tconst Tmul fxmin Tconst Tadd
+ dup Tf Ta2bb Tq exch Tq exch
+ i 0 eq { moveto } { lineto } ifelse
+ } for
+ stroke
+ end
+} def
--- /dev/null
+% Math utilities.
+
+/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
+/sqrteps eps sqrt def
+/cbrteps eps 1 3 div exp def
+
+/ln1p % x => log(1+x)
+{
+ dup 1.0 add % x 1+x
+ dup 1.0 eq { pop } {
+ dup ln % x 1+x log(1+x)
+ 3 -1 roll % 1+x log(1+x) x
+ mul % 1+x x*log(1+x)
+ exch % x*log(1+x) 1+x
+ 1.0 sub % x*log(1+x) (1+x)-1
+ div % x*log(1+x)/[(1+x)-1]
+ } ifelse
+} def
+
+/expm1
+{
+ dup abs cbrteps gt {
+ % e^x - 1
+ 2.718281828 exch exp 1 sub
+ } { dup abs sqrteps gt {
+ % x + x^2/2 + x^3/3
+ dup dup dup dup % x x x x
+ mul mul 6 div % x x x^3/6
+ exch dup mul 2 div % x x^3/6 x^2/2
+ add add % x+x^2/2+x^3/6
+ } { dup abs eps gt {
+ % x + x^2/2
+ dup dup mul 2 div add
+ } {
+ % nothing
+ } ifelse } ifelse } ifelse
+} def
+
+/logistic % x => 1/(1+e^{-x})
+{
+ 0 exch sub 2.718281828 exch exp 1 add 1 exch div
+} def
+
+/logit % p => log(p/(1-p))
+{
+ dup -1.0 logistic lt % p (p<logistic(-1))
+ 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
+ or {
+ dup 1.0 exch sub % p 1-p
+ div ln % log(p/(1-p))
+ } {
+ dup 2.0 mul % p 2p
+ 1.0 exch sub % p 1-2p
+ exch div % (1-2p)/p
+ ln1p % log1p((1-2p)/p)
+ 0.0 exch sub % -log1p((1-2p)/p)
+ } ifelse
+} def
+
+/logithalf % p => log((1/2+p)/(1/2-p))
+{
+ dup abs 0.5 1.0 3.718281828 div sub le {
+ dup 2 mul % p 2p
+ exch 0.5 exch sub % 2p 1/2-p
+ div ln1p % log(1+2p/(1/2-p))
+ } {
+ dup 0.5 add % p 1/2+p
+ exch 0.5 exch sub % 1/2+p 1/2-p
+ div ln % log((1/2+p)/(1/2-p))
+ } ifelse
+} def
--- /dev/null
+% Newton-Raphson root-finding.
+/newton % n x0 Tf => x
+{
+ 0 1 5 -1 roll % x0 Tf start step n
+ {
+ pop % x0 Tf
+ 1 index Tvar % x0 Tf Tx0
+ 1 index % x0 Tf Tx0 Tf
+ exec % x0 Tf Ty0
+ Tqv % x0 Tf y0 dy0
+ dup 0.0 eq { pop pop exit } if
+ div % x0 Tf y0/dy0
+ 2 index exch % x0 Tf x0 y0/dy0
+ sub % x0 Tf x1
+ dup 0.0 ne {
+ dup 3 index sub % x0 Tf x1 x1-x0
+ 1 index div abs % x0 Tf x1 |(x1-x0)/x1|
+ eps le { pop exit } if
+ } if
+ exch % x0 x1 Tf
+ 3 -1 roll % x1 Tf x0
+ pop % x1 Tf
+ } for
+ pop % x
+} def
--- /dev/null
+% Derived axis parameters.
+/awidth axmax axmin sub def
+/aheight aymax aymin sub def
+
+% 1pt margin on either side for arrow heads.
+/a2bbx bbwidth 2 sub awidth div def
+/a2bby bbheight 2 sub aheight div def
+
+/a2bb % xa ya => xbb ybb
+{
+ exch a2bbx mul % ya xbb
+ exch a2bby mul % xbb ybb
+} def
+
+/Ta2bb % Txa Tya => Txbb Tybb
+{
+ exch a2bbx Tconst Tmul % Tyf Tx2bb
+ exch a2bby Tconst Tmul % Txbb Tybb
+} def
+
+/arrowhead % x y angle => ---
+{
+ gsave
+ 1 setlinecap
+ newpath
+ 3 1 roll % angle x y
+ moveto rotate
+ -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
+ stroke
+ grestore
+} def
+
+/ticku 5 def % tick size in pt
+
+/tick % xa ya tx ty => ---
+{
+ exch ticku mul % xa ya ty dxbb
+ exch ticku mul % xa ya dxbb dybb
+ 4 2 roll a2bb % dxbb dybb xbb ybb
+ gsave newpath moveto rlineto stroke grestore
+} def
+
+/ticklabel % v xa ya dxbb dybb => ---
+{
+ 4 2 roll a2bb % v dxbb dybb xbb ybb
+ newpath moveto rlineto show
+} def
+
+/xticklabel % x ty => ---
+{
+ exch dup % ty x x
+ 4 string cvs % ty x xs
+ gsave
+ newpath 0 0 moveto
+ dup true charpath
+ pathbbox % ty x xs llx lly urx ury
+ grestore
+ 3 -1 roll sub % ty x xs llx urx ury-lly
+ 3 1 roll exch add % ty x xs ury-lly urx+llx
+ 2 div % ty x xs ury-lly (urx+llx)/2
+ 0 exch sub % ty x xs ury-lly -(urx+llx)/2
+ exch % ty x xs -(urx+llx)/2 ury-lly
+ 4 index 0 lt {
+ ticku 2 mul add
+ } {
+ pop ticku 2 mul
+ } ifelse % ty x xs -w/2 h
+ 5 -1 roll mul % x xs -w/2 h*ty
+ 4 -1 roll % xs -w/2 h*ty x
+ 0 % xs -w/2 h*ty x y
+ 4 2 roll % xs x y -w/2 h*ty
+ ticklabel
+} def
+
+/yticklabel % y tx => ---
+{
+ exch dup % tx y y
+ 4 string cvs % tx y ys
+ gsave
+ newpath 0 0 moveto
+ dup true charpath
+ pathbbox % tx y ys llx lly urx ury
+ grestore
+ 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
+ 0 exch sub % tx y ys llx urx -(ury+lly)/2
+ 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
+ 4 index 0 lt {
+ ticku 2 mul add
+ } {
+ pop ticku 2 mul
+ } ifelse % tx y ys -h/2 w
+ 5 -1 roll mul % y ys -h/2 w*tx
+ exch % y ys w*tx -h/2
+ 4 -1 roll % ys w*tx -h/2 y
+ 0 % ys w*tx -h/2 y x
+ exch % ys w*tx -h/2 x y
+ 4 2 roll % ys x y w*tx -h/2
+ ticklabel
+} def
+
+/reencodefont % name font proc => font'
+{
+ exch dup length dict copy % name proc font'
+ dup /Encoding get % name proc font' encoding
+ dup length array copy % name proc font' encoding'
+ 3 -1 roll exec % name font' encoding'
+ 1 index exch % name font' encoding' font'
+ /Encoding exch put % name font'
+ definefont
+} def
+
+/setupplotbbox
+{
+ % Clip to bounding box.
