ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2020 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  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
  43 //////////
  44 // complex conjugate
  45 //////////
  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) {
  98                 case info_flags::polynomial:
  99                 case info_flags::integer_polynomial:
 100                 case info_flags::cinteger_polynomial:
 101                 case info_flags::rational_polynomial:
 102                 case info_flags::real:
 103                 case info_flags::rational:
 104                 case info_flags::integer:
 105                 case info_flags::crational:
 106                 case info_flags::cinteger:
 107                 case info_flags::even:
 108                 case info_flags::odd:
 109                 case info_flags::prime:
 110                 case info_flags::crational_polynomial:
 111                 case info_flags::rational_function:
 112                 case info_flags::positive:
 113                 case info_flags::negative:
 114                 case info_flags::nonnegative:
 115                 case info_flags::posint:
 116                 case info_flags::negint:
 117                 case info_flags::nonnegint:
 118                 case info_flags::has_indices:
 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
 139 //////////
 140 // real part
 141 //////////
 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
 197 //////////
 198 // imag part
 199 //////////
 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
 255 //////////
 256 // absolute value
 257 //////////
 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 {
 302         if ((options & expand_options::expand_transcendental)
 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) {
 307                         if (options & expand_options::expand_function_args)
 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
 315         if (options & expand_options::expand_function_args)
 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         if (arg.info(info_flags::real))
 325                 return diff_arg*(2*step(arg) - 1);
 326         return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
 327 }
 328
 329 static void abs_print_latex(const ex & arg, const print_context & c)
 330 {
 331         c.s << "{|"; arg.print(c); c.s << "|}";
 332 }
 333
 334 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 335 {
 336         c.s << "fabs("; arg.print(c); c.s << ")";
 337 }
 338
 339 static ex abs_conjugate(const ex & arg)
 340 {
 341         return abs(arg).hold();
 342 }
 343
 344 static ex abs_real_part(const ex & arg)
 345 {
 346         return abs(arg).hold();
 347 }
 348
 349 static ex abs_imag_part(const ex& arg)
 350 {
 351         return 0;
 352 }
 353
 354 static ex abs_power(const ex & arg, const ex & exp)
 355 {
 356         if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
 357                 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
 358                         return pow(arg, exp);
 359                 else
 360                         return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
 361         } else
 362                 return power(abs(arg), exp).hold();
 363 }
 364
 365 bool abs_info(const ex & arg, unsigned inf)
 366 {
 367         switch (inf) {
 368                 case info_flags::integer:
 369                 case info_flags::even:
 370                 case info_flags::odd:
 371                 case info_flags::prime:
 372                         return arg.info(inf);
 373                 case info_flags::nonnegint:
 374                         return arg.info(info_flags::integer);
 375                 case info_flags::nonnegative:
 376                 case info_flags::real:
 377                         return true;
 378                 case info_flags::negative:
 379                         return false;
 380                 case info_flags::positive:
 381                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
 382                 case info_flags::has_indices: {
 383                         if (arg.info(info_flags::has_indices))
 384                                 return true;
 385                         else
 386                                 return false;
 387                 }
 388         }
 389         return false;
 390 }
 391
 392 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 393                        evalf_func(abs_evalf).
 394                        expand_func(abs_expand).
 395                        expl_derivative_func(abs_expl_derivative).
 396                        info_func(abs_info).
 397                        print_func<print_latex>(abs_print_latex).
 398                        print_func<print_csrc_float>(abs_print_csrc_float).
 399                        print_func<print_csrc_double>(abs_print_csrc_float).
 400                        conjugate_func(abs_conjugate).
 401                        real_part_func(abs_real_part).
 402                        imag_part_func(abs_imag_part).
