summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/dvips/pst-func/pst-func.pro
blob: 1eca86df516a2e4f51718c6f72650c4f742769b9 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
%% $Id: pst-func.pro 861 2018-12-13 20:40:06Z herbert $
%%
%% This is file `pst-func.pro',
%%
%% IMPORTANT NOTICE:
%%
%% Package `pst-func'
%%
%% Herbert Voss <hvoss@tug.org>
%%
%% This program can be redistributed and/or modified under the terms
%% of the LaTeX Project Public License Distributed from CTAN archives
%% in directory macros/latex/base/lppl.txt.
%%
%% DESCRIPTION:
%%   `pst-func' is a PSTricks package to plot special math functions
%%
%%
%% version 0.17 / 2018-12-13  Herbert Voss
%
/tx@FuncDict 100 dict def
tx@FuncDict begin
%
/eps1 1.0e-05 def
/eps2 1.0e-04 def
/eps5 1.0e-05 def
/eps8 1.0e-08 def
%
/PiHalf 1.57079632679489661925640 def
/CEuler 0.5772156649 def % Euler-Mascheroni constant
%
/factorial { % n on stack, returns n! 
  dup 32 gt { pop 1e32 } {
    dup 0 eq { 1 }{ 
      dup 1 gt { dup 1 sub factorial mul } if }
    ifelse } ifelse 
} def 
%
/MoverN { % m n on stack, returns the binomial coefficient m over n
  2 dict begin
  /n exch def /m exch def
  n 0 eq { 1 }{
    m n eq { 1 }{
      m factorial n factorial m n sub factorial mul div } ifelse } ifelse 
  end
} def
%
/Pascal [
  [                   1                   ] % 0
  [                 1   1                 ] % 1
  [               1   2   1               ] % 2
  [             1   3   3   1             ] % 3
  [           1   4   6   4   1           ] % 4
  [         1   5  10  10   5   1         ] % 5
  [       1   6  15  20  15   6   1       ] % 6
  [     1   7  21  35  35  21   7   1     ] % 7
  [   1   8  28  56  70  56  28  8    1   ] % 8
  [ 1   9  36  84 126 126  84  36  9    1 ] % 9
] def
%
/GetBezierCoor { 				% t on stack
  10 dict begin					% hold all local
  /t ED
  /t1 1 t sub def				% t1=1-t
  /Coeff Pascal BezierType get def		% get the coefficients
    0 0						% initial values for x y
    BezierType -1 0 {				% BezierType,...,2,1,0
      /I ED					% I=BezierType,...,2,1,0
      /J BezierType I sub def			% J=0,1,2,...,BezierType
      /T t I exp Coeff J get mul def		% coeff(J)*t^I
      /T1 t1 J exp def				% t1^J  
      Points I dup add 1 add get		% y(2*I+1)
      T mul T1 mul add				% the y coordinate
      exch					% y x	
      Points I dup add get			% x(2*I)
      T mul T1 mul add				% the x coordinate
      exch					% x y	
    } for					% x y on stack
  end
} def

/BezierCurve { % on stack [ coors psk@plotpoints BezierType
%  10 dict begin
  /BezierType ED                        % 2,3,4,5,6,...
  1 exch div /epsilon ED                % step for Bezier =1/plotpoints
  ]                                     % [ yi xi ... y3 x3 y2 x2 y1 x1 y0 x0]
  ps@ReverseOrderOfPoints               % [y0 x0 y1 x1 ... yi xi]
  /Points ED 				% save Points array
  epsilon GetBezierCoor 		% next Bezier point
  Points 0 get Points 1 get 		% starting point
  ArrowA lineto 
  epsilon epsilon 1 epsilon sub { % on stack is the loop variable
    GetBezierCoor lineto 
  } for 
  1 epsilon sub GetBezierCoor  
  1 GetBezierCoor  
  ArrowB lineto moveto
%  end
} def

/Bernstein { % on stack tStart tEnd plotpoints i n
  12 dict begin				% hold all local
  /envelope ED				% plot envelope?
