ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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::algebraic:
 113                 case info_flags::positive:
 114                 case info_flags::negative:
 115                 case info_flags::nonnegative:
 116                 case info_flags::posint:
 117                 case info_flags::negint:
 118                 case info_flags::nonnegint:
 119                 case info_flags::has_indices:
 120                         return arg.info(inf);
 121         }
 122         return false;
 123 }
 124
 125 static bool conjugate_info(const ex & arg, unsigned inf)
 126 {
 127         return func_arg_info(arg, inf);
 128 }
 129
 130 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
 131                                       evalf_func(conjugate_evalf).
 132                                       expl_derivative_func(conjugate_expl_derivative).
 133                                       info_func(conjugate_info).
 134                                       print_func<print_latex>(conjugate_print_latex).
 135                                       conjugate_func(conjugate_conjugate).
 136                                       real_part_func(conjugate_real_part).
 137                                       imag_part_func(conjugate_imag_part).
 138                                       set_name("conjugate","conjugate"));
 139
 140 //////////
 141 // real part
 142 //////////
 143
 144 static ex real_part_evalf(const ex & arg)
 145 {
 146         if (is_exactly_a<numeric>(arg)) {
 147                 return ex_to<numeric>(arg).real();
 148         }
 149         return real_part_function(arg).hold();
 150 }
 151
 152 static ex real_part_eval(const ex & arg)
 153 {
 154         return arg.real_part();
 155 }
 156
 157 static void real_part_print_latex(const ex & arg, const print_context & c)
 158 {
 159         c.s << "\\Re"; arg.print(c); c.s << "";
 160 }
 161
 162 static ex real_part_conjugate(const ex & arg)
 163 {
 164         return real_part_function(arg).hold();
 165 }
 166
 167 static ex real_part_real_part(const ex & arg)
 168 {
 169         return real_part_function(arg).hold();
 170 }
 171
 172 static ex real_part_imag_part(const ex & arg)
 173 {
 174         return 0;
 175 }
 176
 177 // If x is real then Re(e).diff(x) is equal to Re(e.diff(x))
 178 static ex real_part_expl_derivative(const ex & arg, const symbol & s)
 179 {
 180         if (s.info(info_flags::real))
 181                 return real_part_function(arg.diff(s));
 182         else {
 183                 exvector vec_arg;
 184                 vec_arg.push_back(arg);
 185                 return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
 186         }
 187 }
 188
 189 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
 190                                       evalf_func(real_part_evalf).
 191                                       expl_derivative_func(real_part_expl_derivative).
 192                                       print_func<print_latex>(real_part_print_latex).
 193                                       conjugate_func(real_part_conjugate).
 194                                       real_part_func(real_part_real_part).
 195                                       imag_part_func(real_part_imag_part).
 196                                       set_name("real_part","real_part"));
 197
 198 //////////
 199 // imag part
 200 //////////
 201
 202 static ex imag_part_evalf(const ex & arg)
 203 {
 204         if (is_exactly_a<numeric>(arg)) {
 205                 return ex_to<numeric>(arg).imag();
 206         }
 207         return imag_part_function(arg).hold();
 208 }
 209
 210 static ex imag_part_eval(const ex & arg)
 211 {
 212         return arg.imag_part();
 213 }
 214
 215 static void imag_part_print_latex(const ex & arg, const print_context & c)
 216 {
 217         c.s << "\\Im"; arg.print(c); c.s << "";
 218 }
 219
 220 static ex imag_part_conjugate(const ex & arg)
 221 {
 222         return imag_part_function(arg).hold();
 223 }
 224
 225 static ex imag_part_real_part(const ex & arg)
 226 {
 227         return imag_part_function(arg).hold();
 228 }
 229
 230 static ex imag_part_imag_part(const ex & arg)
 231 {
 232         return 0;
 233 }
 234
 235 // If x is real then Im(e).diff(x) is equal to Im(e.diff(x))
 236 static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
 237 {
 238         if (s.info(info_flags::real))
 239                 return imag_part_function(arg.diff(s));
 240         else {
 241                 exvector vec_arg;
 242                 vec_arg.push_back(arg);
 243                 return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
 244         }
 245 }
 246
 247 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
 248                                       evalf_func(imag_part_evalf).
 249                                       expl_derivative_func(imag_part_expl_derivative).
 250                                       print_func<print_latex>(imag_part_print_latex).
 251                                       conjugate_func(imag_part_conjugate).
 252                                       real_part_func(imag_part_real_part).
 253                                       imag_part_func(imag_part_imag_part).