 403                        power_func(abs_power));
 404
 405 //////////
 406 // Step function
 407 //////////
 408
 409 static ex step_evalf(const ex & arg)
 410 {
 411         if (is_exactly_a<numeric>(arg))
 412                 return step(ex_to<numeric>(arg));
 413
 414         return step(arg).hold();
 415 }
 416
 417 static ex step_eval(const ex & arg)
 418 {
 419         if (is_exactly_a<numeric>(arg))
 420                 return step(ex_to<numeric>(arg));
 421
 422         else if (is_exactly_a<mul>(arg) &&
 423                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 424                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 425                 if (oc.is_real()) {
 426                         if (oc > 0)
 427                                 // step(42*x) -> step(x)
 428                                 return step(arg/oc).hold();
 429                         else
 430                                 // step(-42*x) -> step(-x)
 431                                 return step(-arg/oc).hold();
 432                 }
 433                 if (oc.real().is_zero()) {
 434                         if (oc.imag() > 0)
 435                                 // step(42*I*x) -> step(I*x)
 436                                 return step(I*arg/oc).hold();
 437                         else
 438                                 // step(-42*I*x) -> step(-I*x)
 439                                 return step(-I*arg/oc).hold();
 440                 }
 441         }
 442
 443         return step(arg).hold();
 444 }
 445
 446 static ex step_series(const ex & arg,
 447                       const relational & rel,
 448                       int order,
 449                       unsigned options)
 450 {
 451         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 452         if (arg_pt.info(info_flags::numeric)
 453             && ex_to<numeric>(arg_pt).real().is_zero()
 454             && !(options & series_options::suppress_branchcut))
 455                 throw (std::domain_error("step_series(): on imaginary axis"));
 456
 457         epvector seq { expair(step(arg_pt), _ex0) };
 458         return pseries(rel, std::move(seq));
 459 }
 460
 461 static ex step_conjugate(const ex& arg)
 462 {
 463         return step(arg).hold();
 464 }
 465
 466 static ex step_real_part(const ex& arg)
 467 {
 468         return step(arg).hold();
 469 }
 470
 471 static ex step_imag_part(const ex& arg)
 472 {
 473         return 0;
 474 }
 475
 476 REGISTER_FUNCTION(step, eval_func(step_eval).
 477                         evalf_func(step_evalf).
 478                         series_func(step_series).
 479                         conjugate_func(step_conjugate).
 480                         real_part_func(step_real_part).
 481                         imag_part_func(step_imag_part));
 482
 483 //////////
 484 // Complex sign
 485 //////////
 486
 487 static ex csgn_evalf(const ex & arg)
 488 {
 489         if (is_exactly_a<numeric>(arg))
 490                 return csgn(ex_to<numeric>(arg));
 491
 492         return csgn(arg).hold();
 493 }
 494
 495 static ex csgn_eval(const ex & arg)
 496 {
 497         if (is_exactly_a<numeric>(arg))
 498                 return csgn(ex_to<numeric>(arg));
 499
 500         else if (is_exactly_a<mul>(arg) &&
 501                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 502                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 503                 if (oc.is_real()) {
 504                         if (oc > 0)
 505                                 // csgn(42*x) -> csgn(x)
 506                                 return csgn(arg/oc).hold();
 507                         else
 508                                 // csgn(-42*x) -> -csgn(x)
 509                                 return -csgn(arg/oc).hold();
 510                 }
 511                 if (oc.real().is_zero()) {
 512                         if (oc.imag() > 0)
 513                                 // csgn(42*I*x) -> csgn(I*x)
 514                                 return csgn(I*arg/oc).hold();
 515                         else
 516                                 // csgn(-42*I*x) -> -csgn(I*x)
 517                                 return -csgn(I*arg/oc).hold();
 518                 }
 519         }
 520
 521         return csgn(arg).hold();
 522 }
 523
 524 static ex csgn_series(const ex & arg,
 525                       const relational & rel,
 526                       int order,
 527                       unsigned options)
 528 {
 529         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 530         if (arg_pt.info(info_flags::numeric)
 531             && ex_to<numeric>(arg_pt).real().is_zero()
 532             && !(options & series_options::suppress_branchcut))
 533                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 534
 535         epvector seq { expair(csgn(arg_pt), _ex0) };
 536         return pseries(rel, std::move(seq));
 537 }
 538
 539 static ex csgn_conjugate(const ex& arg)
 540 {
 541         return csgn(arg).hold();
 542 }
 543
 544 static ex csgn_real_part(const ex& arg)
 545 {
 546         return csgn(arg).hold();
 547 }
 548
 549 static ex csgn_imag_part(const ex& arg)
 550 {
 551         return 0;
 552 }
 553
 554 static ex csgn_power(const ex & arg, const ex & exp)
 555 {
 556         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 557                 if (ex_to<numeric>(exp).is_odd())
 558                         return csgn(arg).hold();
 559                 else
 560                         return power(csgn(arg), _ex2).hold();
 561         } else
 562                 return power(csgn(arg), exp).hold();
 563 }
 564
 565
 566 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 567                         evalf_func(csgn_evalf).
