ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2016 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         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);
 371                 case info_flags::nonnegint:
 372                         return arg.info(info_flags::integer);
 373                 case info_flags::nonnegative:
 374                 case info_flags::real:
 375                         return true;
 376                 case info_flags::negative:
 377                         return false;
 378                 case info_flags::positive:
 379                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
 380                 case info_flags::has_indices: {
 381                         if (arg.info(info_flags::has_indices))
 382                                 return true;
 383                         else
 384                                 return false;
 385                 }
 386         }
 387         return false;
 388 }
 389
 390 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 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
 403 //////////
 404 // Step function
 405 //////////
 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()
 452             && !(options & series_options::suppress_branchcut))
 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
 474 REGISTER_FUNCTION(step, eval_func(step_eval).
 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
 481 //////////
 482 // Complex sign
 483 //////////
 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()
 530             && !(options & series_options::suppress_branchcut))
 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
 564 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 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
 573 //////////
 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).
 577 //////////
 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.
 583         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 584                 return _ex0;
 585
 586         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 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
 607         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 608                 return _ex0;
 609
 610         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 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)) ||
 637             (y_pt.info(info_flags::numeric) && y_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
 669 //////////
 670 // dilogarithm
 671 //////////
 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)
 703                 if (!x.info(info_flags::crational))
 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)
 774                 if (!(options & series_options::suppress_branchcut) &&
 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
 812 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 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
 819 //////////
 820 // trilogarithm
 821 //////////
 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
 833 //////////
 834 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 835 //////////
 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
 864 //////////
 865 // factorial
 866 //////////
 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
 907 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 908                              evalf_func(factorial_evalf).
 909                              print_func<print_dflt>(factorial_print_dflt_latex).
 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
 915 //////////
 916 // binomial
 917 //////////
 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
 971 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 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
 977 //////////
 978 // Order term function (for truncated power series)
 979 //////////
 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
1037 //////////
1038 // Solve linear system
1039 //////////
1040
1041 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1042 {
1043         // solve a system of linear equations
1044         if (eqns.info(info_flags::relation_equal)) {
1045                 if (!symbols.info(info_flags::symbol))
1046                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1047                 const ex sol = lsolve(lst{eqns}, lst{symbols});
1048
1049                 GINAC_ASSERT(sol.nops()==1);
1050                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1051
1052                 return sol.op(0).op(1); // return rhs of first solution
1053         }
1054
1055         // syntax checks
1056         if (!eqns.info(info_flags::list)) {
1057                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1058         }
1059         for (size_t i=0; i<eqns.nops(); i++) {
1060                 if (!eqns.op(i).info(info_flags::relation_equal)) {
1061                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1062                 }
1063         }
1064         if (!symbols.info(info_flags::list)) {
1065                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1066         }
1067         for (size_t i=0; i<symbols.nops(); i++) {
1068                 if (!symbols.op(i).info(info_flags::symbol)) {
1069                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1070                 }
1071         }
1072
1073         // build matrix from equation system
1074         matrix sys(eqns.nops(),symbols.nops());
1075         matrix rhs(eqns.nops(),1);
1076         matrix vars(symbols.nops(),1);
1077
1078         for (size_t r=0; r<eqns.nops(); r++) {
1079                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1080                 ex linpart = eq;
1081                 for (size_t c=0; c<symbols.nops(); c++) {
1082                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1083                         linpart -= co*symbols.op(c);
1084                         sys(r,c) = co;
1085                 }
1086                 linpart = linpart.expand();
1087                 rhs(r,0) = -linpart;
1088         }
1089
1090         // test if system is linear and fill vars matrix
1091         for (size_t i=0; i<symbols.nops(); i++) {
1092                 vars(i,0) = symbols.op(i);
1093                 if (sys.has(symbols.op(i)))
1094                         throw(std::logic_error("lsolve: system is not linear"));
1095                 if (rhs.has(symbols.op(i)))
1096                         throw(std::logic_error("lsolve: system is not linear"));
1097         }
1098
1099         matrix solution;
1100         try {
1101                 solution = sys.solve(vars,rhs,options);
1102         } catch (const std::runtime_error & e) {
1103                 // Probably singular matrix or otherwise overdetermined system:
1104                 // It is consistent to return an empty list
1105                 return lst{};
1106         }
1107         GINAC_ASSERT(solution.cols()==1);
1108         GINAC_ASSERT(solution.rows()==symbols.nops());
1109
1110         // return list of equations of the form lst{var1==sol1,var2==sol2,...}
1111         lst sollist;
1112         for (size_t i=0; i<symbols.nops(); i++)
1113                 sollist.append(symbols.op(i)==solution(i,0));
1114
1115         return sollist;
1116 }
1117
1118 //////////
1119 // Find real root of f(x) numerically
1120 //////////
1121
1122 const numeric
1123 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1124 {
1125         if (!x1.is_real() || !x2.is_real()) {
1126                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1127         }
1128         if (x1==x2) {
1129                 throw std::runtime_error("fsolve(): vanishing interval");
1130         }
1131         // xx[0] == left interval limit, xx[1] == right interval limit.
