GiNaC  1.8.0
inifcns.cpp
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1 
5 /*
6  * GiNaC Copyright (C) 1999-2020 Johannes Gutenberg University Mainz, Germany
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22 
23 #include "inifcns.h"
24 #include "ex.h"
25 #include "constant.h"
26 #include "lst.h"
27 #include "fderivative.h"
28 #include "matrix.h"
29 #include "mul.h"
30 #include "power.h"
31 #include "operators.h"
32 #include "relational.h"
33 #include "pseries.h"
34 #include "symbol.h"
35 #include "symmetry.h"
36 #include "utils.h"
37 
38 #include <stdexcept>
39 #include <vector>
40 
41 namespace GiNaC {
42 
44 // complex conjugate
46 
47 static ex conjugate_evalf(const ex & arg)
48 {
49  if (is_exactly_a<numeric>(arg)) {
50  return ex_to<numeric>(arg).conjugate();
51  }
52  return conjugate_function(arg).hold();
53 }
54 
55 static ex conjugate_eval(const ex & arg)
56 {
57  return arg.conjugate();
58 }
59 
60 static void conjugate_print_latex(const ex & arg, const print_context & c)
61 {
62  c.s << "\\bar{"; arg.print(c); c.s << "}";
63 }
64 
65 static ex conjugate_conjugate(const ex & arg)
66 {
67  return arg;
68 }
69 
70 // If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x)
71 // and conjugate((U+I*V).diff(x))
72 static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
73 {
74  if (s.info(info_flags::real))
75  return conjugate(arg.diff(s));
76  else {
77  exvector vec_arg;
78  vec_arg.push_back(arg);
79  return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
80  }
81 }
82 
83 static ex conjugate_real_part(const ex & arg)
84 {
85  return arg.real_part();
86 }
87 
88 static ex conjugate_imag_part(const ex & arg)
89 {
90  return -arg.imag_part();
91 }
92 
93 static bool func_arg_info(const ex & arg, unsigned inf)
94 {
95  // for some functions we can return the info() of its argument
96  // (think of conjugate())
97  switch (inf) {
102  case info_flags::real:
104  case info_flags::integer:
107  case info_flags::even:
108  case info_flags::odd:
109  case info_flags::prime:
115  case info_flags::posint:
116  case info_flags::negint:
119  return arg.info(inf);
120  }
121  return false;
122 }
123 
124 static bool conjugate_info(const ex & arg, unsigned inf)
125 {
126  return func_arg_info(arg, inf);
127 }
128 
129 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
130  evalf_func(conjugate_evalf).
131  expl_derivative_func(conjugate_expl_derivative).
132  info_func(conjugate_info).
133  print_func<print_latex>(conjugate_print_latex).
134  conjugate_func(conjugate_conjugate).
135  real_part_func(conjugate_real_part).
136  imag_part_func(conjugate_imag_part).
137  set_name("conjugate","conjugate"));
138 
140 // real part
142 
143 static ex real_part_evalf(const ex & arg)
144 {
145  if (is_exactly_a<numeric>(arg)) {
146  return ex_to<numeric>(arg).real();
147  }
148  return real_part_function(arg).hold();
149 }
150 
151 static ex real_part_eval(const ex & arg)
152 {
153  return arg.real_part();
154 }
155 
156 static void real_part_print_latex(const ex & arg, const print_context & c)
157 {
158  c.s << "\\Re"; arg.print(c); c.s << "";
159 }
160 
161 static ex real_part_conjugate(const ex & arg)
162 {
163  return real_part_function(arg).hold();
164 }
165 
166 static ex real_part_real_part(const ex & arg)
167 {
168  return real_part_function(arg).hold();
169 }
170 
171 static ex real_part_imag_part(const ex & arg)
172 {
173  return 0;
174 }
175 
176 // If x is real then Re(e).diff(x) is equal to Re(e.diff(x))
177 static ex real_part_expl_derivative(const ex & arg, const symbol & s)
178 {
179  if (s.info(info_flags::real))
180  return real_part_function(arg.diff(s));
181  else {
182  exvector vec_arg;
183  vec_arg.push_back(arg);
184  return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
185  }
186 }
187 
188 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
189  evalf_func(real_part_evalf).
190  expl_derivative_func(real_part_expl_derivative).
191  print_func<print_latex>(real_part_print_latex).
192  conjugate_func(real_part_conjugate).
193  real_part_func(real_part_real_part).
194  imag_part_func(real_part_imag_part).
195  set_name("real_part","real_part"));
196 
198 // imag part
200 
201 static ex imag_part_evalf(const ex & arg)
202 {
203  if (is_exactly_a<numeric>(arg)) {
204  return ex_to<numeric>(arg).imag();
205  }
206  return imag_part_function(arg).hold();
207 }
208 
209 static ex imag_part_eval(const ex & arg)
210 {
211  return arg.imag_part();
212 }
213 
214 static void imag_part_print_latex(const ex & arg, const print_context & c)
215 {
216  c.s << "\\Im"; arg.print(c); c.s << "";
217 }
218 
219 static ex imag_part_conjugate(const ex & arg)
220 {
221  return imag_part_function(arg).hold();
222 }
223 
224 static ex imag_part_real_part(const ex & arg)
225 {
226  return imag_part_function(arg).hold();
227 }
228 
229 static ex imag_part_imag_part(const ex & arg)
230 {
231  return 0;
232 }
233 
234 // If x is real then Im(e).diff(x) is equal to Im(e.diff(x))
235 static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
236 {
237  if (s.info(info_flags::real))
238  return imag_part_function(arg.diff(s));
239  else {
240  exvector vec_arg;
241  vec_arg.push_back(arg);
242  return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
243  }
244 }
245 
246 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
247  evalf_func(imag_part_evalf).
248  expl_derivative_func(imag_part_expl_derivative).
249  print_func<print_latex>(imag_part_print_latex).
250  conjugate_func(imag_part_conjugate).
251  real_part_func(imag_part_real_part).
252  imag_part_func(imag_part_imag_part).
253  set_name("imag_part","imag_part"));
254 
256 // absolute value
258 
259 static ex abs_evalf(const ex & arg)
260 {
261  if (is_exactly_a<numeric>(arg))
262  return abs(ex_to<numeric>(arg));
263 
264  return abs(arg).hold();
265 }
266 
267 static ex abs_eval(const ex & arg)
268 {
269  if (is_exactly_a<numeric>(arg))
270  return abs(ex_to<numeric>(arg));
271 
272  if (arg.info(info_flags::nonnegative))
273  return arg;
274 
275  if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
276  return -arg;
277 
278  if (is_ex_the_function(arg, abs))
279  return arg;
280 
281  if (is_ex_the_function(arg, exp))
282  return exp(arg.op(0).real_part());
283 
284  if (is_exactly_a<power>(arg)) {
285  const ex& base = arg.op(0);
286  const ex& exponent = arg.op(1);
287  if (base.info(info_flags::positive) || exponent.info(info_flags::real))
288  return pow(abs(base), exponent.real_part());
289  }
290 
291  if (is_ex_the_function(arg, conjugate_function))
292  return abs(arg.op(0));
293 
294  if (is_ex_the_function(arg, step))
295  return arg;
296 
297  return abs(arg).hold();
298 }
299 
300 static ex abs_expand(const ex & arg, unsigned options)
301 {
303  && is_exactly_a<mul>(arg)) {
304  exvector prodseq;
305  prodseq.reserve(arg.nops());
306  for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
308  prodseq.push_back(abs(i->expand(options)));
309  else
310  prodseq.push_back(abs(*i));
311  }
312  return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
313  }
314 
316  return abs(arg.expand(options)).hold();
317  else
318  return abs(arg).hold();
319 }
320 
321 static ex abs_expl_derivative(const ex & arg, const symbol & s)
322 {
323  ex diff_arg = arg.diff(s);
324  return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
325 }
326 
327 static void abs_print_latex(const ex & arg, const print_context & c)
328 {
329  c.s << "{|"; arg.print(c); c.s << "|}";
330 }
331 
332 static void abs_print_csrc_float(const ex & arg, const print_context & c)
333 {
334  c.s << "fabs("; arg.print(c); c.s << ")";
335 }
336 
337 static ex abs_conjugate(const ex & arg)
338 {
339  return abs(arg).hold();
340 }
341 
342 static ex abs_real_part(const ex & arg)
343 {
344  return abs(arg).hold();
345 }
346 
347 static ex abs_imag_part(const ex& arg)
348 {
349  return 0;
350 }
351 
352 static ex abs_power(const ex & arg, const ex & exp)
353 {
354  if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
355  if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
356  return pow(arg, exp);
357  else
358  return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
359  } else
360  return power(abs(arg), exp).hold();
361 }
362 
363 bool abs_info(const ex & arg, unsigned inf)
364 {
365  switch (inf) {
366  case info_flags::integer:
367  case info_flags::even:
368  case info_flags::odd:
369  case info_flags::prime:
370  return arg.info(inf);
372  return arg.info(info_flags::integer);
374  case info_flags::real:
375  return true;
377  return false;
381  if (arg.info(info_flags::has_indices))
382  return true;
383  else
384  return false;
385  }
386  }
387  return false;
388 }
389 
391  evalf_func(abs_evalf).