 568                         series_func(csgn_series).
 569                         conjugate_func(csgn_conjugate).
 570                         real_part_func(csgn_real_part).
 571                         imag_part_func(csgn_imag_part).
 572                         power_func(csgn_power));
 573
 574
 575 //////////
 576 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 577 // This function is closely related to the unwinding number K, sometimes found
 578 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 579 //////////
 580
 581 static ex eta_evalf(const ex &x, const ex &y)
 582 {
 583         // It seems like we basically have to replicate the eval function here,
 584         // since the expression might not be fully evaluated yet.
 585         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 586                 return _ex0;
 587
 588         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 589                 const numeric nx = ex_to<numeric>(x);
 590                 const numeric ny = ex_to<numeric>(y);
 591                 const numeric nxy = ex_to<numeric>(x*y);
 592                 int cut = 0;
 593                 if (nx.is_real() && nx.is_negative())
 594                         cut -= 4;
 595                 if (ny.is_real() && ny.is_negative())
 596                         cut -= 4;
 597                 if (nxy.is_real() && nxy.is_negative())
 598                         cut += 4;
 599                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 600                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 601         }
 602
 603         return eta(x,y).hold();
 604 }
 605
 606 static ex eta_eval(const ex &x, const ex &y)
 607 {
 608         // trivial:  eta(x,c) -> 0  if c is real and positive
 609         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 610                 return _ex0;
 611
 612         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 613                 // don't call eta_evalf here because it would call Pi.evalf()!
 614                 const numeric nx = ex_to<numeric>(x);
 615                 const numeric ny = ex_to<numeric>(y);
 616                 const numeric nxy = ex_to<numeric>(x*y);
 617                 int cut = 0;
 618                 if (nx.is_real() && nx.is_negative())
 619                         cut -= 4;
 620                 if (ny.is_real() && ny.is_negative())
 621                         cut -= 4;
 622                 if (nxy.is_real() && nxy.is_negative())
 623                         cut += 4;
 624                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 625                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 626         }
 627
 628         return eta(x,y).hold();
 629 }
 630
 631 static ex eta_series(const ex & x, const ex & y,
 632                      const relational & rel,
 633                      int order,
 634                      unsigned options)
 635 {
 636         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 637         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 638         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 639             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 640             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 641                         throw (std::domain_error("eta_series(): on discontinuity"));
 642         epvector seq { expair(eta(x_pt,y_pt), _ex0) };
 643         return pseries(rel, std::move(seq));
 644 }
 645
 646 static ex eta_conjugate(const ex & x, const ex & y)
 647 {
 648         return -eta(x, y).hold();
 649 }
 650
 651 static ex eta_real_part(const ex & x, const ex & y)
 652 {
 653         return 0;
 654 }
 655
 656 static ex eta_imag_part(const ex & x, const ex & y)
 657 {
 658         return -I*eta(x, y).hold();
 659 }
 660
 661 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 662                        evalf_func(eta_evalf).
 663                        series_func(eta_series).
 664                        latex_name("\\eta").
 665                        set_symmetry(sy_symm(0, 1)).
 666                        conjugate_func(eta_conjugate).
 667                        real_part_func(eta_real_part).