 254                                       set_name("imag_part","imag_part"));
 255
 256 //////////
 257 // absolute value
 258 //////////
 259
 260 static ex abs_evalf(const ex & arg)
 261 {
 262         if (is_exactly_a<numeric>(arg))
 263                 return abs(ex_to<numeric>(arg));
 264
 265         return abs(arg).hold();
 266 }
 267
 268 static ex abs_eval(const ex & arg)
 269 {
 270         if (is_exactly_a<numeric>(arg))
 271                 return abs(ex_to<numeric>(arg));
 272
 273         if (arg.info(info_flags::nonnegative))
 274                 return arg;
 275
 276         if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
 277                 return -arg;
 278
 279         if (is_ex_the_function(arg, abs))
 280                 return arg;
 281
 282         if (is_ex_the_function(arg, exp))
 283                 return exp(arg.op(0).real_part());
 284
 285         if (is_exactly_a<power>(arg)) {
 286                 const ex& base = arg.op(0);
 287                 const ex& exponent = arg.op(1);
 288                 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
 289                         return pow(abs(base), exponent.real_part());
 290         }
 291
 292         if (is_ex_the_function(arg, conjugate_function))
 293                 return abs(arg.op(0));
 294
 295         if (is_ex_the_function(arg, step))
 296                 return arg;
 297
 298         return abs(arg).hold();
 299 }
 300
 301 static ex abs_expand(const ex & arg, unsigned options)
 302 {
 303         if ((options & expand_options::expand_transcendental)
 304                 && is_exactly_a<mul>(arg)) {
 305                 exvector prodseq;
 306                 prodseq.reserve(arg.nops());
 307                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 308                         if (options & expand_options::expand_function_args)
 309                                 prodseq.push_back(abs(i->expand(options)));
 310                         else
 311                                 prodseq.push_back(abs(*i));
 312                 }
 313                 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
 314         }
 315
 316         if (options & expand_options::expand_function_args)
 317                 return abs(arg.expand(options)).hold();
 318         else
 319                 return abs(arg).hold();
 320 }
 321
 322 static ex abs_expl_derivative(const ex & arg, const symbol & s)
 323 {
 324         ex diff_arg = arg.diff(s);
 325         return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
 326 }
 327
 328 static void abs_print_latex(const ex & arg, const print_context & c)
 329 {
 330         c.s << "{|"; arg.print(c); c.s << "|}";
 331 }
 332
 333 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 334 {
 335         c.s << "fabs("; arg.print(c); c.s << ")";
 336 }
 337
 338 static ex abs_conjugate(const ex & arg)
 339 {
 340         return abs(arg).hold();
 341 }
 342
 343 static ex abs_real_part(const ex & arg)
 344 {
 345         return abs(arg).hold();
 346 }
 347
 348 static ex abs_imag_part(const ex& arg)
 349 {
 350         return 0;
 351 }
 352
 353 static ex abs_power(const ex & arg, const ex & exp)
 354 {
 355         if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
 356                 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
 357                         return power(arg, exp);
 358                 else
 359                         return power(arg, exp/2)*power(arg.conjugate(), exp/2);
 360         } else
 361                 return power(abs(arg), exp).hold();
 362 }
 363
 364 bool abs_info(const ex & arg, unsigned inf)
 365 {
 366         switch (inf) {
 367                 case info_flags::integer:
 368                 case info_flags::even:
 369                 case info_flags::odd:
 370                 case info_flags::prime:
 371                         return arg.info(inf);
 372                 case info_flags::nonnegint:
 373                         return arg.info(info_flags::integer);
 374                 case info_flags::nonnegative:
 375                 case info_flags::real:
 376                         return true;
 377                 case info_flags::negative:
 378                         return false;
 379                 case info_flags::positive:
 380                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
 381                 case info_flags::has_indices: {
 382                         if (arg.info(info_flags::has_indices))
 383                                 return true;
 384                         else
 385                                 return false;
 386                 }
 387         }
 388         return false;
 389 }
 390
 391 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 392                        evalf_func(abs_evalf).
 393                        expand_func(abs_expand).
 394                        expl_derivative_func(abs_expl_derivative).
 395                        info_func(abs_info).
 396                        print_func<print_latex>(abs_print_latex).
 397                        print_func<print_csrc_float>(abs_print_csrc_float).
 398                        print_func<print_csrc_double>(abs_print_csrc_float).
 399                        conjugate_func(abs_conjugate).
 400                        real_part_func(abs_real_part).
 401                        imag_part_func(abs_imag_part).