  /n ED 
  /i ED
  /ni n i sub def  
  /epsilon ED				% step=1/plotpoints
  /tEnd ED
  /tStart ED
%
% B_{i,n}(t)=\binom{n}{i}t^i(1-t)^{n-i}      (Bernstein)
% f_n(x)=\frac{1}{\sqrt{\pi n\cdot x(1-x)}}  (envelope)
%
  n i MoverN /noveri ED			% \binom{n}{i}
  [					% for the array of points
  tStart epsilon tEnd {
    dup dup /t ED			% leave one on stack
    neg 1 add /t1 ED			% t1=1-t
    envelope 
      { t t1 mul 4 mul PiHalf mul n mul sqrt 1 exch Div }	% envelope
      { noveri t i exp mul t1 ni exp mul } ifelse 	% t f(t)
    ScreenCoor				% convert to screen coor
  } for
  end
  false /Lineto /lineto load def Line
} def
%%
/Si { % integral sin from 0 to x (arg on stack)
  10 dict begin				% hold all local
  dup 0 eq 
  { pop 0 } 
  { 
    /arg exch def  % x
    /arg2 arg dup mul def
    /Sum arg def   %
    /sign -1 def   
    /I 3 def
    /Frac arg2 arg mul 6 div def
    { %  a sequence of x - x^3/(3*3!) + x^5/(5*5!) -...+...
      Frac I div sign mul
      dup abs eps5 lt { pop exit } if
      Sum add /Sum exch def
      /sign sign neg def
      /I I 2 add def
      Frac arg2 mul I 1 sub I mul div /Frac ED
%    arg I Power dup abs 1e30 gt { pop exit } if
%    I factorial div I div sign mul 
%    dup abs eps8 lt { pop exit } if 
%    Sum add /Sum exch def
%    /sign sign neg def
%    /I I 2 add def
   } loop
   Sum
  } ifelse
  end
} def
%
/si { % integral sin from x to infty -> si(x)=Si(x)-pi/2
  Si PiHalf sub
} def
%
/Ci { % integral cosin from x to infty (arg on stack)
  10 dict begin				% hold all local
  abs /arg exch def
  arg 0 eq { 0 } { 
    /arg2 arg dup mul def 
    /Sum CEuler arg ln add def
    /sign -1 def
    /I 2 def
    /Frac arg2 2 div def % first fraction
    { Frac I div sign mul
      dup abs eps5 lt { pop exit } if
      Sum add /Sum exch def
     /sign sign neg def
     /I I 2 add def
     Frac arg2 mul I 1 sub I mul div /Frac ED
    } loop
    Sum
  } ifelse
  end
} def
%
/ci { % integral cosin from x to infty -> ci(x)=-Ci(x)+ln(x)+CEuler
  dup Ci neg exch abs ln add CEuler add
} def
%
/MaxIter 255 def
/func { coeff Derivation FuncValue } def
/func' { coeff Derivation 1 add FuncValue } def
/func'' { coeff Derivation 2 add FuncValue } def
%
/NewtonMehrfach {% the start value must be on top of the stack
  /Nx exch def 
  /Iter 0 def
  {
    /Iter Iter 1 add def
    Nx func /F exch def % f(Nx)
    F abs eps2 lt { exit } if
    Nx func' /FS exch def % f'(Nx) 
    FS 0 eq { /FS 1.0e-06 def } if
    Nx func'' /F2S exch def % f''(Nx)
    1.0 1.0 F F2S mul FS dup mul div sub div /J exch def
    J F mul FS div /Diff exch def 
    /Nx Nx Diff sub def
    Diff abs eps1 lt Iter MaxIter gt or { exit } if 
  } loop 
  Nx % the returned value ist the zero point
} def

/Steffensen {% the start value must be on top of the stack