392  expand_func(abs_expand).
393  expl_derivative_func(abs_expl_derivative).
394  info_func(abs_info).
395  print_func<print_latex>(abs_print_latex).
396  print_func<print_csrc_float>(abs_print_csrc_float).
397  print_func<print_csrc_double>(abs_print_csrc_float).
398  conjugate_func(abs_conjugate).
399  real_part_func(abs_real_part).
400  imag_part_func(abs_imag_part).
401  power_func(abs_power));
402 
404 // Step function
406 
407 static ex step_evalf(const ex & arg)
408 {
409  if (is_exactly_a<numeric>(arg))
410  return step(ex_to<numeric>(arg));
411 
412  return step(arg).hold();
413 }
414 
415 static ex step_eval(const ex & arg)
416 {
417  if (is_exactly_a<numeric>(arg))
418  return step(ex_to<numeric>(arg));
419 
420  else if (is_exactly_a<mul>(arg) &&
421  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
422  numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
423  if (oc.is_real()) {
424  if (oc > 0)
425  // step(42*x) -> step(x)
426  return step(arg/oc).hold();
427  else
428  // step(-42*x) -> step(-x)
429  return step(-arg/oc).hold();
430  }
431  if (oc.real().is_zero()) {
432  if (oc.imag() > 0)
433  // step(42*I*x) -> step(I*x)
434  return step(I*arg/oc).hold();
435  else
436  // step(-42*I*x) -> step(-I*x)
437  return step(-I*arg/oc).hold();
438  }
439  }
440 
441  return step(arg).hold();
442 }
443 
444 static ex step_series(const ex & arg,
445  const relational & rel,
446  int order,
447  unsigned options)
448 {
449  const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
450  if (arg_pt.info(info_flags::numeric)
451  && ex_to<numeric>(arg_pt).real().is_zero()
453  throw (std::domain_error("step_series(): on imaginary axis"));
454 
455  epvector seq { expair(step(arg_pt), _ex0) };
456  return pseries(rel, std::move(seq));
457 }
458 
459 static ex step_conjugate(const ex& arg)
460 {
461  return step(arg).hold();
462 }
463 
464 static ex step_real_part(const ex& arg)
465 {
466  return step(arg).hold();
467 }
468 
469 static ex step_imag_part(const ex& arg)
470 {
471  return 0;
472 }
473 
475  evalf_func(step_evalf).
476  series_func(step_series).
477  conjugate_func(step_conjugate).
478  real_part_func(step_real_part).
479  imag_part_func(step_imag_part));
480 
482 // Complex sign
484 
485 static ex csgn_evalf(const ex & arg)
486 {
487  if (is_exactly_a<numeric>(arg))
488  return csgn(ex_to<numeric>(arg));
489 
490  return csgn(arg).hold();
491 }
492 
493 static ex csgn_eval(const ex & arg)
494 {
495  if (is_exactly_a<numeric>(arg))
496  return csgn(ex_to<numeric>(arg));
497 
498  else if (is_exactly_a<mul>(arg) &&
499  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
500  numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
501  if (oc.is_real()) {
502  if (oc > 0)
503  // csgn(42*x) -> csgn(x)
504  return csgn(arg/oc).hold();
505  else
506  // csgn(-42*x) -> -csgn(x)
507  return -csgn(arg/oc).hold();
508  }
509  if (oc.real().is_zero()) {
510  if (oc.imag() > 0)
511  // csgn(42*I*x) -> csgn(I*x)
512  return csgn(I*arg/oc).hold();
513  else
514  // csgn(-42*I*x) -> -csgn(I*x)
515  return -csgn(I*arg/oc).hold();
516  }
517  }
518 
519  return csgn(arg).hold();
520 }
521 
522 static ex csgn_series(const ex & arg,
523  const relational & rel,
524  int order,
525  unsigned options)
526 {
527  const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
528  if (arg_pt.info(info_flags::numeric)
529  && ex_to<numeric>(arg_pt).real().is_zero()
531  throw (std::domain_error("csgn_series(): on imaginary axis"));
532 
533  epvector seq { expair(csgn(arg_pt), _ex0) };
534  return pseries(rel, std::move(seq));
535 }
536 
537 static ex csgn_conjugate(const ex& arg)
538 {
539  return csgn(arg).hold();
540 }
541 
542 static ex csgn_real_part(const ex& arg)
543 {
544  return csgn(arg).hold();
545 }
546 
547 static ex csgn_imag_part(const ex& arg)
548 {
549  return 0;
550 }
551 
552 static ex csgn_power(const ex & arg, const ex & exp)
553 {
554  if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
555  if (ex_to<numeric>(exp).is_odd())
556  return csgn(arg).hold();
557  else
558  return power(csgn(arg), _ex2).hold();
559  } else
560  return power(csgn(arg), exp).hold();
561 }
562 
563 
565  evalf_func(csgn_evalf).
566  series_func(csgn_series).
567  conjugate_func(csgn_conjugate).
568  real_part_func(csgn_real_part).
569  imag_part_func(csgn_imag_part).
570  power_func(csgn_power));
571 
572 
574 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
575 // This function is closely related to the unwinding number K, sometimes found
576 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
578 
579 static ex eta_evalf(const ex &x, const ex &y)
580 {
581  // It seems like we basically have to replicate the eval function here,
582  // since the expression might not be fully evaluated yet.
584  return _ex0;
585 
587  const numeric nx = ex_to<numeric>(x);
588  const numeric ny = ex_to<numeric>(y);
589  const numeric nxy = ex_to<numeric>(x*y);
590  int cut = 0;
591  if (nx.is_real() && nx.is_negative())
592  cut -= 4;
593  if (ny.is_real() && ny.is_negative())
594  cut -= 4;
595  if (nxy.is_real() && nxy.is_negative())
596  cut += 4;
597  return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
598  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
599  }
600 
601  return eta(x,y).hold();
602 }
603 
604 static ex eta_eval(const ex &x, const ex &y)
605 {
606  // trivial: eta(x,c) -> 0 if c is real and positive
608  return _ex0;
609 
611  // don't call eta_evalf here because it would call Pi.evalf()!