 402                        power_func(abs_power));
 403
 404 //////////
 405 // Step function
 406 //////////
 407
 408 static ex step_evalf(const ex & arg)
 409 {
 410         if (is_exactly_a<numeric>(arg))
 411                 return step(ex_to<numeric>(arg));
 412
 413         return step(arg).hold();
 414 }
 415
 416 static ex step_eval(const ex & arg)
 417 {
 418         if (is_exactly_a<numeric>(arg))
 419                 return step(ex_to<numeric>(arg));
 420
 421         else if (is_exactly_a<mul>(arg) &&
 422                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 423                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 424                 if (oc.is_real()) {
 425                         if (oc > 0)
 426                                 // step(42*x) -> step(x)
 427                                 return step(arg/oc).hold();
 428                         else
 429                                 // step(-42*x) -> step(-x)
 430                                 return step(-arg/oc).hold();
 431                 }
 432                 if (oc.real().is_zero()) {
 433                         if (oc.imag() > 0)
 434                                 // step(42*I*x) -> step(I*x)
 435                                 return step(I*arg/oc).hold();
 436                         else
 437                                 // step(-42*I*x) -> step(-I*x)
 438                                 return step(-I*arg/oc).hold();
 439                 }
 440         }
 441
 442         return step(arg).hold();
 443 }
 444
 445 static ex step_series(const ex & arg,
 446                       const relational & rel,
 447                       int order,
 448                       unsigned options)
 449 {
 450         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 451         if (arg_pt.info(info_flags::numeric)
 452             && ex_to<numeric>(arg_pt).real().is_zero()
 453             && !(options & series_options::suppress_branchcut))
 454                 throw (std::domain_error("step_series(): on imaginary axis"));
 455
 456         epvector seq { expair(step(arg_pt), _ex0) };
 457         return pseries(rel, std::move(seq));
 458 }
 459
 460 static ex step_conjugate(const ex& arg)
 461 {
 462         return step(arg).hold();
 463 }
 464
 465 static ex step_real_part(const ex& arg)
 466 {
 467         return step(arg).hold();
 468 }
 469
 470 static ex step_imag_part(const ex& arg)
 471 {
 472         return 0;
 473 }
 474
 475 REGISTER_FUNCTION(step, eval_func(step_eval).
 476                         evalf_func(step_evalf).
 477                         series_func(step_series).
 478                         conjugate_func(step_conjugate).
 479                         real_part_func(step_real_part).
 480                         imag_part_func(step_imag_part));
 481
 482 //////////
 483 // Complex sign
 484 //////////
 485
 486 static ex csgn_evalf(const ex & arg)
 487 {
 488         if (is_exactly_a<numeric>(arg))
 489                 return csgn(ex_to<numeric>(arg));
 490
 491         return csgn(arg).hold();
 492 }
 493
 494 static ex csgn_eval(const ex & arg)
 495 {
 496         if (is_exactly_a<numeric>(arg))
 497                 return csgn(ex_to<numeric>(arg));
 498
 499         else if (is_exactly_a<mul>(arg) &&
 500                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 501                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 502                 if (oc.is_real()) {
 503                         if (oc > 0)
 504                                 // csgn(42*x) -> csgn(x)
 505                                 return csgn(arg/oc).hold();
 506                         else
 507                                 // csgn(-42*x) -> -csgn(x)
 508                                 return -csgn(arg/oc).hold();
 509                 }
 510                 if (oc.real().is_zero()) {
 511                         if (oc.imag() > 0)
 512                                 // csgn(42*I*x) -> csgn(I*x)
 513                                 return csgn(I*arg/oc).hold();
 514                         else
 515                                 // csgn(-42*I*x) -> -csgn(I*x)
 516                                 return -csgn(I*arg/oc).hold();
 517                 }
 518         }
 519
 520         return csgn(arg).hold();
 521 }
 522
 523 static ex csgn_series(const ex & arg,
 524                       const relational & rel,
 525                       int order,
 526                       unsigned options)
 527 {
 528         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 529         if (arg_pt.info(info_flags::numeric)
 530             && ex_to<numeric>(arg_pt).real().is_zero()
 531             && !(options & series_options::suppress_branchcut))
 532                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 533
 534         epvector seq { expair(csgn(arg_pt), _ex0) };
 535         return pseries(rel, std::move(seq));
 536 }
 537
 538 static ex csgn_conjugate(const ex& arg)
 539 {
 540         return csgn(arg).hold();
 541 }
 542
 543 static ex csgn_real_part(const ex& arg)
 544 {
 545         return csgn(arg).hold();
 546 }
 547
 548 static ex csgn_imag_part(const ex& arg)
 549 {
 550         return 0;
 551 }
 552
 553 static ex csgn_power(const ex & arg, const ex & exp)
 554 {
 555         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 556                 if (ex_to<numeric>(exp).is_odd())
 557                         return csgn(arg).hold();
 558                 else
 559                         return power(csgn(arg), _ex2).hold();
 560         } else
 561                 return power(csgn(arg), exp).hold();
 562 }
 563
 564
 565 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 566                         evalf_func(csgn_evalf).