  /y0 exch def % the start value
  /Iter 0 def   /MaxIter 200 def
  { pstack
    y0 func /F exch def
    F abs eps2 lt { exit } if
    y0 F sub /Phi exch def
    Phi func /F2 exch def
    F2 abs eps2 le { exit }{
      Phi y0 sub dup mul Phi F2 sub 2 Phi mul sub y0 add Div /Diff exch def
      y0 Diff sub /y0 exch def
      Diff abs eps1 le { exit } if
    } ifelse
    /Iter Iter 1 add def
    Iter MaxIter gt { exit } if
  } loop
  y0 28 mul % the returned value ist the zero point
  0  
  3 0 360 arc gsave 0 0 1 setrgbcolor fill grestore 1 setlinewidth stroke
} def 
%
/Horner {% x [coeff] must be on top of the stack
  aload length
  dup 2 add -1 roll
  exch 1 sub {
    dup 4 1 roll
    mul add exch
  } repeat
  pop % the y value is on top of the stack
} def
%
/FuncValue {% x [coeff] Derivation must be on top of the stack
  {
    aload 			% a0 a1 a2 ... a(n-1) [array]
    length                      % a0 a1 a2 ... a(n-1) n
    1 sub /grad exch def        % a0 a1 a2 ... a(n-1) 
    grad -1 1 {                 % for n=grad step -1 until 1
      /n exch def               % Laufvariable speichern
      n                         % a0 a1 a2 ... a(n-1) n
      mul                       % a0 a1 a2 ... a(n-1)*n 
      grad 1 add                % a0 a1 a2 ... a(n-1)*n grad+1 
      1 roll                    % an*na0 a1 a2 ... a(n-2)
    } for
    pop                         % loesche a0
    grad array astore           % [ a1 a2 ... a(n-2)]
  } repeat
  Horner
} def
%
/FindZeros { % dxN dxZ must be on top of the stack (x0..x1 the intervall) => []
  12 dict begin 
  /dxZ exch def /dxN exch def
  /pstZeros [] def 
  x0 dxZ x1 { % suche Nullstellen
    /xWert exch def
    xWert NewtonMehrfach 
    %xWert Steffensen 
    /xNull exch def 
    pstZeros aload length /Laenge exch def % now test if value is a new one
    Laenge 0 eq 
      { xNull 1 }
      { /newZero true def
        Laenge {
	  xNull sub abs dxN lt { /newZero false def } if
        } repeat
	pstZeros aload pop
        newZero { xNull Laenge 1 add } { Laenge } ifelse } ifelse
    array astore 
    /pstZeros exch def
  } for
  pstZeros  % the end array is now on the stack
  end
} def
%
/Simpson { % on stack must be a b M useXVal   --- simple version ---
% /SFunc must be defined 
  /useX ED  % for algebraic functions which uses f(x)
  /M ED /b ED /a ED
  /h b a sub M 2 mul div def
  /s1 0 def
  /s2 0 def
  1 1 M {
    /k exch def
    /xVal k 2 mul 1 sub h mul a add def
    /s1 s1 xVal useX { /x exch def } if SFunc add def
  } for
  1 1 M 1 sub {
    /k exch def
    /xVal k 2 mul h mul a add def
    /s2 s2 xVal useX { /x exch def } if SFunc add def
  } for
  /I a useX { /x exch def } if SFunc b useX { /x exch def } if SFunc add s1 4 mul add s2 2 mul add 3 div h mul def
} def
%
%