612  const numeric nx = ex_to<numeric>(x);
613  const numeric ny = ex_to<numeric>(y);
614  const numeric nxy = ex_to<numeric>(x*y);
615  int cut = 0;
616  if (nx.is_real() && nx.is_negative())
617  cut -= 4;
618  if (ny.is_real() && ny.is_negative())
619  cut -= 4;
620  if (nxy.is_real() && nxy.is_negative())
621  cut += 4;
622  return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
623  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
624  }
625 
626  return eta(x,y).hold();
627 }
628 
629 static ex eta_series(const ex & x, const ex & y,
630  const relational & rel,
631  int order,
632  unsigned options)
633 {
634  const ex x_pt = x.subs(rel, subs_options::no_pattern);
635  const ex y_pt = y.subs(rel, subs_options::no_pattern);
636  if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
638  ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
639  throw (std::domain_error("eta_series(): on discontinuity"));
640  epvector seq { expair(eta(x_pt,y_pt), _ex0) };
641  return pseries(rel, std::move(seq));
642 }
643 
644 static ex eta_conjugate(const ex & x, const ex & y)
645 {
646  return -eta(x, y).hold();
647 }
648 
649 static ex eta_real_part(const ex & x, const ex & y)
650 {
651  return 0;
652 }
653 
654 static ex eta_imag_part(const ex & x, const ex & y)
655 {
656  return -I*eta(x, y).hold();
657 }
658 
659 REGISTER_FUNCTION(eta, eval_func(eta_eval).
660  evalf_func(eta_evalf).
661  series_func(eta_series).
662  latex_name("\\eta").
663  set_symmetry(sy_symm(0, 1)).
664  conjugate_func(eta_conjugate).
665  real_part_func(eta_real_part).
666  imag_part_func(eta_imag_part));
667 
668 
670 // dilogarithm
672 
673 static ex Li2_evalf(const ex & x)
674 {
675  if (is_exactly_a<numeric>(x))
676  return Li2(ex_to<numeric>(x));
677 
678  return Li2(x).hold();
679 }
680 
681 static ex Li2_eval(const ex & x)
682 {
683  if (x.info(info_flags::numeric)) {
684  // Li2(0) -> 0
685  if (x.is_zero())
686  return _ex0;
687  // Li2(1) -> Pi^2/6
688  if (x.is_equal(_ex1))
689  return power(Pi,_ex2)/_ex6;
690  // Li2(1/2) -> Pi^2/12 - log(2)^2/2
691  if (x.is_equal(_ex1_2))
692  return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
693  // Li2(-1) -> -Pi^2/12
694  if (x.is_equal(_ex_1))
695  return -power(Pi,_ex2)/_ex12;
696  // Li2(I) -> -Pi^2/48+Catalan*I
697  if (x.is_equal(I))
698  return power(Pi,_ex2)/_ex_48 + Catalan*I;
699  // Li2(-I) -> -Pi^2/48-Catalan*I
700  if (x.is_equal(-I))
701  return power(Pi,_ex2)/_ex_48 - Catalan*I;
702  // Li2(float)
704  return Li2(ex_to<numeric>(x));
705  }
706 
707  return Li2(x).hold();
708 }
709 
710 static ex Li2_deriv(const ex & x, unsigned deriv_param)
711 {
712  GINAC_ASSERT(deriv_param==0);
713 
714  // d/dx Li2(x) -> -log(1-x)/x
715  return -log(_ex1-x)/x;
716 }
717 
718 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
719 {
720  const ex x_pt = x.subs(rel, subs_options::no_pattern);
721  if (x_pt.info(info_flags::numeric)) {
722  // First special case: x==0 (derivatives have poles)
723  if (x_pt.is_zero()) {
724  // method:
725  // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
726  // simply substitute x==0. The limit, however, exists: it is 1.
727  // We also know all higher derivatives' limits:
728  // (d/dx)^n Li2(x) == n!/n^2.
729  // So the primitive series expansion is
730  // Li2(x==0) == x + x^2/4 + x^3/9 + ...
731  // and so on.
732  // We first construct such a primitive series expansion manually in
733  // a dummy symbol s and then insert the argument's series expansion
734  // for s. Reexpanding the resulting series returns the desired
735  // result.
736  const symbol s;
737  ex ser;
738  // manually construct the primitive expansion
739  for (int i=1; i<order; ++i)
740  ser += pow(s,i) / pow(numeric(i), *_num2_p);
741  // substitute the argument's series expansion
742  ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
743  // maybe that was terminating, so add a proper order term
744  epvector nseq { expair(Order(_ex1), order) };
745  ser += pseries(rel, std::move(nseq));
746  // reexpanding it will collapse the series again
747  return ser.series(rel, order);
748  // NB: Of course, this still does not allow us to compute anything
749  // like sin(Li2(x)).series(x==0,2), since then this code here is
750  // not reached and the derivative of sin(Li2(x)) doesn't allow the
751  // substitution x==0. Probably limits *are* needed for the general
752  // cases. In case L'Hospital's rule is implemented for limits and
753  // basic::series() takes care of this, this whole block is probably
754  // obsolete!
755  }
756  // second special case: x==1 (branch point)
757  if (x_pt.is_equal(_ex1)) {
758  // method:
759  // construct series manually in a dummy symbol s
760  const symbol s;
761  ex ser = zeta(_ex2);
762  // manually construct the primitive expansion
763  for (int i=1; i<order; ++i)
764  ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
765  // substitute the argument's series expansion
766  ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
767  // maybe that was terminating, so add a proper order term
768  epvector nseq { expair(Order(_ex1), order) };
769  ser += pseries(rel, std::move(nseq));
770  // reexpanding it will collapse the series again
771  return ser.series(rel, order);
772  }
773  // third special case: x real, >=1 (branch cut)
775  ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
776  // method:
777  // This is the branch cut: assemble the primitive series manually
778  // and then add the corresponding complex step function.
779  const symbol &s = ex_to<symbol>(rel.lhs());
780  const ex point = rel.rhs();
781  const symbol foo;
782  epvector seq;
783  // zeroth order term:
784  seq.push_back(expair(Li2(x_pt), _ex0));
785  // compute the intermediate terms:
786  ex replarg = series(Li2(x), s==foo, order);
787  for (size_t i=1; i<replarg.nops()-1; ++i)
788  seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
789  // append an order term:
790  seq.push_back(expair(Order(_ex1), replarg.nops()-1));
791  return pseries(rel, std::move(seq));
792  }
793  }
794  // all other cases should be safe, by now:
795  throw do_taylor(); // caught by function::series()
796 }
797 
798 static ex Li2_conjugate(const ex & x)
799 {
800  // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
801  // run along the positive real axis beginning at 1.
802  if (x.info(info_flags::negative)) {
803  return Li2(x).hold();
804  }
805  if (is_exactly_a<numeric>(x) &&
806  (!x.imag_part().is_zero() || x < *_num1_p)) {
807  return Li2(x.conjugate());
808  }
809  return conjugate_function(Li2(x)).hold();
810 }
811 
813  evalf_func(Li2_evalf).
814  derivative_func(Li2_deriv).
815  series_func(Li2_series).
816  conjugate_func(Li2_conjugate).
817  latex_name("\\mathrm{Li}_2"));
818 
820 // trilogarithm
822 
823 static ex Li3_eval(const ex & x)
824 {
825  if (x.is_zero())
826  return x;
827  return Li3(x).hold();
828 }
829 
830 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
831  latex_name("\\mathrm{Li}_3"));
832 
834 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
836 
837 static ex zetaderiv_eval(const ex & n, const ex & x)
838 {
839  if (n.info(info_flags::numeric)) {
840  // zetaderiv(0,x) -> zeta(x)
841  if (n.is_zero())
842  return zeta(x).hold();
843  }
844 
845  return zetaderiv(n, x).hold();
846 }
847 
848 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
849 {
850  GINAC_ASSERT(deriv_param<2);
851 
852  if (deriv_param==0) {
853  // d/dn zeta(n,x)
854  throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
855  }
856  // d/dx psi(n,x)
857  return zetaderiv(n+1,x);
858 }
859 
860 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
861  derivative_func(zetaderiv_deriv).