 567                         series_func(csgn_series).
 568                         conjugate_func(csgn_conjugate).
 569                         real_part_func(csgn_real_part).
 570                         imag_part_func(csgn_imag_part).
 571                         power_func(csgn_power));
 572
 573
 574 //////////
 575 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 576 // This function is closely related to the unwinding number K, sometimes found
 577 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 578 //////////
 579
 580 static ex eta_evalf(const ex &x, const ex &y)
 581 {
 582         // It seems like we basically have to replicate the eval function here,
 583         // since the expression might not be fully evaluated yet.
 584         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 585                 return _ex0;
 586
 587         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 588                 const numeric nx = ex_to<numeric>(x);
 589                 const numeric ny = ex_to<numeric>(y);
 590                 const numeric nxy = ex_to<numeric>(x*y);
 591                 int cut = 0;
 592                 if (nx.is_real() && nx.is_negative())
 593                         cut -= 4;
 594                 if (ny.is_real() && ny.is_negative())
 595                         cut -= 4;
 596                 if (nxy.is_real() && nxy.is_negative())
 597                         cut += 4;
 598                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 599                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 600         }
 601
 602         return eta(x,y).hold();
 603 }
 604
 605 static ex eta_eval(const ex &x, const ex &y)
 606 {
 607         // trivial:  eta(x,c) -> 0  if c is real and positive
 608         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 609                 return _ex0;
 610
 611         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 612                 // don't call eta_evalf here because it would call Pi.evalf()!
 613                 const numeric nx = ex_to<numeric>(x);
 614                 const numeric ny = ex_to<numeric>(y);
 615                 const numeric nxy = ex_to<numeric>(x*y);
 616                 int cut = 0;
 617                 if (nx.is_real() && nx.is_negative())
 618                         cut -= 4;
 619                 if (ny.is_real() && ny.is_negative())
 620                         cut -= 4;
 621                 if (nxy.is_real() && nxy.is_negative())
 622                         cut += 4;
 623                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 624                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 625         }
 626
 627         return eta(x,y).hold();
 628 }
 629
 630 static ex eta_series(const ex & x, const ex & y,
 631                      const relational & rel,
 632                      int order,
 633                      unsigned options)
 634 {
 635         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 636         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 637         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 638             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 639             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 640                         throw (std::domain_error("eta_series(): on discontinuity"));
 641         epvector seq { expair(eta(x_pt,y_pt), _ex0) };
 642         return pseries(rel, std::move(seq));
 643 }
 644
 645 static ex eta_conjugate(const ex & x, const ex & y)
 646 {
 647         return -eta(x, y).hold();
 648 }
 649
 650 static ex eta_real_part(const ex & x, const ex & y)
 651 {
 652         return 0;
 653 }
 654
 655 static ex eta_imag_part(const ex & x, const ex & y)
 656 {
 657         return -I*eta(x, y).hold();
 658 }
 659
 660 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 661                        evalf_func(eta_evalf).
 662                        series_func(eta_series).
 663                        latex_name("\\eta").
 664                        set_symmetry(sy_symm(0, 1)).
 665                        conjugate_func(eta_conjugate).
 666                        real_part_func(eta_real_part).
 667                        imag_part_func(eta_imag_part));
 668
 669
 670 //////////
 671 // dilogarithm
 672 //////////
 673
 674 static ex Li2_evalf(const ex & x)
 675 {
 676         if (is_exactly_a<numeric>(x))
 677                 return Li2(ex_to<numeric>(x));
 678
 679         return Li2(x).hold();
 680 }
 681
 682 static ex Li2_eval(const ex & x)
 683 {
 684         if (x.info(info_flags::numeric)) {
 685                 // Li2(0) -> 0
 686                 if (x.is_zero())
 687                         return _ex0;
 688                 // Li2(1) -> Pi^2/6
 689                 if (x.is_equal(_ex1))
 690                         return power(Pi,_ex2)/_ex6;
 691                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 692                 if (x.is_equal(_ex1_2))
 693                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 694                 // Li2(-1) -> -Pi^2/12
 695                 if (x.is_equal(_ex_1))
 696                         return -power(Pi,_ex2)/_ex12;
 697                 // Li2(I) -> -Pi^2/48+Catalan*I
 698                 if (x.is_equal(I))
 699                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 700                 // Li2(-I) -> -Pi^2/48-Catalan*I
 701                 if (x.is_equal(-I))
 702                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 703                 // Li2(float)
 704                 if (!x.info(info_flags::crational))
 705                         return Li2(ex_to<numeric>(x));
 706         }
 707
 708         return Li2(x).hold();
 709 }
 710
 711 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 712 {
 713         GINAC_ASSERT(deriv_param==0);
 714
 715         // d/dx Li2(x) -> -log(1-x)/x
 716         return -log(_ex1-x)/x;
 717 }
 718
 719 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 720 {
 721         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 722         if (x_pt.info(info_flags::numeric)) {
 723                 // First special case: x==0 (derivatives have poles)
 724                 if (x_pt.is_zero()) {
 725                         // method:
 726                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 727                         // simply substitute x==0.  The limit, however, exists: it is 1.