 668                        imag_part_func(eta_imag_part));
 669
 670
 671 //////////
 672 // dilogarithm
 673 //////////
 674
 675 static ex Li2_evalf(const ex & x)
 676 {
 677         if (is_exactly_a<numeric>(x))
 678                 return Li2(ex_to<numeric>(x));
 679
 680         return Li2(x).hold();
 681 }
 682
 683 static ex Li2_eval(const ex & x)
 684 {
 685         if (x.info(info_flags::numeric)) {
 686                 // Li2(0) -> 0
 687                 if (x.is_zero())
 688                         return _ex0;
 689                 // Li2(1) -> Pi^2/6
 690                 if (x.is_equal(_ex1))
 691                         return power(Pi,_ex2)/_ex6;
 692                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 693                 if (x.is_equal(_ex1_2))
 694                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 695                 // Li2(-1) -> -Pi^2/12
 696                 if (x.is_equal(_ex_1))
 697                         return -power(Pi,_ex2)/_ex12;
 698                 // Li2(I) -> -Pi^2/48+Catalan*I
 699                 if (x.is_equal(I))
 700                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 701                 // Li2(-I) -> -Pi^2/48-Catalan*I
 702                 if (x.is_equal(-I))
 703                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 704                 // Li2(float)
 705                 if (!x.info(info_flags::crational))
 706                         return Li2(ex_to<numeric>(x));
 707         }
 708
 709         return Li2(x).hold();
 710 }
 711
 712 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 713 {
 714         GINAC_ASSERT(deriv_param==0);
 715
 716         // d/dx Li2(x) -> -log(1-x)/x
 717         return -log(_ex1-x)/x;
 718 }
 719
 720 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 721 {
 722         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 723         if (x_pt.info(info_flags::numeric)) {
 724                 // First special case: x==0 (derivatives have poles)
 725                 if (x_pt.is_zero()) {
 726                         // method:
 727                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 728                         // simply substitute x==0.  The limit, however, exists: it is 1.
 729                         // We also know all higher derivatives' limits:
 730                         // (d/dx)^n Li2(x) == n!/n^2.
 731                         // So the primitive series expansion is
 732                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 733                         // and so on.
 734                         // We first construct such a primitive series expansion manually in
 735                         // a dummy symbol s and then insert the argument's series expansion
 736                         // for s.  Reexpanding the resulting series returns the desired
 737                         // result.
 738                         const symbol s;
 739                         ex ser;
 740                         // manually construct the primitive expansion
 741                         for (int i=1; i<order; ++i)
 742                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 743                         // substitute the argument's series expansion
 744                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 745                         // maybe that was terminating, so add a proper order term
 746                         epvector nseq { expair(Order(_ex1), order) };
 747                         ser += pseries(rel, std::move(nseq));
 748                         // reexpanding it will collapse the series again
 749                         return ser.series(rel, order);
 750                         // NB: Of course, this still does not allow us to compute anything
 751                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 752                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 753                         // substitution x==0.  Probably limits *are* needed for the general
 754                         // cases.  In case L'Hospital's rule is implemented for limits and
 755                         // basic::series() takes care of this, this whole block is probably
 756                         // obsolete!
 757                 }
 758                 // second special case: x==1 (branch point)
 759                 if (x_pt.is_equal(_ex1)) {
 760                         // method:
 761                         // construct series manually in a dummy symbol s
 762                         const symbol s;
 763                         ex ser = zeta(_ex2);
 764                         // manually construct the primitive expansion
 765                         for (int i=1; i<order; ++i)
 766                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 767                         // substitute the argument's series expansion
 768                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 769                         // maybe that was terminating, so add a proper order term
 770                         epvector nseq { expair(Order(_ex1), order) };
 771                         ser += pseries(rel, std::move(nseq));
 772                         // reexpanding it will collapse the series again
 773                         return ser.series(rel, order);
 774                 }
 775                 // third special case: x real, >=1 (branch cut)
 776                 if (!(options & series_options::suppress_branchcut) &&
 777                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 778                         // method:
 779                         // This is the branch cut: assemble the primitive series manually
 780                         // and then add the corresponding complex step function.