+ newpath
+ llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
+ clip
+
+ % Center coordinates with 1pt margin for arrow heads.
+ 0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
+} def
+
+/setupasymptote
+{
+ 0.5 setgray
+ 0.4 setlinewidth
+ [4 4] 1 setdash
+} def
+
+/setupaxes
+{
+ 0.4 setlinewidth
+ /Times-Roman-Numeric
+ /Times-Roman findfont
+ { dup 45 /minus put } reencodefont % replace hyphen by minus
+ 10 scalefont setfont
+} def
+
+/drawaxes
+{
+ newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
+ axmax 0 a2bb 0 arrowhead
+ axmin 0 a2bb 180 arrowhead
+ newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
+ 0 aymax a2bb 90 arrowhead
+ 0 aymin a2bb -90 arrowhead
+} def
+
+/setupfunctionplot
+{
+ 0 0 1 setrgbcolor
+ 0.6 setlinewidth
+ 1 setlinecap
+} def
+
+/setupconditionnumberplot
+{
+ 1 0 0 setrgbcolor
+ 0.6 setlinewidth
+ 1 setlinecap
+} def
--- /dev/null
+% Interpolation points.
+
+/linpoint { div } def % i n => s_{n,i}
+
+/chebypoint % i n => s_{n,i}
+{
+ exch % n i
+ dup 0 eq {
+ pop pop 0.0
+ } {
+ 2 copy eq {
+ pop pop 1.0
+ } {
+ 2 mul 1 sub % n 2i-1
+ exch 2 mul % 2i-1 2n
+ div % (2i-1)/(2i)
+ 180 mul cos % cos(pi*(2i-1)/(2n))
+ 1 exch sub % 1-cos(pi*(2i-1)/(2n))
+ 2 div % [1-cos(pi*(2i-1)/(2n))]/2
+ } ifelse
+ } ifelse
+} def
+
+% Spline interpolation.
+
+% Compute the tangent vector of a piecewise linear interpolation
+% between two consecutive spline interpolation nodes:
+%
+% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
+% c_i(1) = c_{i-1}(0)
+%
+/splinter % t i n => T(c_i(t))
+{
+ 2 copy % t i n i n
+ splinterpoint % t i n x0
+ 3 1 roll % t x0 i n
+ exch 1 add exch % t x0 i+1 n
+ splinterpoint % t x0 x1
+ 1 index sub Tconst % t x0 T(x1-x0)
+ 3 -1 roll Tvar % x0 T(x1-x0) Tt
+ Tmul % x0 T((x1-x0)*t)
+ exch Tconst % T((x1-x0)*t) Tx0
+ Tadd % T(x0+(x1-x0)*t)
+} def
+
+% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
+/splinterpoval % fxmin fxmax Tf t i n => x0 y0
+{
+ splinter % fxmin fxmax Tf Tc
+ 4 2 roll % Tf Tc fxmin fxmax
+ Tlerp % Tf Tx0a
+ dup 3 -1 roll % Tx0a Tx0a Tf
+ exec % Tx0a Ty0a
+ Ta2bb % Tx0 Ty0
+} def
+
+% Compute control points of a cubic spline matching the starting and
+% ending tangent vectors. Pops the starting vectors, keeps the ending.
+%
+% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
+%
+% Note that
+%
+% c(0) = p0, c'(0) = 3 p1 - 3 p0,
+% c(1) = p1, c'(1) = 3 p3 - 3 p2,
+%
+% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
+%
+% p1 = p0 + c'(0)/3,
+% p2 = p3 - c'(1)/3.
+%
+/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
+{
+ 4 2 roll % Tx3 Ty3 Tx0 Ty0
+ % Compute x1 = x1 + dx1/3.
+ exch Tqv % Tx3 Ty3 Ty0 x0 dx0
+ 3 div add % Tx3 Ty3 Ty0 x1
+ % Compute y1 = y0 + dy0/3.
+ exch Tqv % Tx3 Ty3 x1 y0 dy0
+ 3 div add % Tx3 Ty3 x1 y1
+ % Compute x2 = x3 - dx3/3.
+ 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
+ 3 div sub % Tx3 Ty3 x1 y1 x2
+ % Compute y2 = y3 - dy3/3.