862  latex_name("\\zeta^\\prime"));
863 
865 // factorial
867 
868 static ex factorial_evalf(const ex & x)
869 {
870  return factorial(x).hold();
871 }
872 
873 static ex factorial_eval(const ex & x)
874 {
875  if (is_exactly_a<numeric>(x))
876  return factorial(ex_to<numeric>(x));
877  else
878  return factorial(x).hold();
879 }
880 
881 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
882 {
883  if (is_exactly_a<symbol>(x) ||
884  is_exactly_a<constant>(x) ||
885  is_exactly_a<function>(x)) {
886  x.print(c); c.s << "!";
887  } else {
888  c.s << "("; x.print(c); c.s << ")!";
889  }
890 }
891 
892 static ex factorial_conjugate(const ex & x)
893 {
894  return factorial(x).hold();
895 }
896 
897 static ex factorial_real_part(const ex & x)
898 {
899  return factorial(x).hold();
900 }
901 
902 static ex factorial_imag_part(const ex & x)
903 {
904  return 0;
905 }
906 
908  evalf_func(factorial_evalf).
910  print_func<print_latex>(factorial_print_dflt_latex).
911  conjugate_func(factorial_conjugate).
912  real_part_func(factorial_real_part).
913  imag_part_func(factorial_imag_part));
914 
916 // binomial
918 
919 static ex binomial_evalf(const ex & x, const ex & y)
920 {
921  return binomial(x, y).hold();
922 }
923 
924 static ex binomial_sym(const ex & x, const numeric & y)
925 {
926  if (y.is_integer()) {
927  if (y.is_nonneg_integer()) {
928  const unsigned N = y.to_int();
929  if (N == 0) return _ex1;
930  if (N == 1) return x;
931  ex t = x.expand();
932  for (unsigned i = 2; i <= N; ++i)
933  t = (t * (x + i - y - 1)).expand() / i;
934  return t;
935  } else
936  return _ex0;
937  }
938 
939  return binomial(x, y).hold();
940 }
941 
942 static ex binomial_eval(const ex & x, const ex &y)
943 {
944  if (is_exactly_a<numeric>(y)) {
945  if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
946  return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
947  else
948  return binomial_sym(x, ex_to<numeric>(y));
949  } else
950  return binomial(x, y).hold();
951 }
952 
953 // At the moment the numeric evaluation of a binomial function always
954 // gives a real number, but if this would be implemented using the gamma
955 // function, also complex conjugation should be changed (or rather, deleted).
956 static ex binomial_conjugate(const ex & x, const ex & y)
957 {
958  return binomial(x,y).hold();
959 }
960 
961 static ex binomial_real_part(const ex & x, const ex & y)
962 {
963  return binomial(x,y).hold();
964 }
965 
966 static ex binomial_imag_part(const ex & x, const ex & y)
967 {
968  return 0;
969 }
970 
972  evalf_func(binomial_evalf).
973  conjugate_func(binomial_conjugate).
974  real_part_func(binomial_real_part).
975  imag_part_func(binomial_imag_part));
976 
978 // Order term function (for truncated power series)
980 
981 static ex Order_eval(const ex & x)
982 {
983  if (is_exactly_a<numeric>(x)) {
984  // O(c) -> O(1) or 0
985  if (!x.is_zero())
986  return Order(_ex1).hold();
987  else
988  return _ex0;
989  } else if (is_exactly_a<mul>(x)) {
990  const mul &m = ex_to<mul>(x);
991  // O(c*expr) -> O(expr)
992  if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
993  return Order(x / m.op(m.nops() - 1)).hold();
994  }
995  return Order(x).hold();
996 }
997 
998 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
999 {
1000  // Just wrap the function into a pseries object
1001  GINAC_ASSERT(is_a<symbol>(r.lhs()));
1002  const symbol &s = ex_to<symbol>(r.lhs());
1003  epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
1004  return pseries(r, std::move(new_seq));
1005 }
1006 
1007 static ex Order_conjugate(const ex & x)
1008 {
1009  return Order(x).hold();
1010 }
1011 
1012 static ex Order_real_part(const ex & x)
1013 {
1014  return Order(x).hold();
1015 }
1016 
1017 static ex Order_imag_part(const ex & x)
1018 {
1019  if(x.info(info_flags::real))
1020  return 0;
1021  return Order(x).hold();
1022 }
1023 
1024 static ex Order_expl_derivative(const ex & arg, const symbol & s)
1025 {
1026  return Order(arg.diff(s));
1027 }
1028 
1029 REGISTER_FUNCTION(Order, eval_func(Order_eval).
1030  series_func(Order_series).
1031  latex_name("\\mathcal{O}").
1032  expl_derivative_func(Order_expl_derivative).
1033  conjugate_func(Order_conjugate).
1034  real_part_func(Order_real_part).
1035  imag_part_func(Order_imag_part));
1036 
1038 // Solve linear system
1040 
1041 class symbolset {
1043  void insert_symbols(const ex &e)
1044  {
1045  if (is_a<symbol>(e)) {
1046  s.insert(e);
1047  } else {
1048  for (const ex &sube : e) {
1049  insert_symbols(sube);
1050  }
1051  }
1052  }
1053 public:
1054  explicit symbolset(const ex &e)
1055  {
1056  insert_symbols(e);
1057  }
1058  bool has(const ex &e) const
1059  {
1060  return s.find(e) != s.end();
1061  }
1062 };
1063 
1064 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1065 {
1066  // solve a system of linear equations
1067  if (eqns.info(info_flags::relation_equal)) {
1068  if (!symbols.info(info_flags::symbol))
1069  throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1070  const ex sol = lsolve(lst{eqns}, lst{symbols});
1071 
1072  GINAC_ASSERT(sol.nops()==1);
1073  GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1074 
1075  return sol.op(0).op(1); // return rhs of first solution
1076  }
1077 
1078  // syntax checks
1079  if (!(eqns.info(info_flags::list) || eqns.info(info_flags::exprseq))) {
1080  throw(std::invalid_argument("lsolve(): 1st argument must be a list, a sequence, or an equation"));
1081  }
1082  for (size_t i=0; i<eqns.nops(); i++) {
1083  if (!eqns.op(i).info(info_flags::relation_equal)) {
1084  throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1085  }
1086  }
1087  if (!(symbols.info(info_flags::list) || symbols.info(info_flags::exprseq))) {
1088  throw(std::invalid_argument("lsolve(): 2nd argument must be a list, a sequence, or a symbol"));
1089  }
1090  for (size_t i=0; i<symbols.nops(); i++) {
1091  if (!symbols.op(i).info(info_flags::symbol)) {
1092  throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a sequence of symbols"));
1093  }
1094  }
1095 
1096  // build matrix from equation system
1097  matrix sys(eqns.nops(),symbols.nops());
1098  matrix rhs(eqns.nops(),1);
1099  matrix vars(symbols.nops(),1);
1100 
1101  for (size_t r=0; r<eqns.nops(); r++) {
1102  const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1103  const symbolset syms(eq);
1104  ex linpart = eq;
1105  for (size_t c=0; c<symbols.nops(); c++) {
1106  if (!syms.has(symbols.op(c)))
1107  continue;
1108  const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1109  linpart -= co*symbols.op(c);
1110  sys(r,c) = co;
1111  }
1112  linpart = linpart.expand();
1113  rhs(r,0) = -linpart;
1114  }
1115 
1116  // test if system is linear and fill vars matrix
1117  const symbolset sys_syms(sys);
1118  const symbolset rhs_syms(rhs);
1119  for (size_t i=0; i<symbols.nops(); i++) {
1120  vars(i,0) = symbols.op(i);
1121  if (sys_syms.has(symbols.op(i)))
1122  throw(std::logic_error("lsolve: system is not linear"));
1123  if (rhs_syms.has(symbols.op(i)))
1124  throw(std::logic_error("lsolve: system is not linear"));
1125  }
1126 
1127  matrix solution;
1128  try {
1129  solution = sys.solve(vars,rhs,options);
1130  } catch (const std::runtime_error & e) {
1131  // Probably singular matrix or otherwise overdetermined system:
1132  // It is consistent to return an empty list
1133  return lst{};
1134  }
1135  GINAC_ASSERT(solution.cols()==1);
1136  GINAC_ASSERT(solution.rows()==symbols.nops());
1137 
1138  // return list of equations of the form lst{var1==sol1,var2==sol2,...}
1139  lst sollist;
1140  for (size_t i=0; i<symbols.nops(); i++)
1141  sollist.append(symbols.op(i)==solution(i,0));
1142 
1143  return sollist;
1144 }
1145 
1147 // Find real root of f(x) numerically
1149 
1150 const numeric
1151 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1152 {
1153  if (!x1.is_real() || !x2.is_real()) {
1154  throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1155  }
1156  if (x1==x2) {
1157  throw std::runtime_error("fsolve(): vanishing interval");
1158  }
1159  // xx[0] == left interval limit, xx[1] == right interval limit.