 781                         const symbol &s = ex_to<symbol>(rel.lhs());
 782                         const ex point = rel.rhs();
 783                         const symbol foo;
 784                         epvector seq;
 785                         // zeroth order term:
 786                         seq.push_back(expair(Li2(x_pt), _ex0));
 787                         // compute the intermediate terms:
 788                         ex replarg = series(Li2(x), s==foo, order);
 789                         for (size_t i=1; i<replarg.nops()-1; ++i)
 790                                 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));
 791                         // append an order term:
 792                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 793                         return pseries(rel, std::move(seq));
 794                 }
 795         }
 796         // all other cases should be safe, by now:
 797         throw do_taylor();  // caught by function::series()
 798 }
 799
 800 static ex Li2_conjugate(const ex & x)
 801 {
 802         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 803         // run along the positive real axis beginning at 1.
 804         if (x.info(info_flags::negative)) {
 805                 return Li2(x).hold();
 806         }
 807         if (is_exactly_a<numeric>(x) &&
 808             (!x.imag_part().is_zero() || x < *_num1_p)) {
 809                 return Li2(x.conjugate());
 810         }
 811         return conjugate_function(Li2(x)).hold();
 812 }
 813
 814 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 815                        evalf_func(Li2_evalf).
 816                        derivative_func(Li2_deriv).
 817                        series_func(Li2_series).
 818                        conjugate_func(Li2_conjugate).
 819                        latex_name("\\mathrm{Li}_2"));
 820
 821 //////////
 822 // trilogarithm
 823 //////////
 824
 825 static ex Li3_eval(const ex & x)
 826 {
 827         if (x.is_zero())
 828                 return x;
 829         return Li3(x).hold();
 830 }
 831
 832 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 833                        latex_name("\\mathrm{Li}_3"));
 834
 835 //////////
 836 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 837 //////////
 838
 839 static ex zetaderiv_eval(const ex & n, const ex & x)
 840 {
 841         if (n.info(info_flags::numeric)) {
 842                 // zetaderiv(0,x) -> zeta(x)
 843                 if (n.is_zero())
 844                         return zeta(x).hold();
 845         }
 846
 847         return zetaderiv(n, x).hold();
 848 }
 849
 850 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 851 {
 852         GINAC_ASSERT(deriv_param<2);
 853
 854         if (deriv_param==0) {
 855                 // d/dn zeta(n,x)
 856                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 857         }
 858         // d/dx psi(n,x)
 859         return zetaderiv(n+1,x);
 860 }
 861
 862 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 863                                  derivative_func(zetaderiv_deriv).
 864                                  latex_name("\\zeta^\\prime"));
 865
 866 //////////
 867 // factorial
 868 //////////
 869
 870 static ex factorial_evalf(const ex & x)
 871 {
 872         return factorial(x).hold();
 873 }
 874
 875 static ex factorial_eval(const ex & x)
 876 {
 877         if (is_exactly_a<numeric>(x))
 878                 return factorial(ex_to<numeric>(x));
 879         else
 880                 return factorial(x).hold();
 881 }
 882
 883 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 884 {
 885         if (is_exactly_a<symbol>(x) ||
 886             is_exactly_a<constant>(x) ||
 887                 is_exactly_a<function>(x)) {
 888                 x.print(c); c.s << "!";
 889         } else {
 890                 c.s << "("; x.print(c); c.s << ")!";
 891         }
 892 }
 893
 894 static ex factorial_conjugate(const ex & x)
 895 {
 896         return factorial(x).hold();
 897 }
 898
 899 static ex factorial_real_part(const ex & x)
 900 {
 901         return factorial(x).hold();
 902 }
 903
 904 static ex factorial_imag_part(const ex & x)
 905 {
 906         return 0;
 907 }
 908
 909 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 910                              evalf_func(factorial_evalf).
 911                              print_func<print_dflt>(factorial_print_dflt_latex).
 912                              print_func<print_latex>(factorial_print_dflt_latex).
 913                              conjugate_func(factorial_conjugate).
 914                              real_part_func(factorial_real_part).