/LogGamma { 5 dict begin	% z on stack
  /z ED
  /sum 0 def
  /k 1 def
  {
    z k div dup 1 add ln sub dup
    abs eps8 lt { pop exit } if
    sum add /sum exch def
    /k k 1 add def
  } loop
  sum z ln sub CEuler z mul sub
  end 
} def
%
/ChebyshevT { 5 dict begin	% z on stack
  /xtmp exch def
  /n exch def
  0 0 1 n .5 mul floor {
     /k exch def
     xtmp xtmp mul 1 sub k exp
     xtmp n 2 k mul sub exp mul
     n 2 k mul MoverN mul
     add
  } for
  end
} def
%
/ChebyshevU {5 dict begin	% z on stack
  /xtmp exch def
  /n exch def
  0 0 1 n .5 mul floor {
    /k exch def
    xtmp xtmp mul 1 sub k exp
    xtmp n 2 k mul sub exp mul
    n 1 add 2 k mul 1 add MoverN mul
    add
  } for
  end
} def
%
/vasicek{           %density=sqrt((1-R2)/R2)*exp(1/2*(norminv(x)2 - (1/sqrt(R2)*((sqrt(1-R2)*norminv(x)-norminv(pd)))2))
  2 dict begin
  /pd where { pop }{ /pd 0.22 def } ifelse    % element of (0,1) probability of default of portfolio  
  /R2 where { pop }{ /R2 0.11 def } ifelse    % element of (0,1) R_Squared of portfolio
  dup                 % x   x
  norminv             % x   norminv(x)
  dup mul             % x   norminv(x)^2
  exch                % norminv(x)2   x
  norminv             % norminv(x)2   norminv(x)
  1 R2 sub sqrt mul   % norminv(x)2   sqrt(1-R2)*norminv(x)
  pd norminv sub      % norminv(x)2   sqrt(1-R2)*norminv(x)-norminv(pd)
  R2 sqrt div         % norminv(x)2   1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd))
  dup mul             % norminv(x)2  (1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2
  sub                 % norminv(x)2 -(1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2
  2 div               % 1/2*(norminv(x)2 -(1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2)
  ENeperian exch exp  % exp(1/2*(norminv(x)2 -(1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2)
  1 R2 sub            % exp(1/2*(norminv(x)2 -(1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2)   1-R2
  R2 div              % exp(1/2*(norminv(x)2 -(1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2)   (1-R2)/R2
  sqrt                % exp(1/2*(norminv(x)2 -(1/sqrt(R2)*(sqrt(1-R2)*norminv(x)-norminv(pd)))2)   sqrt((1-R2)/R2)
  mul                 % sqrt((1-R2)/R2)*exp(1/2*(norminv(x)2 - (1/sqrt(R2)*((sqrt(1-R2)*norminv(x)-norminv(pd)))2))
  end
} def
%end{vasicek density}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% subroutines for complex numbers, given as an array [a b] 
% which is a+bi = Real+i Imag
%
/cxadd {		% [a1 b1] [a2 b2] = [a1+a2 b1+b2]
  dup 0 get		% [a1 b1] [a2 b2] a2
  3 -1 roll		% [a2 b2] a2 [a1 b1]
  dup 0 get		% [a2 b2] a2 [a1 b1] a1
  3 -1 roll		% [a2 b2] [a1 b1] a1 a2
  add			% [a2 b2] [a1 b1] a1+a2
  3 1 roll		% a1+a2 [a2 b2] [a1 b1]
  1 get			% a1+a2 [a2 b2] b1
  exch 1 get		% a1+a2 b1 b2