1160  // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1161  // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1162  numeric xx[2] = { x1<x2 ? x1 : x2,
1163  x1<x2 ? x2 : x1 };
1164  ex f;
1165  if (is_a<relational>(f_in)) {
1166  f = f_in.lhs()-f_in.rhs();
1167  } else {
1168  f = f_in;
1169  }
1170  const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1171  f.subs(x==xx[1]).evalf() };
1172  if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1173  throw std::runtime_error("fsolve(): function does not evaluate numerically");
1174  }
1175  numeric fx[2] = { ex_to<numeric>(fx_[0]),
1176  ex_to<numeric>(fx_[1]) };
1177  if (!fx[0].is_real() || !fx[1].is_real()) {
1178  throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1179  }
1180  if (fx[0]*fx[1]>=0) {
1181  throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1182  }
1183 
1184  // The Newton-Raphson method has quadratic convergence! Simply put, it
1185  // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
1186  const ex ff = normal(-f/f.diff(x));
1187  int side = 0; // Start at left interval limit.
1188  numeric xxprev;
1189  numeric fxprev;
1190  do {
1191  xxprev = xx[side];
1192  fxprev = fx[side];
1193  ex dx_ = ff.subs(x == xx[side]).evalf();
1194  if (!is_a<numeric>(dx_))
1195  throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1196  xx[side] += ex_to<numeric>(dx_);
1197  // Now check if Newton-Raphson method shot out of the interval
1198  bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1199  (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1200  if (!bad_shot) {
1201  // Compute f(x) only if new x is inside the interval.
1202  // The function might be difficult to compute numerically
1203  // or even ill defined outside the interval. Also it's
1204  // a small optimization.
1205  ex f_x = f.subs(x == xx[side]).evalf();
1206  if (!is_a<numeric>(f_x))
1207  throw std::runtime_error("fsolve(): function does not evaluate numerically");
1208  fx[side] = ex_to<numeric>(f_x);
1209  }
1210  if (bad_shot) {
1211  // Oops, Newton-Raphson method shot out of the interval.
1212  // Restore, and try again with the other side instead!
1213  xx[side] = xxprev;
1214  fx[side] = fxprev;
1215  side = !side;
1216  xxprev = xx[side];
1217  fxprev = fx[side];
1218 
1219  ex dx_ = ff.subs(x == xx[side]).evalf();
1220  if (!is_a<numeric>(dx_))
1221  throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1222  xx[side] += ex_to<numeric>(dx_);
1223 
1224  ex f_x = f.subs(x==xx[side]).evalf();
1225  if (!is_a<numeric>(f_x))
1226  throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1227  fx[side] = ex_to<numeric>(f_x);
1228  }
1229  if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1230  // Oops, the root isn't bracketed any more.
1231  // Restore, and perform a bisection!
1232  xx[side] = xxprev;
1233  fx[side] = fxprev;
1234 
1235  // Ah, the bisection! Bisections converge linearly. Unfortunately,
1236  // they occur pretty often when Newton-Raphson arrives at an x too
1237  // close to the result on one side of the interval and
1238  // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1239  // precision errors! Recall that this function does not have a
1240  // precision goal as one of its arguments but instead relies on
1241  // x converging to a fixed point. We speed up the (safe but slow)
1242  // bisection method by mixing in a dash of the (unsafer but faster)
1243  // secant method: Instead of splitting the interval at the
1244  // arithmetic mean (bisection), we split it nearer to the root as
1245  // determined by the secant between the values xx[0] and xx[1].
1246  // Don't set the secant_weight to one because that could disturb
1247  // the convergence in some corner cases!
1248  constexpr double secant_weight = 0.984375; // == 63/64 < 1
1249  numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1250  + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1251  ex fxmid_ = f.subs(x == xxmid).evalf();
1252  if (!is_a<numeric>(fxmid_))
1253  throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1254  numeric fxmid = ex_to<numeric>(fxmid_);
1255  if (fxmid.is_zero()) {
1256  // Luck strikes...
1257  return xxmid;
1258  }
1259  if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1260  side = !side;
1261  }
1262  xxprev = xx[side];
1263  fxprev = fx[side];
1264  xx[side] = xxmid;
1265  fx[side] = fxmid;
1266  }
1267  } while (xxprev!=xx[side]);
1268  return xxprev;
1269 }
1270 
1271 
1272 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1273  * for static lib (so ginsh will see them). */
1274 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1276 
1277 } // namespace GiNaC
GiNaC::eta_evalf
static ex eta_evalf(const ex &x, const ex &y)
Definition: inifcns.cpp:579
inifcns.h
Interface to GiNaC's initially known functions.
GiNaC::eta_conjugate
static ex eta_conjugate(const ex &x, const ex &y)
Definition: inifcns.cpp:644
GiNaC::container::append
container & append(const ex &b)
Add element at back.
Definition: container.h:391
GiNaC::epvector
std::vector< expair > epvector
expair-vector
Definition: expairseq.h:33
GiNaC::ex::expand
ex expand(unsigned options=0) const
Definition: ex.cpp:73
GiNaC::info_flags::real
@ real
Definition: flags.h:221
GiNaC::numeric::to_int
int to_int() const
Converts numeric types to machine's int.
Definition: numeric.cpp:1303
GiNaC::zetaderiv_eval
static ex zetaderiv_eval(const ex &n, const ex &x)
Definition: inifcns.cpp:837
GiNaC::const_iterator
Definition: ex.h:370
constant.h
Interface to GiNaC's constant types and some special constants.
GiNaC::eta_eval
static ex eta_eval(const ex &x, const ex &y)
Definition: inifcns.cpp:604
GiNaC::lsolve
ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
Factorial function.
Definition: inifcns.cpp:1064
GiNaC::imag_part_real_part
static ex imag_part_real_part(const ex &arg)
Definition: inifcns.cpp:224
GiNaC::real_part
ex real_part(const ex &thisex)
Definition: ex.h:721
GiNaC::rhs
ex rhs(const ex &thisex)
Definition: ex.h:817
GiNaC::info_flags::exprseq
@ exprseq
Definition: flags.h:252
GiNaC::ex::end
const_iterator end() const noexcept
Definition: ex.h:652
GiNaC::info_flags::integer
@ integer
Definition: flags.h:223
GiNaC::is_odd
bool is_odd(const numeric &x)
Definition: numeric.h:284
GiNaC::numeric::is_integer
bool is_integer() const
True if object is a non-complex integer.