 728                         // We also know all higher derivatives' limits:
 729                         // (d/dx)^n Li2(x) == n!/n^2.
 730                         // So the primitive series expansion is
 731                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 732                         // and so on.
 733                         // We first construct such a primitive series expansion manually in
 734                         // a dummy symbol s and then insert the argument's series expansion
 735                         // for s.  Reexpanding the resulting series returns the desired
 736                         // result.
 737                         const symbol s;
 738                         ex ser;
 739                         // manually construct the primitive expansion
 740                         for (int i=1; i<order; ++i)
 741                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 742                         // substitute the argument's series expansion
 743                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 744                         // maybe that was terminating, so add a proper order term
 745                         epvector nseq { expair(Order(_ex1), order) };
 746                         ser += pseries(rel, std::move(nseq));
 747                         // reexpanding it will collapse the series again
 748                         return ser.series(rel, order);
 749                         // NB: Of course, this still does not allow us to compute anything
 750                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 751                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 752                         // substitution x==0.  Probably limits *are* needed for the general
 753                         // cases.  In case L'Hospital's rule is implemented for limits and
 754                         // basic::series() takes care of this, this whole block is probably
 755                         // obsolete!
 756                 }
 757                 // second special case: x==1 (branch point)
 758                 if (x_pt.is_equal(_ex1)) {
 759                         // method:
 760                         // construct series manually in a dummy symbol s
 761                         const symbol s;
 762                         ex ser = zeta(_ex2);
 763                         // manually construct the primitive expansion
 764                         for (int i=1; i<order; ++i)
 765                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 766                         // substitute the argument's series expansion
 767                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 768                         // maybe that was terminating, so add a proper order term
 769                         epvector nseq { expair(Order(_ex1), order) };
 770                         ser += pseries(rel, std::move(nseq));
 771                         // reexpanding it will collapse the series again
 772                         return ser.series(rel, order);
 773                 }
 774                 // third special case: x real, >=1 (branch cut)
 775                 if (!(options & series_options::suppress_branchcut) &&
 776                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 777                         // method:
 778                         // This is the branch cut: assemble the primitive series manually
 779                         // and then add the corresponding complex step function.
 780                         const symbol &s = ex_to<symbol>(rel.lhs());
 781                         const ex point = rel.rhs();
 782                         const symbol foo;
 783                         epvector seq;
 784                         // zeroth order term:
 785                         seq.push_back(expair(Li2(x_pt), _ex0));
 786                         // compute the intermediate terms:
 787                         ex replarg = series(Li2(x), s==foo, order);
 788                         for (size_t i=1; i<replarg.nops()-1; ++i)
 789                                 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));
 790                         // append an order term:
 791                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 792                         return pseries(rel, std::move(seq));
 793                 }
 794         }
 795         // all other cases should be safe, by now:
 796         throw do_taylor();  // caught by function::series()
 797 }
 798
 799 static ex Li2_conjugate(const ex & x)
 800 {
 801         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 802         // run along the positive real axis beginning at 1.
 803         if (x.info(info_flags::negative)) {
 804                 return Li2(x).hold();
 805         }
 806         if (is_exactly_a<numeric>(x) &&
 807             (!x.imag_part().is_zero() || x < *_num1_p)) {
 808                 return Li2(x.conjugate());
 809         }
 810         return conjugate_function(Li2(x)).hold();
 811 }
 812
 813 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 814                        evalf_func(Li2_evalf).
 815                        derivative_func(Li2_deriv).
 816                        series_func(Li2_series).
 817                        conjugate_func(Li2_conjugate).