+ 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
+ 3 div sub % Tx3 Ty3 x1 y1 x2 y2
+} def
+
+/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
+{
+ 0.0 0 % n fxmin fxmax Tf t i
+ 6 -1 roll % fxmin fxmax Tf t i n
+ splinterpoval % Tx0 Ty0
+ 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
+} def
+
+/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
+ % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
+{
+ 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
+ 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
+ splinterpoval % Tx0 Ty0 Tx3 Ty3
+ cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
+ 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
+} def
+
+/cubicsplinterpostop % Tx3 Ty3 => ---
+{
+ pop pop
+} def
+
+/cubicsplinterpolate % n fxmin fxmax Tf => ---
+{
+ 5 dict begin
+ /Tf exch def
+ /fxmax exch def
+ /fxmin exch def
+ /n exch def
+ /fxwidth fxmax fxmin sub def
+ newpath
+ n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
+ 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
+ cubicsplinterpostop
+ stroke
+ end
+} def
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
-
-% Interpolation parameters.
-/nspline 20 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin -1 a2bb moveto
- axmax -1 a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 2 1 0 tick 2 1 yticklabel
- 0 1 1 0 tick
- 0 -1 1 0 tick
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax {Texpm1} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
+
+% Interpolation parameters.
+/nspline 20 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin -1 a2bb moveto
+ axmax -1 a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 2 1 0 tick 2 1 yticklabel
+ 0 1 1 0 tick
+ 0 -1 1 0 tick
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin axmax {Texpm1} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 0.5 def /aymin -2.8 def /aymax 0.5 def
-
-% Interpolation parameters.
-/nspline 70 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 1 1 0 tick
- 0 -1 1 0 tick -1 1 yticklabel
- 0 -2 1 0 tick -2 1 yticklabel
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin -.001 {Texp Tneg Tln1p} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 0.5 def /aymin -2.8 def /aymax 0.5 def
+
+% Interpolation parameters.
+/nspline 70 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 1 1 0 tick
+ 0 -1 1 0 tick -1 1 yticklabel
+ 0 -2 1 0 tick -2 1 yticklabel
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin -.001 {Texp Tneg Tln1p} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -1.2 def /axmax 2.2 def /aymin -3.8 def /aymax 2.8 def
-
-% Interpolation parameters.
-/nspline 70 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [2 3] 0 setdash
- newpath
- -1 aymin a2bb moveto
- -1 aymax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
-
- % y ticks
- 0 2 -1 0 tick
- 0 1 -1 0 tick 1 -1 yticklabel
- 0 -1 -1 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
- 0 -3 -1 0 tick
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline -0.99 axmax {Tln1p} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -1.2 def /axmax 2.2 def /aymin -3.8 def /aymax 2.8 def
+
+% Interpolation parameters.
+/nspline 70 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ -1 aymin a2bb moveto
+ -1 aymax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+
+ % y ticks
+ 0 2 -1 0 tick
+ 0 1 -1 0 tick 1 -1 yticklabel
+ 0 -1 -1 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+ 0 -3 -1 0 tick
+grestore
+
+gsave
+ setupfunctionplot
+ nspline -0.99 axmax {Tln1p} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
-
-% Interpolation parameters.
-/nspline 20 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont % encoding => encoding
- {
- dup 45 /minus put % replace hyphen by minus
- } reencodefont
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 2 -1 0 tick 2 -1 yticklabel
- 0 1 -1 0 tick
- 0 -1 -1 0 tick -1 -1 yticklabel
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- nspline axmin axmax {Texp Tln1p} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -1.8 def /aymax 2.8 def
+
+% Interpolation parameters.
+/nspline 20 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 2 -1 0 tick 2 -1 yticklabel
+ 0 1 -1 0 tick
+ 0 -1 -1 0 tick -1 -1 yticklabel
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin axmax {Texp Tln1p} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -0.5 def /aymax 1.2 def
-
-% Interpolation parameters.
-/nspline 20 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin 1 a2bb moveto
- axmax 1 a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 1 -1 0 tick 1 -1 yticklabel
- 0 0.75 -0.5 0 tick
- 0 0.5 -0.5 0 tick
- 0 0.25 -0.5 0 tick
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax {Tlogistic} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -0.5 def /aymax 1.2 def
+
+% Interpolation parameters.
+/nspline 20 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin 1 a2bb moveto
+ axmax 1 a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 1 -1 0 tick 1 -1 yticklabel
+ 0 0.75 -0.5 0 tick
+ 0 0.5 -0.5 0 tick
+ 0 0.25 -0.5 0 tick
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin axmax {Tlogistic} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto
-llx ury lineto
-urx ury lineto
-urx lly lineto
-closepath
-clip
-
-% Axes.
-/axmin -4.5 def /axmax 4.5 def /aymin -0.75 def /aymax 0.75 def
-
-% Interpolation parameters.
-/nspline 8 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin 0.5 a2bb moveto
- axmax 0.5 a2bb lineto
- stroke
- newpath
- axmin -0.5 a2bb moveto
- axmax -0.5 a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 4 0 0 -1 tick 4 -1 xticklabel
- 3 0 0 -1 tick
- 2 0 0 -1 tick 2 -1 xticklabel
- 1 0 0 -1 tick
- -1 0 0 1 tick
- -2 0 0 1 tick -2 1 xticklabel
- -3 0 0 1 tick
- -4 0 0 1 tick -4 1 xticklabel
-
- % y ticks
- 0 0.5 -1 0 tick 0.5 -1 yticklabel
- 0 0.25 -0.5 0 tick
- 0 -0.25 -0.5 0 tick
- 0 -0.5 -1 0 tick -0.5 -1 yticklabel
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax {Tlogistic 0.5 Tconst Tsub} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -4.5 def /axmax 4.5 def /aymin -0.75 def /aymax 0.75 def
+
+% Interpolation parameters.