Definition: numeric.cpp:1154
GiNaC::ex::evalf
ex evalf() const
Definition: ex.h:121
r
size_t r
Definition: factor.cpp:770
GiNaC::info_flags::odd
@ odd
Definition: flags.h:233
GiNaC::info_flags::negative
@ negative
Definition: flags.h:227
mul.h
Interface to GiNaC's products of expressions.
GiNaC::numeric::info
bool info(unsigned inf) const override
Information about the object.
Definition: numeric.cpp:684
GiNaC::pseries
This class holds a extended truncated power series (positive and negative integer powers).
Definition: pseries.h:36
GiNaC::abs_imag_part
static ex abs_imag_part(const ex &arg)
Definition: inifcns.cpp:347
GiNaC::abs_expand
static ex abs_expand(const ex &arg, unsigned options)
Definition: inifcns.cpp:300
GiNaC::real_part_evalf
static ex real_part_evalf(const ex &arg)
Definition: inifcns.cpp:143
GiNaC::real_part_imag_part
static ex real_part_imag_part(const ex &arg)
Definition: inifcns.cpp:171
GiNaC::ex::coeff
ex coeff(const ex &s, int n=1) const
Definition: ex.h:175
GiNaC::status_flags::expanded
@ expanded
.expand(0) has already done its job (other expand() options ignore this flag)
Definition: flags.h:204
GiNaC::sy_symm
symmetry sy_symm()
Definition: symmetry.h:121
GiNaC::_ex0
const ex _ex0
Definition: utils.cpp:177
GiNaC::symbolset::symbolset
symbolset(const ex &e)
Definition: inifcns.cpp:1054
GiNaC::relational
This class holds a relation consisting of two expressions and a logical relation between them.
Definition: relational.h:35
GiNaC::do_taylor
Exception class thrown by classes which provide their own series expansion to signal that ordinary Ta...
Definition: function.h:668
GiNaC::ex::subs
ex subs(const exmap &m, unsigned options=0) const
Definition: ex.h:826
GiNaC::print_context
Base class for print_contexts.
Definition: print.h:103
GiNaC::info_flags::even
@ even
Definition: flags.h:232
symmetry.h
Interface to GiNaC's symmetry definitions.
GiNaC::csgn_conjugate
static ex csgn_conjugate(const ex &arg)
Definition: inifcns.cpp:537
GiNaC::numeric::imag
const numeric imag() const
Imaginary part of a number.
Definition: numeric.cpp:1346
GiNaC::evalf
ex evalf(const ex &thisex)
Definition: ex.h:769
GiNaC::symbolset::s
exset s
Definition: inifcns.cpp:1042
GiNaC::info_flags::relation_equal
@ relation_equal
Definition: flags.h:238
GiNaC::abs_evalf
static ex abs_evalf(const ex &arg)
Definition: inifcns.cpp:259
GiNaC::ex::conjugate
ex conjugate() const
Definition: ex.h:146
power.h
Interface to GiNaC's symbolic exponentiation (basis^exponent).
GiNaC::expand_options::expand_transcendental
@ expand_transcendental
expands transcendental functions like log and exp
Definition: flags.h:35
GiNaC::exvector
std::vector< ex > exvector
Definition: basic.h:46
GiNaC::ex::nops
size_t nops() const
Definition: ex.h:135
GiNaC::info_flags::rational_polynomial
@ rational_polynomial
Definition: flags.h:258
GiNaC::series
ex series(const ex &thisex, const ex &r, int order, unsigned options=0)
Definition: ex.h:781
GiNaC::info_flags::cinteger_polynomial
@ cinteger_polynomial
Definition: flags.h:257
GiNaC::power
This class holds a two-component object, a basis and and exponent representing exponentiation.
Definition: power.h:39
GiNaC::step_conjugate
static ex step_conjugate(const ex &arg)
Definition: inifcns.cpp:459
GiNaC::imag_part_print_latex
static void imag_part_print_latex(const ex &arg, const print_context &c)
Definition: inifcns.cpp:214
GiNaC::binomial_evalf
static ex binomial_evalf(const ex &x, const ex &y)
Definition: inifcns.cpp:919
GiNaC::numeric::is_real
bool is_real() const
True if object is a real integer, rational or float (but not complex).
Definition: numeric.cpp:1208
GiNaC::info_flags::nonnegative
@ nonnegative
Definition: flags.h:228
GiNaC::Order_series
static ex Order_series(const ex &x, const relational &r, int order, unsigned options)
Definition: inifcns.cpp:998
GiNaC::csgn_power
static ex csgn_power(const ex &arg, const ex &exp)
Definition: inifcns.cpp:552
GiNaC::_ex1
const ex _ex1
Definition: utils.cpp:193
relational.h
Interface to relations between expressions.
GiNaC::zeta1_SERIAL::serial
static unsigned serial
Definition: inifcns.h:109
GiNaC::info_flags::positive
@ positive
Definition: flags.h:226
GiNaC::abs_info
bool abs_info(const ex &arg, unsigned inf)
Definition: inifcns.cpp:363
GiNaC::eta_series
static ex eta_series(const ex &x, const ex &y, const relational &rel, int order, unsigned options)
Definition: inifcns.cpp:629
GiNaC::Li2_deriv
static ex Li2_deriv(const ex &x, unsigned deriv_param)
Definition: inifcns.cpp:710
GiNaC::abs_eval
static ex abs_eval(const ex &arg)
Definition: inifcns.cpp:267
options
unsigned options
Definition: factor.cpp:2480
GiNaC::ex::is_equal
bool is_equal(const ex &other) const
Definition: ex.h:345
GiNaC::Order_conjugate
static ex Order_conjugate(const ex &x)
Definition: inifcns.cpp:1007
m
mvec m
Definition: factor.cpp:771
GiNaC::Order_eval
static ex Order_eval(const ex &x)
Definition: inifcns.cpp:981
GiNaC::factorial_real_part
static ex factorial_real_part(const ex &x)
Definition: inifcns.cpp:897
GiNaC::factorial
const numeric factorial(const numeric &n)
Factorial combinatorial function.
Definition: numeric.cpp:2113
GiNaC::Li2
const numeric Li2(const numeric &x)
Definition: numeric.cpp:1705
GiNaC::matrix::solve
matrix solve(const matrix &vars, const matrix &rhs, unsigned algo=solve_algo::automatic) const
Solve a linear system consisting of a m x n matrix and a m x p right hand side by applying an elimina...
Definition: matrix.cpp:995
GiNaC::print_func< print_dflt >
print_func< print_dflt >(&diracone::do_print). print_func< print_latex >(&diracone
Definition: clifford.cpp:51
GiNaC::real_part_expl_derivative
static ex real_part_expl_derivative(const ex &arg, const symbol &s)
Definition: inifcns.cpp:177
GiNaC::matrix::cols
unsigned cols() const
Get number of columns.
Definition: matrix.h:77
GiNaC::force_include_zeta1
unsigned force_include_zeta1
Definition: inifcns.cpp:1275
GiNaC::symbolset::insert_symbols
void insert_symbols(const ex &e)
Definition: inifcns.cpp:1043
GiNaC::conjugate_conjugate
static ex conjugate_conjugate(const ex &arg)
Definition: inifcns.cpp:65
GiNaC
Definition: add.cpp:38
syms
exset syms
Definition: factor.cpp:2434
GiNaC::_ex6
const ex _ex6
Definition: utils.cpp:213
GiNaC::is_real
bool is_real(const numeric &x)
Definition: numeric.h:293
GiNaC::conjugate
ex conjugate(const ex &thisex)
Definition: ex.h:718
GiNaC::abs_conjugate
static ex abs_conjugate(const ex &arg)
Definition: inifcns.cpp:337
GiNaC::info_flags::symbol
@ symbol
Definition: flags.h:246
GiNaC::info_flags::nonnegint
@ nonnegint
Definition: flags.h:231
GiNaC::binomial_sym
static ex binomial_sym(const ex &x, const numeric &y)
Definition: inifcns.cpp:924
GiNaC::symbol::info
bool info(unsigned inf) const override
Information about the object.