 818                        latex_name("\\mathrm{Li}_2"));
 819
 820 //////////
 821 // trilogarithm
 822 //////////
 823
 824 static ex Li3_eval(const ex & x)
 825 {
 826         if (x.is_zero())
 827                 return x;
 828         return Li3(x).hold();
 829 }
 830
 831 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 832                        latex_name("\\mathrm{Li}_3"));
 833
 834 //////////
 835 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 836 //////////
 837
 838 static ex zetaderiv_eval(const ex & n, const ex & x)
 839 {
 840         if (n.info(info_flags::numeric)) {
 841                 // zetaderiv(0,x) -> zeta(x)
 842                 if (n.is_zero())
 843                         return zeta(x).hold();
 844         }
 845
 846         return zetaderiv(n, x).hold();
 847 }
 848
 849 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 850 {
 851         GINAC_ASSERT(deriv_param<2);
 852
 853         if (deriv_param==0) {
 854                 // d/dn zeta(n,x)
 855                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 856         }
 857         // d/dx psi(n,x)
 858         return zetaderiv(n+1,x);
 859 }
 860
 861 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 862                                  derivative_func(zetaderiv_deriv).
 863                                  latex_name("\\zeta^\\prime"));
 864
 865 //////////
 866 // factorial
 867 //////////
 868
 869 static ex factorial_evalf(const ex & x)
 870 {
 871         return factorial(x).hold();
 872 }
 873
 874 static ex factorial_eval(const ex & x)
 875 {
 876         if (is_exactly_a<numeric>(x))
 877                 return factorial(ex_to<numeric>(x));
 878         else
 879                 return factorial(x).hold();
 880 }
 881
 882 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 883 {
 884         if (is_exactly_a<symbol>(x) ||
 885             is_exactly_a<constant>(x) ||
 886                 is_exactly_a<function>(x)) {
 887                 x.print(c); c.s << "!";
 888         } else {
 889                 c.s << "("; x.print(c); c.s << ")!";
 890         }
 891 }
 892
 893 static ex factorial_conjugate(const ex & x)
 894 {
 895         return factorial(x).hold();
 896 }
 897
 898 static ex factorial_real_part(const ex & x)
 899 {
 900         return factorial(x).hold();
 901 }
 902
 903 static ex factorial_imag_part(const ex & x)
 904 {
 905         return 0;
 906 }
 907
 908 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 909                              evalf_func(factorial_evalf).
 910                              print_func<print_dflt>(factorial_print_dflt_latex).
 911                              print_func<print_latex>(factorial_print_dflt_latex).
 912                              conjugate_func(factorial_conjugate).
 913                              real_part_func(factorial_real_part).
 914                              imag_part_func(factorial_imag_part));
 915
 916 //////////
 917 // binomial
 918 //////////
 919
 920 static ex binomial_evalf(const ex & x, const ex & y)
 921 {
 922         return binomial(x, y).hold();
 923 }
 924
 925 static ex binomial_sym(const ex & x, const numeric & y)
 926 {
 927         if (y.is_integer()) {
 928                 if (y.is_nonneg_integer()) {
 929                         const unsigned N = y.to_int();
 930                         if (N == 0) return _ex1;
 931                         if (N == 1) return x;
 932                         ex t = x.expand();
 933                         for (unsigned i = 2; i <= N; ++i)
 934                                 t = (t * (x + i - y - 1)).expand() / i;
 935                         return t;
 936                 } else
 937                         return _ex0;
 938         }
 939
 940         return binomial(x, y).hold();
 941 }
 942
 943 static ex binomial_eval(const ex & x, const ex &y)
 944 {
 945         if (is_exactly_a<numeric>(y)) {
 946                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 947                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 948                 else
 949                         return binomial_sym(x, ex_to<numeric>(y));
 950         } else
 951                 return binomial(x, y).hold();
 952 }
 953
 954 // At the moment the numeric evaluation of a binomial function always
 955 // gives a real number, but if this would be implemented using the gamma
 956 // function, also complex conjugation should be changed (or rather, deleted).
 957 static ex binomial_conjugate(const ex & x, const ex & y)
 958 {
 959         return binomial(x,y).hold();
 960 }
 961
 962 static ex binomial_real_part(const ex & x, const ex & y)
 963 {
 964         return binomial(x,y).hold();
 965 }
 966
 967 static ex binomial_imag_part(const ex & x, const ex & y)
 968 {
 969         return 0;
 970 }
 971
 972 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 973                             evalf_func(binomial_evalf).
 974                             conjugate_func(binomial_conjugate).
 975                             real_part_func(binomial_real_part).