+/nspline 8 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin 0.5 a2bb moveto
+ axmax 0.5 a2bb lineto
+ stroke
+ newpath
+ axmin -0.5 a2bb moveto
+ axmax -0.5 a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 4 0 0 -1 tick 4 -1 xticklabel
+ 3 0 0 -1 tick
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1 0 0 -1 tick
+ -1 0 0 1 tick
+ -2 0 0 1 tick -2 1 xticklabel
+ -3 0 0 1 tick
+ -4 0 0 1 tick -4 1 xticklabel
+
+ % y ticks
+ 0 0.5 -1 0 tick 0.5 -1 yticklabel
+ 0 0.25 -0.5 0 tick
+ 0 -0.25 -0.5 0 tick
+ 0 -0.5 -1 0 tick -0.5 -1 yticklabel
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin axmax {Tlogistic 0.5 Tconst Tsub} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -0.2 def /axmax 1.2 def /aymin -4.2 def /aymax 4.2 def
-
-% Interpolation parameters.
-/nspline 75 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-/newton % n x0 Tf => x
-{
- 0 1 5 -1 roll % x0 Tf start step n
- {
- pop % x0 Tf
- 1 index Tvar % x0 Tf Tx0
- 1 index % x0 Tf Tx0 Tf
- exec % x0 Tf Ty0
- Tqv % x0 Tf y0 dy0
- dup 0.0 eq { pop pop exit } if
- div % x0 Tf y0/dy0
- 2 index exch % x0 Tf x0 y0/dy0
- sub % x0 Tf x1
- dup 0.0 ne {
- dup 3 index sub % x0 Tf x1 x1-x0
- 1 index div abs % x0 Tf x1 |(x1-x0)/x1|
- eps le { pop exit } if
- } if
- exch % x0 x1 Tf
- 3 -1 roll % x1 Tf x0
- pop % x1 Tf
- } for
- pop % x
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- 1 aymin a2bb moveto
- 1 aymax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.875 0 0 -0.25 tick
- 0.75 0 0 -0.25 tick
- 0.625 0 0 -0.25 tick
- 0.5 0 0 -0.5 tick 0.5 -1 xticklabel
- 0.375 0 0 -0.25 tick
- 0.25 0 0 -0.25 tick
- 0.125 0 0 -0.25 tick
-
- % y ticks
- 0 3 -0.5 0 tick
- 0 2 -1 0 tick 2 -1 yticklabel
- 0 1 -0.5 0 tick
- 0 -1 -0.5 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
- 0 -3 -0.5 0 tick
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline 0.00001 0.999 {Tlogit} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -0.2 def /axmax 1.2 def /aymin -4.2 def /aymax 4.2 def
+
+% Interpolation parameters.
+/nspline 75 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ 1 aymin a2bb moveto
+ 1 aymax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.875 0 0 -0.25 tick
+ 0.75 0 0 -0.25 tick
+ 0.625 0 0 -0.25 tick
+ 0.5 0 0 -0.5 tick 0.5 -1 xticklabel
+ 0.375 0 0 -0.25 tick
+ 0.25 0 0 -0.25 tick
+ 0.125 0 0 -0.25 tick
+
+ % y ticks
+ 0 3 -0.5 0 tick
+ 0 2 -1 0 tick 2 -1 yticklabel
+ 0 1 -0.5 0 tick
+ 0 -1 -0.5 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+ 0 -3 -0.5 0 tick
+grestore
+
+gsave
+ setupfunctionplot
+ nspline 0.00001 0.999 {Tlogit} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -3.2 def /axmax 0.5 def /aymin -4.5 def /aymax 4.5 def
-
-% Interpolation parameters.
-/nspline 40 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
- -2.25 0 0 -0.5 tick
- -2.5 0 0 -0.5 tick
- -2.75 0 0 -0.5 tick
- -3 0 0 -1 tick -3 -1 xticklabel
-
- % y ticks
- 0 3 -0.5 0 tick
- 0 2 -1 0 tick 2 0.5 yticklabel
- 0 1 -0.5 0 tick
- 0 -1 -0.5 0 tick
- 0 -2 -1 0 tick -2 0.5 yticklabel
- 0 -3 -0.5 0 tick
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin -.001 {Texp Tlogit} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -3.2 def /axmax 0.5 def /aymin -4.5 def /aymax 4.5 def
+
+% Interpolation parameters.
+/nspline 40 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+ -2.25 0 0 -0.5 tick
+ -2.5 0 0 -0.5 tick
+ -2.75 0 0 -0.5 tick
+ -3 0 0 -1 tick -3 -1 xticklabel
+
+ % y ticks
+ 0 3 -0.5 0 tick
+ 0 2 -1 0 tick 2 0.5 yticklabel
+ 0 1 -0.5 0 tick
+ 0 -1 -0.5 0 tick
+ 0 -2 -1 0 tick -2 0.5 yticklabel
+ 0 -3 -0.5 0 tick
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin -.001 {Texp Tlogit} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -0.75 def /axmax 0.75 def /aymin -3.2 def /aymax 3.2 def
-
-% Interpolation parameters.
-/nspline 90 def % must be even here to avoid zero
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptotes.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- -0.5 aymin a2bb moveto
- -0.5 aymax a2bb lineto
- stroke
- newpath
- 0.5 aymin a2bb moveto
- 0.5 aymax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
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- 0.25 0 0 -0.5 tick
- 0.125 0 0 -0.25 tick
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- -0.25 0 0 -0.5 tick
- -0.375 0 0 -0.25 tick
- -0.5 0 0 -1 tick -0.5 -1 xticklabel
-
- % y ticks
- 0 5 -0.5 0 tick
- 0 4 -1 0 tick 4 -1 yticklabel
- 0 3 -0.5 0 tick
- 0 2 -1 0 tick 2 -1 yticklabel
- 0 1 -0.5 0 tick
- 0 -1 -0.5 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline -.499 .499 {Tlogithalf} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -0.75 def /axmax 0.75 def /aymin -3.2 def /aymax 3.2 def
+
+% Interpolation parameters.