 915                              imag_part_func(factorial_imag_part));
 916
 917 //////////
 918 // binomial
 919 //////////
 920
 921 static ex binomial_evalf(const ex & x, const ex & y)
 922 {
 923         return binomial(x, y).hold();
 924 }
 925
 926 static ex binomial_sym(const ex & x, const numeric & y)
 927 {
 928         if (y.is_integer()) {
 929                 if (y.is_nonneg_integer()) {
 930                         const unsigned N = y.to_int();
 931                         if (N == 0) return _ex1;
 932                         if (N == 1) return x;
 933                         ex t = x.expand();
 934                         for (unsigned i = 2; i <= N; ++i)
 935                                 t = (t * (x + i - y - 1)).expand() / i;
 936                         return t;
 937                 } else
 938                         return _ex0;
 939         }
 940
 941         return binomial(x, y).hold();
 942 }
 943
 944 static ex binomial_eval(const ex & x, const ex &y)
 945 {
 946         if (is_exactly_a<numeric>(y)) {
 947                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 948                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 949                 else
 950                         return binomial_sym(x, ex_to<numeric>(y));
 951         } else
 952                 return binomial(x, y).hold();
 953 }
 954
 955 // At the moment the numeric evaluation of a binomial function always
 956 // gives a real number, but if this would be implemented using the gamma
 957 // function, also complex conjugation should be changed (or rather, deleted).
 958 static ex binomial_conjugate(const ex & x, const ex & y)
 959 {
 960         return binomial(x,y).hold();
 961 }
 962
 963 static ex binomial_real_part(const ex & x, const ex & y)
 964 {
 965         return binomial(x,y).hold();
 966 }
 967
 968 static ex binomial_imag_part(const ex & x, const ex & y)
 969 {
 970         return 0;
 971 }
 972
 973 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 974                             evalf_func(binomial_evalf).
 975                             conjugate_func(binomial_conjugate).
 976                             real_part_func(binomial_real_part).
 977                             imag_part_func(binomial_imag_part));
 978
 979 //////////
 980 // Order term function (for truncated power series)
 981 //////////
 982
 983 static ex Order_eval(const ex & x)
 984 {
 985         if (is_exactly_a<numeric>(x)) {
 986                 // O(c) -> O(1) or 0
 987                 if (!x.is_zero())
 988                         return Order(_ex1).hold();
 989                 else
 990                         return _ex0;
 991         } else if (is_exactly_a<mul>(x)) {
 992                 const mul &m = ex_to<mul>(x);
 993                 // O(c*expr) -> O(expr)
 994                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 995                         return Order(x / m.op(m.nops() - 1)).hold();
 996         }
 997         return Order(x).hold();
 998 }
 999
1000 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
1001 {
1002         // Just wrap the function into a pseries object
1003         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1004         const symbol &s = ex_to<symbol>(r.lhs());
1005         epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
1006         return pseries(r, std::move(new_seq));
1007 }
1008
1009 static ex Order_conjugate(const ex & x)
1010 {
1011         return Order(x).hold();
1012 }
1013
1014 static ex Order_real_part(const ex & x)
1015 {
1016         return Order(x).hold();
1017 }
1018
1019 static ex Order_imag_part(const ex & x)
1020 {
1021         if(x.info(info_flags::real))
1022                 return 0;
1023         return Order(x).hold();
1024 }
1025
1026 static ex Order_expl_derivative(const ex & arg, const symbol & s)
1027 {
1028         return Order(arg.diff(s));
1029 }
1030
1031 REGISTER_FUNCTION(Order, eval_func(Order_eval).
1032                          series_func(Order_series).
1033                          latex_name("\\mathcal{O}").
1034                          expl_derivative_func(Order_expl_derivative).
1035                          conjugate_func(Order_conjugate).
1036                          real_part_func(Order_real_part).