 976                             imag_part_func(binomial_imag_part));
 977
 978 //////////
 979 // Order term function (for truncated power series)
 980 //////////
 981
 982 static ex Order_eval(const ex & x)
 983 {
 984         if (is_exactly_a<numeric>(x)) {
 985                 // O(c) -> O(1) or 0
 986                 if (!x.is_zero())
 987                         return Order(_ex1).hold();
 988                 else
 989                         return _ex0;
 990         } else if (is_exactly_a<mul>(x)) {
 991                 const mul &m = ex_to<mul>(x);
 992                 // O(c*expr) -> O(expr)
 993                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 994                         return Order(x / m.op(m.nops() - 1)).hold();
 995         }
 996         return Order(x).hold();
 997 }
 998
 999 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
1000 {
1001         // Just wrap the function into a pseries object
1002         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1003         const symbol &s = ex_to<symbol>(r.lhs());
1004         epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
1005         return pseries(r, std::move(new_seq));
1006 }
1007
1008 static ex Order_conjugate(const ex & x)
1009 {
1010         return Order(x).hold();
1011 }
1012
1013 static ex Order_real_part(const ex & x)
1014 {
1015         return Order(x).hold();
1016 }
1017
1018 static ex Order_imag_part(const ex & x)
1019 {
1020         if(x.info(info_flags::real))
1021                 return 0;
1022         return Order(x).hold();
1023 }
1024
1025 static ex Order_expl_derivative(const ex & arg, const symbol & s)
1026 {
1027         return Order(arg.diff(s));
1028 }
1029
1030 REGISTER_FUNCTION(Order, eval_func(Order_eval).
1031                          series_func(Order_series).
1032                          latex_name("\\mathcal{O}").
1033                          expl_derivative_func(Order_expl_derivative).
1034                          conjugate_func(Order_conjugate).
1035                          real_part_func(Order_real_part).
1036                          imag_part_func(Order_imag_part));
1037
1038 //////////
1039 // Solve linear system
1040 //////////
1041
1042 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1043 {
1044         // solve a system of linear equations
1045         if (eqns.info(info_flags::relation_equal)) {
1046                 if (!symbols.info(info_flags::symbol))
1047                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1048                 const ex sol = lsolve(lst{eqns}, lst{symbols});
1049
1050                 GINAC_ASSERT(sol.nops()==1);
1051                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1052
1053                 return sol.op(0).op(1); // return rhs of first solution
1054         }
1055
1056         // syntax checks
1057         if (!eqns.info(info_flags::list)) {
1058                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1059         }
1060         for (size_t i=0; i<eqns.nops(); i++) {
1061                 if (!eqns.op(i).info(info_flags::relation_equal)) {
1062                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1063                 }
1064         }
1065         if (!symbols.info(info_flags::list)) {
1066                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1067         }
1068         for (size_t i=0; i<symbols.nops(); i++) {
1069                 if (!symbols.op(i).info(info_flags::symbol)) {
1070                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1071                 }
1072         }
1073
1074         // build matrix from equation system
1075         matrix sys(eqns.nops(),symbols.nops());
1076         matrix rhs(eqns.nops(),1);
1077         matrix vars(symbols.nops(),1);
1078
1079         for (size_t r=0; r<eqns.nops(); r++) {
1080                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1081                 ex linpart = eq;
1082                 for (size_t c=0; c<symbols.nops(); c++) {
1083                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1084                         linpart -= co*symbols.op(c);
1085                         sys(r,c) = co;
1086                 }
1087                 linpart = linpart.expand();
1088                 rhs(r,0) = -linpart;
1089         }
1090
1091         // test if system is linear and fill vars matrix
1092         for (size_t i=0; i<symbols.nops(); i++) {
1093                 vars(i,0) = symbols.op(i);
1094                 if (sys.has(symbols.op(i)))
1095                         throw(std::logic_error("lsolve: system is not linear"));
1096                 if (rhs.has(symbols.op(i)))
1097                         throw(std::logic_error("lsolve: system is not linear"));
1098         }
1099
1100         matrix solution;
1101         try {
1102                 solution = sys.solve(vars,rhs,options);
1103         } catch (const std::runtime_error & e) {
1104                 // Probably singular matrix or otherwise overdetermined system:
1105                 // It is consistent to return an empty list
1106                 return lst{};
1107         }
1108         GINAC_ASSERT(solution.cols()==1);
1109         GINAC_ASSERT(solution.rows()==symbols.nops());
1110
1111         // return list of equations of the form lst{var1==sol1,var2==sol2,...}
1112         lst sollist;
1113         for (size_t i=0; i<symbols.nops(); i++)
1114                 sollist.append(symbols.op(i)==solution(i,0));
1115
1116         return sollist;
1117 }
1118
1119 //////////
1120 // Find real root of f(x) numerically
1121 //////////
1122
1123 const numeric
1124 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1125 {
1126         if (!x1.is_real() || !x2.is_real()) {
1127                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1128         }
1129         if (x1==x2) {
1130                 throw std::runtime_error("fsolve(): vanishing interval");
1131         }
1132         // xx[0] == left interval limit, xx[1] == right interval limit.