  add 2 array astore
} def
%
/cxneg {		% [a b]
  dup 1 get		% [a b] b
  exch 0 get		% b a
  neg exch neg		% -a -b
  2 array astore
} def
%
/cxsub { cxneg cxadd } def  % same as negative addition
%
% [a1 b1][a2 b2] = [a1a2-b1b2 a1b2+b1a2] = [a3 b3]
/cxmul {		% [a1 b1] [a2 b2]
  dup 0 get		% [a1 b1] [a2 b2] a2
  exch 1 get		% [a1 b1] a2 b2
  3 -1 roll		% a2 b2 [a1 b1]
  dup 0 get		% a2 b2 [a1 b1] a1
  exch 1 get		% a2 b2 a1 b1
  dup			% a2 b2 a1 b1 b1
  5 -1 roll dup		% b2 a1 b1 b1 a2 a2
  3 1 roll mul		% b2 a1 b1 a2 b1a2
  5 -2 roll dup		% b1 a2 b1a2 b2 a1 a1
  3 -1 roll dup		% b1 a2 b1a2 a1 a1 b2 b2
  3 1 roll mul		% b1 a2 b1a2 a1 b2 a1b2
  4 -1 roll add		% b1 a2 a1 b2 b3
  4 2 roll mul		% b1 b2 b3 a1a2
  4 2 roll mul sub	% b3 a3
  exch 2 array astore
} def
%
% [a b]^2 = [a^2-b^2 2ab] = [a2 b2]
/cxsqr {		% [a b]   square root
  dup 0 get exch 1 get	% a b
  dup dup mul		% a b b^2
  3 -1 roll		% b b^2 a
  dup dup mul 		% b b^2 a a^2
  3 -1 roll sub		% b a a2
  3 1 roll mul 2 mul	% a2 b2	
  2 array astore
} def
%
/cxsqrt {		% [a b]
%  dup cxnorm sqrt /r exch def
%  cxarg 2 div RadtoDeg dup cos r mul exch sin r mul cxmake2 
  cxlog 		% log[a b]
  2 cxrdiv 		% log[a b]/2
  aload pop exch	% b a
  2.781 exch exp	% b exp(a)
  exch cxconv exch	% [Re +iIm] exp(a)
  cxrmul		%
} def
%
/cxarg { 		% [a b] 
  aload pop 		% a b
  exch atan 		% arctan b/a
  DegtoRad 		% arg(z)=atan(b/a)
} def
%
% log[a b] = [a^2-b^2 2ab] = [a2 b2]
/cxlog {		% [a b]
  dup 			% [a b][a b]
  cxnorm 		% [a b] |z|
  log 			% [a b] log|z|
  exch 			% log|z|[a b]
  cxarg 		% log|z| Theta
  cxmake2 		% [log|z| Theta]
} def
%
% square of magnitude of complex number
/cxnorm2 {		% [a b]
  dup 0 get exch 1 get	% a b
  dup mul			% a b^2
  exch dup mul add	% a^2+b^2
} def
%
/cxnorm {		% [a b]
  cxnorm2 sqrt
} def
%
/cxconj {		% conjugent complex
  dup 0 get exch 1 get	% a b
  neg 2 array astore	% [a -b]
} def
%
/cxre { 0 get } def	% real value
/cxim { 1 get } def	% imag value
%
% 1/[a b] = ([a -b]/(a^2+b^2)
/cxrecip {		% [a b]
  dup cxnorm2 exch	% n2 [a b]
  dup 0 get exch 1 get	% n2 a b
  3 -1 roll		% a b n2
  dup			% a b n2 n2
  4 -1 roll exch div	% b n2 a/n2
  3 1 roll div		% a/n2 b/n2
  neg 2 array astore
} def
%
/cxmake1 { 0 2 array astore } def % make a complex number, real given
/cxmake2 { 2 array astore } def	  % dito, both given
%
/cxdiv { cxrecip cxmul } def
%
% multiplikation by a real number
/cxrmul {		% [a b] r
  exch aload pop	% r a b
  3 -1 roll dup		% a b r r
  3 1 roll mul		% a r b*r
  3 1 roll mul		% b*r a*r
  exch 2 array astore   % [a*r b*r]
} def
%
% division by a real number
/cxrdiv {		% [a b] r
  1 exch div		% [a b] 1/r
  cxrmul
} def
%
% exp(i theta) = cos(theta)+i sin(theta) polar<->cartesian
/cxconv {		% theta
  RadtoDeg dup sin exch cos cxmake2
} def
%
end