+/nspline 90 def % must be even here to avoid zero
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptotes.
+gsave
+ setupasymptote
+ newpath
+ -0.5 aymin a2bb moveto
+ -0.5 aymax a2bb lineto
+ stroke
+ newpath
+ 0.5 aymin a2bb moveto
+ 0.5 aymax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 0.5 0 0 -1 tick 0.5 -1 xticklabel
+ 0.375 0 0 -0.25 tick
+ 0.25 0 0 -0.5 tick
+ 0.125 0 0 -0.25 tick
+ -0.125 0 0 -0.25 tick
+ -0.25 0 0 -0.5 tick
+ -0.375 0 0 -0.25 tick
+ -0.5 0 0 -1 tick -0.5 -1 xticklabel
+
+ % y ticks
+ 0 5 -0.5 0 tick
+ 0 4 -1 0 tick 4 -1 yticklabel
+ 0 3 -0.5 0 tick
+ 0 2 -1 0 tick 2 -1 yticklabel
+ 0 1 -0.5 0 tick
+ 0 -1 -0.5 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+grestore
+
+gsave
+ setupfunctionplot
+ nspline -.499 .499 {Tlogithalf} cubicsplinterpolate
+grestore
+
+showpage
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%BoundingBox: 0 0 180 60
-%%BeginProlog
-
-% Bounding box parameters.
-/llx 0 def /lly 0 def /urx 180 def /ury 60 def
-/bbwidth urx llx sub def /bbheight ury lly sub def
-newpath
-llx lly moveto llx ury lineto urx ury lineto urx lly lineto closepath
-clip
-
-% Axes.
-/axmin -2.2 def /axmax 2.2 def /aymin -3.5 def /aymax 0.5 def
-
-% Interpolation parameters.
-/nspline 40 def
-/splinterpoint { chebypoint } def % i n => s_{i,n}
-
-% Derived axis parameters.
-/awidth axmax axmin sub def
-/aheight aymax aymin sub def
-
-% 1pt margin on either side for arrow heads.
-/a2bbx bbwidth 2 sub awidth div def
-/a2bby bbheight 2 sub aheight div def
-
-/a2bb % xa ya => xbb ybb
-{
- exch a2bbx mul % ya xbb
- exch a2bby mul % xbb ybb
-} def
-
-/Ta2bb % Txa Tya => Txbb Tybb
-{
- exch a2bbx Tconst Tmul % Tyf Tx2bb
- exch a2bby Tconst Tmul % Txbb Tybb
-} def
-
-% Math utilities.
-
-/eps 1.0 { dup 2 div dup 1.0 add 1.0 eq { pop exit } if exch pop } loop def
-/sqrteps eps sqrt def
-/cbrteps eps 1 3 div exp def
-
-/ln1p % x => log(1+x)
-{
- dup 1.0 add % x 1+x
- dup 1.0 eq { pop } {
- dup ln % x 1+x log(1+x)
- 3 -1 roll % 1+x log(1+x) x
- mul % 1+x x*log(1+x)
- exch % x*log(1+x) 1+x
- 1.0 sub % x*log(1+x) (1+x)-1
- div % x*log(1+x)/[(1+x)-1]
- } ifelse
-} def
-
-/expm1
-{
- dup abs cbrteps gt {
- % e^x - 1
- 2.718281828 exch exp 1 sub
- } { dup abs sqrteps gt {
- % x + x^2/2 + x^3/3
- dup dup dup dup % x x x x
- mul mul 6 div % x x x^3/6
- exch dup mul 2 div % x x^3/6 x^2/2
- add add % x+x^2/2+x^3/6
- } { dup abs eps gt {
- % x + x^2/2
- dup dup mul 2 div add
- } {
- % nothing
- } ifelse } ifelse } ifelse
-} def
-
-/logistic % x => 1/(1+e^{-x})
-{
- 0 exch sub 2.718281828 exch exp 1 add 1 exch div
-} def
-
-/logit % p => log(p/(1-p))
-{
- dup -1.0 logistic lt % p (p<logistic(-1))
- 1 index 1.0 logistic gt % p (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 exch sub % p 1-p
- div ln % log(p/(1-p))
- } {
- dup 2.0 mul % p 2p
- 1.0 exch sub % p 1-2p
- exch div % (1-2p)/p
- ln1p % log1p((1-2p)/p)
- 0.0 exch sub % -log1p((1-2p)/p)
- } ifelse
-} def
-
-/logithalf % p => log((1/2+p)/(1/2-p))
-{
- dup abs 0.5 1.0 3.718281828 div sub le {
- dup 2 mul % p 2p
- exch 0.5 exch sub % 2p 1/2-p
- div ln1p % log(1+2p/(1/2-p))
- } {
- dup 0.5 add % p 1/2+p
- exch 0.5 exch sub % 1/2+p 1/2-p
- div ln % log((1/2+p)/(1/2-p))
- } ifelse
-} def
-
-% Automatic differentiation.