1037                          imag_part_func(Order_imag_part));
1038
1039 //////////
1040 // Solve linear system
1041 //////////
1042
1043 class symbolset {
1044         exset s;
1045         void insert_symbols(const ex &e)
1046         {
1047                 if (is_a<symbol>(e)) {
1048                         s.insert(e);
1049                 } else {
1050                         for (const ex &sube : e) {
1051                                 insert_symbols(sube);
1052                         }
1053                 }
1054         }
1055 public:
1056         explicit symbolset(const ex &e)
1057         {
1058                 insert_symbols(e);
1059         }
1060         bool has(const ex &e) const
1061         {
1062                 return s.find(e) != s.end();
1063         }
1064 };
1065
1066 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1067 {
1068         // solve a system of linear equations
1069         if (eqns.info(info_flags::relation_equal)) {
1070                 if (!symbols.info(info_flags::symbol))
1071                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1072                 const ex sol = lsolve(lst{eqns}, lst{symbols});
1073
1074                 GINAC_ASSERT(sol.nops()==1);
1075                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1076
1077                 return sol.op(0).op(1); // return rhs of first solution
1078         }
1079
1080         // syntax checks
1081         if (!(eqns.info(info_flags::list) || eqns.info(info_flags::exprseq))) {
1082                 throw(std::invalid_argument("lsolve(): 1st argument must be a list, a sequence, or an equation"));
1083         }
1084         for (size_t i=0; i<eqns.nops(); i++) {
1085                 if (!eqns.op(i).info(info_flags::relation_equal)) {
1086                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1087                 }
1088         }
1089         if (!(symbols.info(info_flags::list) || symbols.info(info_flags::exprseq))) {
1090                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list, a sequence, or a symbol"));
1091         }
1092         for (size_t i=0; i<symbols.nops(); i++) {
1093                 if (!symbols.op(i).info(info_flags::symbol)) {
1094                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a sequence of symbols"));
1095                 }
1096         }
1097
1098         // build matrix from equation system
1099         matrix sys(eqns.nops(),symbols.nops());
1100         matrix rhs(eqns.nops(),1);
1101         matrix vars(symbols.nops(),1);
1102
1103         for (size_t r=0; r<eqns.nops(); r++) {
1104                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1105                 const symbolset syms(eq);
1106                 ex linpart = eq;
1107                 for (size_t c=0; c<symbols.nops(); c++) {
1108                         if (!syms.has(symbols.op(c)))
1109                                 continue;
1110                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1111                         linpart -= co*symbols.op(c);
1112                         sys(r,c) = co;
1113                 }
1114                 linpart = linpart.expand();
1115                 rhs(r,0) = -linpart;
1116         }
1117
1118         // test if system is linear and fill vars matrix
1119         const symbolset sys_syms(sys);
1120         const symbolset rhs_syms(rhs);
1121         for (size_t i=0; i<symbols.nops(); i++) {
1122                 vars(i,0) = symbols.op(i);
1123                 if (sys_syms.has(symbols.op(i)))
1124                         throw(std::logic_error("lsolve: system is not linear"));
1125                 if (rhs_syms.has(symbols.op(i)))
1126                         throw(std::logic_error("lsolve: system is not linear"));
1127         }
1128
1129         matrix solution;
1130         try {
1131                 solution = sys.solve(vars,rhs,options);
1132         } catch (const std::runtime_error & e) {
1133                 // Probably singular matrix or otherwise overdetermined system:
1134                 // It is consistent to return an empty list
1135                 return lst{};
1136         }
1137         GINAC_ASSERT(solution.cols()==1);
1138         GINAC_ASSERT(solution.rows()==symbols.nops());
1139
1140         // return list of equations of the form lst{var1==sol1,var2==sol2,...}
1141         lst sollist;
1142         for (size_t i=0; i<symbols.nops(); i++)
1143                 sollist.append(symbols.op(i)==solution(i,0));
1144
1145         return sollist;
1146 }
1147
1148 //////////
1149 // Find real root of f(x) numerically
1150 //////////
1151
1152 const numeric
1153 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1154 {
1155         if (!x1.is_real() || !x2.is_real()) {
1156                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1157         }
1158         if (x1==x2) {
1159                 throw std::runtime_error("fsolve(): vanishing interval");
1160         }
1161         // xx[0] == left interval limit, xx[1] == right interval limit.