Definition: symbol.cpp:206
GiNaC::abs_power
static ex abs_power(const ex &arg, const ex &exp)
Definition: inifcns.cpp:352
GiNaC::numeric::is_negative
bool is_negative() const
True if object is not complex and less than zero.
Definition: numeric.cpp:1145
matrix.h
Interface to symbolic matrices.
x
ex x
Definition: factor.cpp:1641
GiNaC::ex::op
ex op(size_t i) const
Definition: ex.h:136
GiNaC::zeta
function zeta(const T1 &p1)
Definition: inifcns.h:111
GiNaC::ex::info
bool info(unsigned inf) const
Definition: ex.h:132
utils.h
Interface to several small and furry utilities needed within GiNaC but not of any interest to the use...
GiNaC::real_part_conjugate
static ex real_part_conjugate(const ex &arg)
Definition: inifcns.cpp:161
GiNaC::info_flags::crational_polynomial
@ crational_polynomial
Definition: flags.h:259
GiNaC::Li2_series
static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
Definition: inifcns.cpp:718
GiNaC::ex
Lightweight wrapper for GiNaC's symbolic objects.
Definition: ex.h:72
GiNaC::step_real_part
static ex step_real_part(const ex &arg)
Definition: inifcns.cpp:464
GiNaC::is_even
bool is_even(const numeric &x)
Definition: numeric.h:281
lst.h
Definition of GiNaC's lst.
GiNaC::ex::begin
const_iterator begin() const noexcept
Definition: ex.h:647
GiNaC::numeric::is_zero
bool is_zero() const
True if object is zero.
Definition: numeric.cpp:1129
GiNaC::csgn
int csgn(const numeric &x)
Definition: numeric.h:260
GiNaC::_ex2
const ex _ex2
Definition: utils.cpp:197
GiNaC::relational::lhs
ex lhs() const
Definition: relational.h:79
GiNaC::normal
ex normal(const ex &thisex)
Definition: ex.h:754
GiNaC::step_imag_part
static ex step_imag_part(const ex &arg)
Definition: inifcns.cpp:469
GiNaC::csgn_eval
static ex csgn_eval(const ex &arg)
Definition: inifcns.cpp:493
GiNaC::ex::imag_part
ex imag_part() const
Definition: ex.h:148
GiNaC::abs_print_csrc_float
static void abs_print_csrc_float(const ex &arg, const print_context &c)
Definition: inifcns.cpp:332
GiNaC::exp
const numeric exp(const numeric &x)
Exponential function.
Definition: numeric.cpp:1439
GiNaC::conjugate_evalf
static ex conjugate_evalf(const ex &arg)
Definition: inifcns.cpp:47
GiNaC::ex::diff
ex diff(const symbol &s, unsigned nth=1) const
Compute partial derivative of an expression.
Definition: ex.cpp:86
GiNaC::info_flags::list
@ list
Definition: flags.h:249
GiNaC::fderivative
This class represents the (abstract) derivative of a symbolic function.
Definition: fderivative.h:38
GiNaC::ex::ldegree
int ldegree(const ex &s) const
Definition: ex.h:174
GiNaC::info_flags::prime
@ prime
Definition: flags.h:234
GiNaC::fsolve
const numeric fsolve(const ex &f_in, const symbol &x, const numeric &x1, const numeric &x2)
Find a real root of real-valued function f(x) numerically within a given interval.
Definition: inifcns.cpp:1151
GiNaC::info_flags::rational
@ rational
Definition: flags.h:222
ex.h
Interface to GiNaC's light-weight expression handles.
GiNaC::basic::hold
const basic & hold() const
Stop further evaluation.
Definition: basic.cpp:887
GiNaC::abs_expl_derivative
static ex abs_expl_derivative(const ex &arg, const symbol &s)
Definition: inifcns.cpp:321
GiNaC::eta_imag_part
static ex eta_imag_part(const ex &x, const ex &y)
Definition: inifcns.cpp:654
GiNaC::factorial_evalf
static ex factorial_evalf(const ex &x)
Definition: inifcns.cpp:868
GiNaC::conjugate_info
static bool conjugate_info(const ex &arg, unsigned inf)
Definition: inifcns.cpp:124
GiNaC::Order_real_part
static ex Order_real_part(const ex &x)
Definition: inifcns.cpp:1012
GiNaC::I
const numeric I
Imaginary unit.
Definition: numeric.cpp:1433
GiNaC::abs_print_latex
static void abs_print_latex(const ex &arg, const print_context &c)
Definition: inifcns.cpp:327
symbol.h
Interface to GiNaC's symbolic objects.
GiNaC::step
numeric step(const numeric &x)
Definition: numeric.h:257
GiNaC::numeric::is_nonneg_integer
bool is_nonneg_integer() const
True if object is an exact integer greater or equal zero.
Definition: numeric.cpp:1168
GiNaC::ex::print
void print(const print_context &c, unsigned level=0) const
Print expression to stream.
Definition: ex.cpp:56
GiNaC::imag_part_conjugate
static ex imag_part_conjugate(const ex &arg)
Definition: inifcns.cpp:219
GiNaC::ex::real_part
ex real_part() const
Definition: ex.h:147
GiNaC::func_arg_info
static bool func_arg_info(const ex &arg, unsigned inf)
Definition: inifcns.cpp:93
GiNaC::imag_part_imag_part
static ex imag_part_imag_part(const ex &arg)
Definition: inifcns.cpp:229
GiNaC::ex::rhs
ex rhs() const
Right hand side of relational expression.
Definition: ex.cpp:233
GiNaC::info_flags::numeric
@ numeric
Definition: flags.h:220
GiNaC::Pi
const constant Pi("Pi", PiEvalf, "\\pi", domain::positive)
Pi.
Definition: constant.h:82
GiNaC::exset
std::set< ex, ex_is_less > exset
Definition: basic.h:49
GiNaC::info_flags::polynomial
@ polynomial
Definition: flags.h:255
GiNaC::_ex_48
const ex _ex_48
Definition: utils.cpp:88
GiNaC::conjugate_eval
static ex conjugate_eval(const ex &arg)
Definition: inifcns.cpp:55
GiNaC::conjugate_real_part
static ex conjugate_real_part(const ex &arg)
Definition: inifcns.cpp:83
GiNaC::op
ex op(const ex &thisex, size_t i)
Definition: ex.h:811
GiNaC::Order_imag_part
static ex Order_imag_part(const ex &x)
Definition: inifcns.cpp:1017
GiNaC::container
Wrapper template for making GiNaC classes out of STL containers.
Definition: container.h:73
GiNaC::real_part_real_part
static ex real_part_real_part(const ex &arg)
Definition: inifcns.cpp:166
GiNaC::relational::rhs
ex rhs() const
Definition: relational.h:80
GiNaC::subs_options::no_pattern
@ no_pattern
disable pattern matching
Definition: flags.h:51
GiNaC::REGISTER_FUNCTION
REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval). evalf_func(conjugate_evalf). expl_derivative_func(conjugate_expl_derivative). info_func(conjugate_info). print_func< print_latex >(conjugate_print_latex). conjugate_func(conjugate_conjugate). real_part_func(conjugate_real_part). imag_part_func(conjugate_imag_part). set_name("conjugate","conjugate"))
GiNaC::imag_part_eval
static ex imag_part_eval(const ex &arg)
Definition: inifcns.cpp:209
GiNaC::factorial_eval
static ex factorial_eval(const ex &x)
Definition: inifcns.cpp:873
GiNaC::real_part_eval
static ex real_part_eval(const ex &arg)
Definition: inifcns.cpp:151
GiNaC::step_series
static ex step_series(const ex &arg, const relational &rel, int order, unsigned options)
Definition: inifcns.cpp:444
GiNaC::factorial_imag_part
static ex factorial_imag_part(const ex &x)
Definition: inifcns.cpp:902
c
size_t c
Definition: factor.cpp:770
GiNaC::imag_part_evalf
static ex imag_part_evalf(const ex &arg)
Definition: inifcns.cpp:201
GiNaC::Li2_eval
static ex Li2_eval(const ex &x)
Definition: inifcns.cpp:681
GiNaC::ex::series
ex series(const ex &r, int order, unsigned options=0) const
Compute the truncated series expansion of an expression.