-
-/T { 2 array astore } def % x dx => [x dx]
-/Tq { 0 get } def % Tx => x (`position')
-/Tv { 1 get } def % Tx => dx (`velocity')
-/Tqv { dup 0 get exch 1 get } def % [x dx] => x dx
-/Tvar { 1.0 T } def % x => [x 1]
-/Tconst { 0.0 T } def % x => [x 0]
-
-/Tadd % Tx Ty => T(x+y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- exch % y dy dx x
- 4 -1 roll add % dy dx x+y
- 3 1 roll add % x+y dx+dy
- T % T(x+y)
-} def
-
-/Tmul % Tx Ty => T(x*y)
-{
- Tqv % Tx y dy
- 3 -1 roll % y dy Tx
- Tqv % y dy x dx
- 3 -1 roll % y x dx dy
- 2 index mul % y x dx x*dy
- exch % y x x*dy dx
- 3 index mul % y x x*dy y*dx
- add % y x y*dx+x*dy
- 3 1 roll mul % y*dx+x*dy x*y
- exch % x*y y*dx+x*dy
- T % T(x*y)
-} def
-
-/Tneg % Tx => T(-x)
-{
- Tqv % x dx
- 0 exch sub % x -dx
- exch % -dx x
- 0 exch sub % -dx -x
- exch % -x -dx
- T % T(-x)
-} def
-
-/Trecip % Tx => T(1/x)
-{
- Tqv exch % x dx
- 1 exch div % dx 1/x
- dup dup % dx 1/x 1/x 1/x
- 4 1 roll % 1/x dx 1/x 1/x
- 0 exch sub % 1/x dx 1/x -1/x
- mul mul % 1/x -dx/x^2
- T % T(1/x)
-} def
-
-/Tsub { Tneg Tadd } def % Tx Ty => T(x-y)
-/Tdiv { Trecip Tmul } def % Tx Ty => T(x/y)
-
-/Tln % Tx => T(log x)
-{
- Tqv % x dx
- exch dup ln % dx x log(x)
- 3 1 roll % log(x) dx x
- div % log(x) dx/x
- T % T(log x)
-} def
-
-/Tln1p % Tx => T(log(1+x))
-{
- Tqv % x dx
- exch dup % dx x x
- ln1p % dx x log(1+x)
- 3 1 roll % log(1+x) dx x
- 1 add % log(1+x) dx 1+x
- div % log(1+x) dx/(1+x)
- T % T(log(1+x))
-} def
-
-/Texp % Tx => T(e^x)
-{
- Tqv % x dx
- exch 2.718281828 exch exp exch % e^x dx
- 1 index mul % e^x e^x*dx
- T % T(e^x)
-} def
-
-/Texpm1 % Tx => T(e^x-1)
-{
- Tqv % x dx
- exch dup expm1 % dx x e^x-1
- 3 1 roll % e^x-1 dx x
- 2.718281828 exch exp % e^x-1 dx e^x
- mul % e^x-1 e^x*dx
- T % T(e^x-1)
-} def
-
-/Tsin % Tx => T(sin(x))
-{
- Tqv % x dx
- exch dup % dx x x
- sin exch cos % dx sin(x) cos(x)
- 3 -1 roll % sin(x) cos(x) dx
- 0.017453292519943295 mul
- mul % sin(x) cos(x)*dx
- T % T(sin(x))
-} def
-
-/Tcos % Tx => T(cos(x))
-{
- Tqv % x dx
- exch dup % dx x x
- cos exch sin % dx cos(x) sin(x)
- 3 -1 roll % cos(x) sin(x) dx
- 0.017453292519943295 mul
- mul 1 exch sub % cos(x) -sin(x)*dx
- T % T(cos(x))
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-/Tlogit % Tp => T(log(p/(1-p)))
-{
- dup Tq -1.0 logistic lt % Tp (p<logistic(-1))
- 1 index Tq 1.0 logistic gt % Tp (p<logistic(-1)) (p>logistic(1))
- or {
- dup 1.0 Tconst exch Tsub % Tp T(1-p)
- Tdiv Tln % T(log(p/(1-p)))
- } {
- dup 2.0 Tconst Tmul % Tp T2p
- 1.0 Tconst exch Tsub % Tp T(1-2p)
- exch Tdiv % T((1-2p)/p)
- Tln1p % T(log1p((1-2p)/p))
- 0.0 Tconst exch Tsub % T(-log1p((1-2p)/p))
- } ifelse
-} def
-
-/Tlogithalf % Tp => T(log((1/2+p)/(1/2-p)))
-{
- dup Tq abs 0.5 1.0 3.718281828 div sub le {
- dup 2.0 Tconst Tmul % Tp T(2p)
- exch 0.5 Tconst exch Tsub % T(2p) T(1/2-p)
- Tdiv Tln1p % T(log(1+2p/(1/2-p)))
- } {
- dup 0.5 Tconst Tadd % T(p) T(1/2+p)
- exch 0.5 Tconst exch Tsub % T(1/2+p) T(1/2-p)
- Tdiv Tln % T(log((1/2+p)/(1/2-p)))
- } ifelse
-} def
-
-/Tlerp % Tt fxmin fxmax => Tx
-{
- 1 index sub Tconst % Tt fxmin T(fxmax-fxmin)
- 3 -1 roll Tmul % fxmin T((fxmax-fxmin)*t)
- exch Tconst Tadd % T(fxmin+(fxmax-fxmin)*t)
-} def
-
-/Tlogistic % Tx => T(1/(1+e^{-x}))
-{
- 0.0 Tconst exch Tsub Texp % T(e^{-x})
- 1.0 Tconst Tadd % T(1+e^{-x})
- 1.0 Tconst exch Tdiv % T(1/(1+e^{-x}))
-} def
-
-% Interpolation points.
-
-/linpoint { div } def % i n => s_{n,i}
-
-/chebypoint % i n => s_{n,i}
-{
- exch % n i
- dup 0 eq {
- pop pop 0.0
- } {
- 2 copy eq {
- pop pop 1.0
- } {
- 2 mul 1 sub % n 2i-1
- exch 2 mul % 2i-1 2n
- div % (2i-1)/(2i)
- 180 mul cos % cos(pi*(2i-1)/(2n))
- 1 exch sub % 1-cos(pi*(2i-1)/(2n))
- 2 div % [1-cos(pi*(2i-1)/(2n))]/2
- } ifelse
- } ifelse
-} def
-
-% Spline interpolation.