1162         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1163         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1164         numeric xx[2] = { x1<x2 ? x1 : x2,
1165                           x1<x2 ? x2 : x1 };
1166         ex f;
1167         if (is_a<relational>(f_in)) {
1168                 f = f_in.lhs()-f_in.rhs();
1169         } else {
1170                 f = f_in;
1171         }
1172         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1173                             f.subs(x==xx[1]).evalf() };
1174         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1175                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1176         }
1177         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1178                           ex_to<numeric>(fx_[1]) };
1179         if (!fx[0].is_real() || !fx[1].is_real()) {
1180                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1181         }
1182         if (fx[0]*fx[1]>=0) {
1183                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1184         }
1185
1186         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1187         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1188         const ex ff = normal(-f/f.diff(x));
1189         int side = 0;  // Start at left interval limit.
1190         numeric xxprev;
1191         numeric fxprev;
1192         do {
1193                 xxprev = xx[side];
1194                 fxprev = fx[side];
1195                 ex dx_ = ff.subs(x == xx[side]).evalf();
1196                 if (!is_a<numeric>(dx_))
1197                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1198                 xx[side] += ex_to<numeric>(dx_);
1199                 // Now check if Newton-Raphson method shot out of the interval
1200                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1201                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1202                 if (!bad_shot) {
1203                         // Compute f(x) only if new x is inside the interval.
1204                         // The function might be difficult to compute numerically
1205                         // or even ill defined outside the interval. Also it's
1206                         // a small optimization.
1207                         ex f_x = f.subs(x == xx[side]).evalf();
1208                         if (!is_a<numeric>(f_x))
1209                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1210                         fx[side] = ex_to<numeric>(f_x);
1211                 }
1212                 if (bad_shot) {
1213                         // Oops, Newton-Raphson method shot out of the interval.
1214                         // Restore, and try again with the other side instead!
1215                         xx[side] = xxprev;
1216                         fx[side] = fxprev;
1217                         side = !side;
1218                         xxprev = xx[side];
1219                         fxprev = fx[side];
1220
1221                         ex dx_ = ff.subs(x == xx[side]).evalf();
1222                         if (!is_a<numeric>(dx_))
1223                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1224                         xx[side] += ex_to<numeric>(dx_);
1225
1226                         ex f_x = f.subs(x==xx[side]).evalf();
1227                         if (!is_a<numeric>(f_x))
1228                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1229                         fx[side] = ex_to<numeric>(f_x);
1230                 }
1231                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1232                         // Oops, the root isn't bracketed any more.
1233                         // Restore, and perform a bisection!
1234                         xx[side] = xxprev;
1235                         fx[side] = fxprev;
1236
1237                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1238                         // they occur pretty often when Newton-Raphson arrives at an x too
1239                         // close to the result on one side of the interval and
1240                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1241                         // precision errors! Recall that this function does not have a
1242                         // precision goal as one of its arguments but instead relies on
1243                         // x converging to a fixed point. We speed up the (safe but slow)
1244                         // bisection method by mixing in a dash of the (unsafer but faster)
1245                         // secant method: Instead of splitting the interval at the
1246                         // arithmetic mean (bisection), we split it nearer to the root as
1247                         // determined by the secant between the values xx[0] and xx[1].
1248                         // Don't set the secant_weight to one because that could disturb
1249                         // the convergence in some corner cases!
1250                         constexpr double secant_weight = 0.984375;  // == 63/64 < 1
1251                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1252                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1253                         ex fxmid_ = f.subs(x == xxmid).evalf();
1254                         if (!is_a<numeric>(fxmid_))
1255                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1256                         numeric fxmid = ex_to<numeric>(fxmid_);
1257                         if (fxmid.is_zero()) {
1258                                 // Luck strikes...
1259                                 return xxmid;
1260                         }
1261                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1262                                 side = !side;
1263                         }
1264                         xxprev = xx[side];
1265                         fxprev = fx[side];
1266                         xx[side] = xxmid;
1267                         fx[side] = fxmid;
1268                 }
1269         } while (xxprev!=xx[side]);
1270         return xxprev;
1271 }
1272
1273
1274 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1275  * for static lib (so ginsh will see them). */
1276 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1277 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1278
1279 } // namespace GiNaC