Definition: pseries.cpp:1259
GiNaC::imag_part
ex imag_part(const ex &thisex)
Definition: ex.h:724
GiNaC::conjugate_imag_part
static ex conjugate_imag_part(const ex &arg)
Definition: inifcns.cpp:88
GiNaC::binomial_conjugate
static ex binomial_conjugate(const ex &x, const ex &y)
Definition: inifcns.cpp:956
GiNaC::_ex1_2
const ex _ex1_2
Definition: utils.cpp:189
GiNaC::symbolset::has
bool has(const ex &e) const
Definition: inifcns.cpp:1058
GiNaC::binomial_imag_part
static ex binomial_imag_part(const ex &x, const ex &y)
Definition: inifcns.cpp:966
GiNaC::_ex12
const ex _ex12
Definition: utils.cpp:237
GiNaC::imag
const numeric imag(const numeric &x)
Definition: numeric.h:314
GiNaC::symbol
Basic CAS symbol.
Definition: symbol.h:39
GiNaC::csgn_evalf
static ex csgn_evalf(const ex &arg)
Definition: inifcns.cpp:485
GiNaC::subs
ex subs(const ex &thisex, const exmap &m, unsigned options=0)
Definition: ex.h:831
n
size_t n
Definition: factor.cpp:1463
GiNaC::info_flags::rational_function
@ rational_function
Definition: flags.h:260
GiNaC::conjugate_expl_derivative
static ex conjugate_expl_derivative(const ex &arg, const symbol &s)
Definition: inifcns.cpp:72
GiNaC::pow
const numeric pow(const numeric &x, const numeric &y)
Definition: numeric.h:251
pseries.h
Interface to class for extended truncated power series.
GiNaC::info_flags::has_indices
@ has_indices
Definition: flags.h:264
GiNaC::binomial_eval
static ex binomial_eval(const ex &x, const ex &y)
Definition: inifcns.cpp:942
GiNaC::csgn_imag_part
static ex csgn_imag_part(const ex &arg)
Definition: inifcns.cpp:547
GiNaC::Li2_evalf
static ex Li2_evalf(const ex &x)
Definition: inifcns.cpp:673
GiNaC::matrix
Symbolic matrices.
Definition: matrix.h:38
GiNaC::binomial
const numeric binomial(const numeric &n, const numeric &k)
The Binomial coefficients.
Definition: numeric.cpp:2143
GiNaC::Li3_eval
static ex Li3_eval(const ex &x)
Definition: inifcns.cpp:823
GiNaC::csgn_real_part
static ex csgn_real_part(const ex &arg)
Definition: inifcns.cpp:542
GiNaC::real_part_print_latex
static void real_part_print_latex(const ex &arg, const print_context &c)
Definition: inifcns.cpp:156
GiNaC::zetaderiv_deriv
static ex zetaderiv_deriv(const ex &n, const ex &x, unsigned deriv_param)
Definition: inifcns.cpp:848
GiNaC::matrix::rows
unsigned rows() const
Get number of rows.
Definition: matrix.h:75
GiNaC::expair
A pair of expressions.
Definition: expair.h:38
GiNaC::info_flags::negint
@ negint
Definition: flags.h:230
GiNaC::step_evalf
static ex step_evalf(const ex &arg)
Definition: inifcns.cpp:407
GiNaC::ex::is_zero
bool is_zero() const
Definition: ex.h:213
GiNaC::abs_real_part
static ex abs_real_part(const ex &arg)
Definition: inifcns.cpp:342
is_ex_the_function
#define is_ex_the_function(OBJ, FUNCNAME)
Definition: function.h:765
GiNaC::binomial_real_part
static ex binomial_real_part(const ex &x, const ex &y)
Definition: inifcns.cpp:961
GiNaC::info_flags::cinteger
@ cinteger
Definition: flags.h:225
GiNaC::symbolset
Definition: inifcns.cpp:1041
GiNaC::info_flags::posint
@ posint
Definition: flags.h:229
GiNaC::log
const numeric log(const numeric &x)
Natural logarithm.
Definition: numeric.cpp:1450
GiNaC::force_include_tgamma
unsigned force_include_tgamma
Definition: inifcns.cpp:1274
GiNaC::conjugate_print_latex
static void conjugate_print_latex(const ex &arg, const print_context &c)
Definition: inifcns.cpp:60
GiNaC::expand_options::expand_function_args
@ expand_function_args
expands the arguments of functions
Definition: flags.h:33
GiNaC::info_flags::crational
@ crational
Definition: flags.h:224
GiNaC::info_flags::integer_polynomial
@ integer_polynomial
Definition: flags.h:256
GiNaC::abs
const numeric abs(const numeric &x)
Absolute value.
Definition: numeric.cpp:2315
GiNaC::numeric::real
const numeric real() const
Real part of a number.
Definition: numeric.cpp:1339
GiNaC::is_integer
bool is_integer(const numeric &x)
Definition: numeric.h:272
GiNaC::Li2_conjugate
static ex Li2_conjugate(const ex &x)
Definition: inifcns.cpp:798
GiNaC::_num1_p
const numeric * _num1_p
Definition: utils.cpp:192
GiNaC::_ex_1_2
const ex _ex_1_2
Definition: utils.cpp:164
order
int order
Definition: integration_kernel.cpp:248
fderivative.h
Interface to abstract derivatives of functions.
GiNaC::imag_part_expl_derivative
static ex imag_part_expl_derivative(const ex &arg, const symbol &s)
Definition: inifcns.cpp:235
operators.h
Interface to GiNaC's overloaded operators.
GiNaC::_ex_1
const ex _ex_1
Definition: utils.cpp:160
GiNaC::numeric
This class is a wrapper around CLN-numbers within the GiNaC class hierarchy.
Definition: numeric.h:82
GiNaC::factorial_print_dflt_latex
static void factorial_print_dflt_latex(const ex &x, const print_context &c)
Definition: inifcns.cpp:881
GiNaC::eta_real_part
static ex eta_real_part(const ex &x, const ex &y)
Definition: inifcns.cpp:649
GINAC_ASSERT
#define GINAC_ASSERT(X)
Assertion macro for checking invariances.
Definition: assertion.h:33
GiNaC::factorial_conjugate
static ex factorial_conjugate(const ex &x)
Definition: inifcns.cpp:892
GiNaC::_num2_p
const numeric * _num2_p
Definition: utils.cpp:196
GiNaC::series_options::suppress_branchcut
@ suppress_branchcut
Suppress branch cuts in series expansion.
Definition: flags.h:83
GiNaC::mul
Product of expressions.
Definition: mul.h:32
GiNaC::ex::lhs
ex lhs() const
Left hand side of relational expression.
Definition: ex.cpp:225
GiNaC::Order_expl_derivative
static ex Order_expl_derivative(const ex &arg, const symbol &s)
Definition: inifcns.cpp:1024
GiNaC::step_eval
static ex step_eval(const ex &arg)
Definition: inifcns.cpp:415
GiNaC::csgn_series
static ex csgn_series(const ex &arg, const relational &rel, int order, unsigned options)
Definition: inifcns.cpp:522
GiNaC::Catalan
const constant Catalan("Catalan", CatalanEvalf, "G", domain::positive)
Catalan's constant.
Definition: constant.h:83

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