1132         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1133         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1134         numeric xx[2] = { x1<x2 ? x1 : x2,
1135                           x1<x2 ? x2 : x1 };
1136         ex f;
1137         if (is_a<relational>(f_in)) {
1138                 f = f_in.lhs()-f_in.rhs();
1139         } else {
1140                 f = f_in;
1141         }
1142         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1143                             f.subs(x==xx[1]).evalf() };
1144         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1145                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1146         }
1147         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1148                           ex_to<numeric>(fx_[1]) };
1149         if (!fx[0].is_real() || !fx[1].is_real()) {
1150                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1151         }
1152         if (fx[0]*fx[1]>=0) {
1153                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1154         }
1155
1156         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1157         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1158         const ex ff = normal(-f/f.diff(x));
1159         int side = 0;  // Start at left interval limit.
1160         numeric xxprev;
1161         numeric fxprev;
1162         do {
1163                 xxprev = xx[side];
1164                 fxprev = fx[side];
1165                 ex dx_ = ff.subs(x == xx[side]).evalf();
1166                 if (!is_a<numeric>(dx_))
1167                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1168                 xx[side] += ex_to<numeric>(dx_);
1169                 // Now check if Newton-Raphson method shot out of the interval
1170                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1171                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1172                 if (!bad_shot) {
1173                         // Compute f(x) only if new x is inside the interval.
1174                         // The function might be difficult to compute numerically
1175                         // or even ill defined outside the interval. Also it's
1176                         // a small optimization.
1177                         ex f_x = f.subs(x == xx[side]).evalf();
1178                         if (!is_a<numeric>(f_x))
1179                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1180                         fx[side] = ex_to<numeric>(f_x);
1181                 }
1182                 if (bad_shot) {
1183                         // Oops, Newton-Raphson method shot out of the interval.
1184                         // Restore, and try again with the other side instead!
1185                         xx[side] = xxprev;
1186                         fx[side] = fxprev;
1187                         side = !side;
1188                         xxprev = xx[side];
1189                         fxprev = fx[side];
1190
1191                         ex dx_ = ff.subs(x == xx[side]).evalf();
1192                         if (!is_a<numeric>(dx_))
1193                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1194                         xx[side] += ex_to<numeric>(dx_);
1195
1196                         ex f_x = f.subs(x==xx[side]).evalf();
1197                         if (!is_a<numeric>(f_x))
1198                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1199                         fx[side] = ex_to<numeric>(f_x);
1200                 }
1201                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1202                         // Oops, the root isn't bracketed any more.
1203                         // Restore, and perform a bisection!
1204                         xx[side] = xxprev;
1205                         fx[side] = fxprev;
1206
1207                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1208                         // they occur pretty often when Newton-Raphson arrives at an x too
1209                         // close to the result on one side of the interval and
1210                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1211                         // precision errors! Recall that this function does not have a
1212                         // precision goal as one of its arguments but instead relies on
1213                         // x converging to a fixed point. We speed up the (safe but slow)
1214                         // bisection method by mixing in a dash of the (unsafer but faster)
1215                         // secant method: Instead of splitting the interval at the
1216                         // arithmetic mean (bisection), we split it nearer to the root as
1217                         // determined by the secant between the values xx[0] and xx[1].
1218                         // Don't set the secant_weight to one because that could disturb
1219                         // the convergence in some corner cases!
1220                         constexpr double secant_weight = 0.984375;  // == 63/64 < 1
1221                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1222                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1223                         ex fxmid_ = f.subs(x == xxmid).evalf();
1224                         if (!is_a<numeric>(fxmid_))
1225                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1226                         numeric fxmid = ex_to<numeric>(fxmid_);
1227                         if (fxmid.is_zero()) {
1228                                 // Luck strikes...
1229                                 return xxmid;
1230                         }
1231                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1232                                 side = !side;
1233                         }
1234                         xxprev = xx[side];
1235                         fxprev = fx[side];
1236                         xx[side] = xxmid;
1237                         fx[side] = fxmid;
1238                 }
1239         } while (xxprev!=xx[side]);
1240         return xxprev;
1241 }
1242
1243
1244 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1245  * for static lib (so ginsh will see them). */
1246 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1247 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1248
1249 } // namespace GiNaC