-
-% Compute the tangent vector of a piecewise linear interpolation
-% between two consecutive spline interpolation nodes:
-%
-% c_i(t) := xmin + (xmax - xmin)*(x_i + (x_{i+1} - x_i)*t)
-% c_i(1) = c_{i-1}(0)
-%
-/splinter % t i n => T(c_i(t))
-{
- 2 copy % t i n i n
- splinterpoint % t i n x0
- 3 1 roll % t x0 i n
- exch 1 add exch % t x0 i+1 n
- splinterpoint % t x0 x1
- 1 index sub Tconst % t x0 T(x1-x0)
- 3 -1 roll Tvar % x0 T(x1-x0) Tt
- Tmul % x0 T((x1-x0)*t)
- exch Tconst % T((x1-x0)*t) Tx0
- Tadd % T(x0+(x1-x0)*t)
-} def
-
-% Evaluate the tangent vector f(fxmin + (fxmax - fxmin)*c_i(t)).
-/splinterpoval % fxmin fxmax Tf t i n => x0 y0
-{
- splinter % fxmin fxmax Tf Tc
- 4 2 roll % Tf Tc fxmin fxmax
- Tlerp % Tf Tx0a
- dup 3 -1 roll % Tx0a Tx0a Tf
- exec % Tx0a Ty0a
- Ta2bb % Tx0 Ty0
-} def
-
-% Compute control points of a cubic spline matching the starting and
-% ending tangent vectors. Pops the starting vectors, keeps the ending.
-%
-% c(t) = (1-t)^3 p0 + 3 t (1-t)^2 p1 + 3 t^2 (1-t) p2 + t^3 p3,
-%
-% Note that
-%
-% c(0) = p0, c'(0) = 3 p1 - 3 p0,
-% c(1) = p1, c'(1) = 3 p3 - 3 p2,
-%
-% so p0 = c(0) = (x0, y0), p3 = c(1) = (x1, y1), and
-%
-% p1 = p0 + c'(0)/3,
-% p2 = p3 - c'(1)/3.
-%
-/cubicsplinterpocontrol % Tx0 Ty0 Tx3 Ty3 => Tx3 Ty3 x1 y1 x2 y2
-{
- 4 2 roll % Tx3 Ty3 Tx0 Ty0
- % Compute x1 = x1 + dx1/3.
- exch Tqv % Tx3 Ty3 Ty0 x0 dx0
- 3 div add % Tx3 Ty3 Ty0 x1
- % Compute y1 = y0 + dy0/3.
- exch Tqv % Tx3 Ty3 x1 y0 dy0
- 3 div add % Tx3 Ty3 x1 y1
- % Compute x2 = x3 - dx3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x3 dx3
- 3 div sub % Tx3 Ty3 x1 y1 x2
- % Compute y2 = y3 - dy3/3.
- 3 index Tqv % Tx3 Ty3 x1 y1 x2 y3 dy3
- 3 div sub % Tx3 Ty3 x1 y1 x2 y2
-} def
-
-/cubicsplinterpostart % n fxmin fxmax Tf => Tx0 Ty0 x0 y0
-{
- 0.0 0 % n fxmin fxmax Tf t i
- 6 -1 roll % fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0
- 1 index Tq 1 index Tq % Tx0 Ty0 x0 y0
-} def
-
-/cubicsplinterpostep % Tx0 Ty0 i n fxmin fxmax Tf
- % => Tx3 Ty3 x1 y1 x2 y2 x3 y3
-{
- 1.0 % Tx0 Ty0 i n fxmin fxmax Tf t
- 6 -2 roll % Tx0 Ty0 fxmin fxmax Tf t i n
- splinterpoval % Tx0 Ty0 Tx3 Ty3
- cubicsplinterpocontrol % Tx3 Ty3 x1 y1 x2 y2
- 5 index Tq 5 index Tq % Tx3 Ty3 x1 y1 x2 y2 x3 y3
-} def
-
-/cubicsplinterpostop % Tx3 Ty3 => ---
-{
- pop pop
-} def
-
-/cubicsplinterpolate % n fxmin fxmax Tf => ---
-{
- 5 dict begin
- /Tf exch def
- /fxmax exch def
- /fxmin exch def
- /n exch def
- /fxwidth fxmax fxmin sub def
- newpath
- n fxmin fxmax {Tf} cubicsplinterpostart moveto % Tx0 Ty0
- 0 1 n 1 sub { n fxmin fxmax {Tf} cubicsplinterpostep curveto } for
- cubicsplinterpostop
- stroke
- end
-} def
-
-% Graphics.
-
-/arrowhead % x y angle => ---
-{
- gsave
- 1 setlinecap
- newpath
- 3 1 roll % angle x y
- moveto rotate
- -2 2 rmoveto 2 -2 rlineto -2 -2 rlineto
- stroke
- grestore
-} def
-
-/ticku 5 def % tick size in pt
-
-/tick % xa ya tx ty => ---
-{
- exch ticku mul % xa ya ty dxbb
- exch ticku mul % xa ya dxbb dybb
- 4 2 roll a2bb % dxbb dybb xbb ybb
- gsave newpath moveto rlineto stroke grestore
-} def
-
-/ticklabel % v xa ya dxbb dybb => ---
-{
- 4 2 roll a2bb % v dxbb dybb xbb ybb
- newpath moveto rlineto show
-} def
-
-/xticklabel % x ty => ---
-{
- exch dup % ty x x
- 4 string cvs % ty x xs
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % ty x xs llx lly urx ury
- grestore
- 3 -1 roll sub % ty x xs llx urx ury-lly
- 3 1 roll exch add % ty x xs ury-lly urx+llx
- 2 div % ty x xs ury-lly (urx+llx)/2
- 0 exch sub % ty x xs ury-lly -(urx+llx)/2
- exch % ty x xs -(urx+llx)/2 ury-lly
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % ty x xs -w/2 h
- 5 -1 roll mul % x xs -w/2 h*ty
- 4 -1 roll % xs -w/2 h*ty x
- 0 % xs -w/2 h*ty x y
- 4 2 roll % xs x y -w/2 h*ty
- ticklabel
-} def
-
-/yticklabel % y tx => ---
-{
- exch dup % tx y y
- 4 string cvs % tx y ys
- gsave
- newpath 0 0 moveto
- dup true charpath
- pathbbox % tx y ys llx lly urx ury
- grestore
- 3 -1 roll add 2 div % tx y ys llx urx (ury+lly)/2
- 0 exch sub % tx y ys llx urx -(ury+lly)/2
- 3 1 roll exch sub % tx y ys -(ury+lly)/2 urx-llx
- 4 index 0 lt {
- ticku 2 mul add
- } {
- pop ticku 2 mul
- } ifelse % tx y ys -h/2 w
- 5 -1 roll mul % y ys -h/2 w*tx
- exch % y ys w*tx -h/2
- 4 -1 roll % ys w*tx -h/2 y
- 0 % ys w*tx -h/2 y x
- exch % ys w*tx -h/2 x y
- 4 2 roll % ys x y w*tx -h/2
- ticklabel
-} def
-
-/reencodefont % name font proc => font'
-{
- exch dup length dict copy % name proc font'
- dup /Encoding get % name proc font' encoding
- dup length array copy % name proc font' encoding'
- 3 -1 roll exec % name font' encoding'
- 1 index exch % name font' encoding' font'
- /Encoding exch put % name font'
- definefont
-} def
-
-%%EndProlog
-
-% Center coordinates with 1pt margin for arrow heads.