1133         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1134         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1135         numeric xx[2] = { x1<x2 ? x1 : x2,
1136                           x1<x2 ? x2 : x1 };
1137         ex f;
1138         if (is_a<relational>(f_in)) {
1139                 f = f_in.lhs()-f_in.rhs();
1140         } else {
1141                 f = f_in;
1142         }
1143         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1144                             f.subs(x==xx[1]).evalf() };
1145         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1146                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1147         }
1148         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1149                           ex_to<numeric>(fx_[1]) };
1150         if (!fx[0].is_real() || !fx[1].is_real()) {
1151                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1152         }
1153         if (fx[0]*fx[1]>=0) {
1154                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1155         }
1156
1157         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1158         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1159         const ex ff = normal(-f/f.diff(x));
1160         int side = 0;  // Start at left interval limit.
1161         numeric xxprev;
1162         numeric fxprev;
1163         do {
1164                 xxprev = xx[side];
1165                 fxprev = fx[side];
1166                 ex dx_ = ff.subs(x == xx[side]).evalf();
1167                 if (!is_a<numeric>(dx_))
1168                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1169                 xx[side] += ex_to<numeric>(dx_);
1170                 // Now check if Newton-Raphson method shot out of the interval
1171                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1172                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1173                 if (!bad_shot) {
1174                         // Compute f(x) only if new x is inside the interval.
1175                         // The function might be difficult to compute numerically
1176                         // or even ill defined outside the interval. Also it's
1177                         // a small optimization.
1178                         ex f_x = f.subs(x == xx[side]).evalf();
1179                         if (!is_a<numeric>(f_x))
1180                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1181                         fx[side] = ex_to<numeric>(f_x);
1182                 }
1183                 if (bad_shot) {
1184                         // Oops, Newton-Raphson method shot out of the interval.
1185                         // Restore, and try again with the other side instead!
1186                         xx[side] = xxprev;
1187                         fx[side] = fxprev;
1188                         side = !side;
1189                         xxprev = xx[side];
1190                         fxprev = fx[side];
1191
1192                         ex dx_ = ff.subs(x == xx[side]).evalf();
1193                         if (!is_a<numeric>(dx_))
1194                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1195                         xx[side] += ex_to<numeric>(dx_);
1196
1197                         ex f_x = f.subs(x==xx[side]).evalf();
1198                         if (!is_a<numeric>(f_x))
1199                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1200                         fx[side] = ex_to<numeric>(f_x);
1201                 }
1202                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1203                         // Oops, the root isn't bracketed any more.
1204                         // Restore, and perform a bisection!
1205                         xx[side] = xxprev;
1206                         fx[side] = fxprev;
1207
1208                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1209                         // they occur pretty often when Newton-Raphson arrives at an x too
1210                         // close to the result on one side of the interval and
1211                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1212                         // precision errors! Recall that this function does not have a
1213                         // precision goal as one of its arguments but instead relies on
1214                         // x converging to a fixed point. We speed up the (safe but slow)
1215                         // bisection method by mixing in a dash of the (unsafer but faster)
1216                         // secant method: Instead of splitting the interval at the
1217                         // arithmetic mean (bisection), we split it nearer to the root as
1218                         // determined by the secant between the values xx[0] and xx[1].
1219                         // Don't set the secant_weight to one because that could disturb
1220                         // the convergence in some corner cases!
1221                         constexpr double secant_weight = 0.984375;  // == 63/64 < 1
1222                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1223                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1224                         ex fxmid_ = f.subs(x == xxmid).evalf();
1225                         if (!is_a<numeric>(fxmid_))
1226                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1227                         numeric fxmid = ex_to<numeric>(fxmid_);
1228                         if (fxmid.is_zero()) {
1229                                 // Luck strikes...
1230                                 return xxmid;
1231                         }
1232                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1233                                 side = !side;
1234                         }
1235                         xxprev = xx[side];
1236                         fxprev = fx[side];
1237                         xx[side] = xxmid;
1238                         fx[side] = fxmid;
1239                 }
1240         } while (xxprev!=xx[side]);
1241         return xxprev;
1242 }
1243
1244
1245 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1246  * for static lib (so ginsh will see them). */
1247 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1248 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1249
1250 } // namespace GiNaC