-0 axmin sub 0 aymin sub a2bb 1 add exch 1 add exch translate
-
-% Draw reference asymptote.
-gsave
- 0.5 setgray
- 0.4 setlinewidth
- [4 4] 0 setdash
- newpath
- axmin axmin a2bb moveto
- axmax axmax a2bb lineto
- stroke
-grestore
-
-% Draw axes.
-gsave
- 0.4 setlinewidth
- /Times-Roman-Numeric
- /Times-Roman findfont
- { dup 45 /minus put } reencodefont % replace hyphen by minus
- 10 scalefont setfont
-
- newpath axmin 0 a2bb moveto axmax 0 a2bb lineto stroke
- axmax 0 a2bb 0 arrowhead
- axmin 0 a2bb 180 arrowhead
- newpath 0 aymin a2bb moveto 0 aymax a2bb lineto stroke
- 0 aymax a2bb 90 arrowhead
- 0 aymin a2bb -90 arrowhead
-
- % x ticks
- 2 0 0 -1 tick 2 -1 xticklabel
- 1.75 0 0 -0.5 tick
- 1.5 0 0 -0.5 tick
- 1.25 0 0 -0.5 tick
- 1 0 0 -1 tick 1 -1 xticklabel
- 0.75 0 0 -0.5 tick
- 0.5 0 0 -0.5 tick
- 0.25 0 0 -0.5 tick
- -0.25 0 0 -0.5 tick
- -0.5 0 0 -0.5 tick
- -0.75 0 0 -0.5 tick
- -1 0 0 -1 tick -1 -1 xticklabel
- -1.25 0 0 -0.5 tick
- -1.5 0 0 -0.5 tick
- -1.75 0 0 -0.5 tick
- -2 0 0 -1 tick -2 -1 xticklabel
-
- % y ticks
- 0 -1 -1 0 tick
- 0 -2 -1 0 tick -2 -1 yticklabel
- 0 -3 -1 0 tick
-grestore
-
-gsave
- 0 0 1 setrgbcolor
- 0.6 setlinewidth
- 1 setlinecap
- nspline axmin axmax {Tlogistic Tln} cubicsplinterpolate
-grestore
-
-showpage
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%BoundingBox: 0 0 180 60
+%%BeginProlog
+
+% Bounding box parameters.
+/llx 0 def /lly 0 def /urx 180 def /ury 60 def
+/bbwidth urx llx sub def /bbheight ury lly sub def
+
+% Axes.
+/axmin -2.2 def /axmax 2.2 def /aymin -3.5 def /aymax 0.5 def
+
+% Interpolation parameters.
+/nspline 40 def
+/splinterpoint { chebypoint } def % i n => s_{i,n}
+
+(common/ad.ps) runlibfile
+(common/math.ps) runlibfile
+(common/plot.ps) runlibfile
+(common/splinterp.ps) runlibfile
+
+%%EndProlog
+
+setupplotbbox
+
+% Draw reference asymptote.
+gsave
+ setupasymptote
+ newpath
+ axmin axmin a2bb moveto
+ axmax axmax a2bb lineto
+ stroke
+grestore
+
+% Draw axes.
+gsave
+ setupaxes
+ drawaxes
+
+ % x ticks
+ 2 0 0 -1 tick 2 -1 xticklabel
+ 1.75 0 0 -0.5 tick
+ 1.5 0 0 -0.5 tick
+ 1.25 0 0 -0.5 tick
+ 1 0 0 -1 tick 1 -1 xticklabel
+ 0.75 0 0 -0.5 tick
+ 0.5 0 0 -0.5 tick
+ 0.25 0 0 -0.5 tick
+ -0.25 0 0 -0.5 tick
+ -0.5 0 0 -0.5 tick
+ -0.75 0 0 -0.5 tick
+ -1 0 0 -1 tick -1 -1 xticklabel
+ -1.25 0 0 -0.5 tick
+ -1.5 0 0 -0.5 tick
+ -1.75 0 0 -0.5 tick
+ -2 0 0 -1 tick -2 -1 xticklabel
+
+ % y ticks
+ 0 -1 -1 0 tick
+ 0 -2 -1 0 tick -2 -1 yticklabel
+ 0 -3 -1 0 tick
+grestore
+
+gsave
+ setupfunctionplot
+ nspline axmin axmax {Tlogistic Tln} cubicsplinterpolate
+grestore
+
+showpage
projects / mit-scheme.git / commitdiff