ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2014 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 "matrix.h"
  28 #include "mul.h"
  29 #include "power.h"
  30 #include "operators.h"
  31 #include "relational.h"
  32 #include "pseries.h"
  33 #include "symbol.h"
  34 #include "symmetry.h"
  35 #include "utils.h"
  36
  37 #include <stdexcept>
  38 #include <vector>
  39
  40 namespace GiNaC {
  41
  42 //////////
  43 // complex conjugate
  44 //////////
  45
  46 static ex conjugate_evalf(const ex & arg)
  47 {
  48         if (is_exactly_a<numeric>(arg)) {
  49                 return ex_to<numeric>(arg).conjugate();
  50         }
  51         return conjugate_function(arg).hold();
  52 }
  53
  54 static ex conjugate_eval(const ex & arg)
  55 {
  56         return arg.conjugate();
  57 }
  58
  59 static void conjugate_print_latex(const ex & arg, const print_context & c)
  60 {
  61         c.s << "\\bar{"; arg.print(c); c.s << "}";
  62 }
  63
  64 static ex conjugate_conjugate(const ex & arg)
  65 {
  66         return arg;
  67 }
  68
  69 static ex conjugate_real_part(const ex & arg)
  70 {
  71         return arg.real_part();
  72 }
  73
  74 static ex conjugate_imag_part(const ex & arg)
  75 {
  76         return -arg.imag_part();
  77 }
  78
  79 static bool func_arg_info(const ex & arg, unsigned inf)
  80 {
  81         // for some functions we can return the info() of its argument
  82         // (think of conjugate())
  83         switch (inf) {
  84                 case info_flags::polynomial:
  85                 case info_flags::integer_polynomial:
  86                 case info_flags::cinteger_polynomial:
  87                 case info_flags::rational_polynomial:
  88                 case info_flags::real:
  89                 case info_flags::rational:
  90                 case info_flags::integer:
  91                 case info_flags::crational:
  92                 case info_flags::cinteger:
  93                 case info_flags::even:
  94                 case info_flags::odd:
  95                 case info_flags::prime:
  96                 case info_flags::crational_polynomial:
  97                 case info_flags::rational_function:
  98                 case info_flags::algebraic:
  99                 case info_flags::positive:
 100                 case info_flags::negative:
 101                 case info_flags::nonnegative:
 102                 case info_flags::posint:
 103                 case info_flags::negint:
 104                 case info_flags::nonnegint:
 105                 case info_flags::has_indices:
 106                         return arg.info(inf);
 107         }
 108         return false;
 109 }
 110
 111 static bool conjugate_info(const ex & arg, unsigned inf)
 112 {
 113         return func_arg_info(arg, inf);
 114 }
 115
 116 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
 117                                       evalf_func(conjugate_evalf).
 118                                       info_func(conjugate_info).
 119                                       print_func<print_latex>(conjugate_print_latex).
 120                                       conjugate_func(conjugate_conjugate).
 121                                       real_part_func(conjugate_real_part).
 122                                       imag_part_func(conjugate_imag_part).
 123                                       set_name("conjugate","conjugate"));
 124
 125 //////////
 126 // real part
 127 //////////
 128
 129 static ex real_part_evalf(const ex & arg)
 130 {
 131         if (is_exactly_a<numeric>(arg)) {
 132                 return ex_to<numeric>(arg).real();
 133         }
 134         return real_part_function(arg).hold();
 135 }
 136
 137 static ex real_part_eval(const ex & arg)
 138 {
 139         return arg.real_part();
 140 }
 141
 142 static void real_part_print_latex(const ex & arg, const print_context & c)
 143 {
 144         c.s << "\\Re"; arg.print(c); c.s << "";
 145 }
 146
 147 static ex real_part_conjugate(const ex & arg)
 148 {
 149         return real_part_function(arg).hold();
 150 }
 151
 152 static ex real_part_real_part(const ex & arg)
 153 {
 154         return real_part_function(arg).hold();
 155 }
 156
 157 static ex real_part_imag_part(const ex & arg)
 158 {
 159         return 0;
 160 }
 161
 162 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
 163                                       evalf_func(real_part_evalf).
 164                                       print_func<print_latex>(real_part_print_latex).
 165                                       conjugate_func(real_part_conjugate).
 166                                       real_part_func(real_part_real_part).
 167                                       imag_part_func(real_part_imag_part).
 168                                       set_name("real_part","real_part"));
 169
 170 //////////
 171 // imag part
 172 //////////
 173
 174 static ex imag_part_evalf(const ex & arg)
 175 {
 176         if (is_exactly_a<numeric>(arg)) {
 177                 return ex_to<numeric>(arg).imag();
 178         }
 179         return imag_part_function(arg).hold();
 180 }
 181
 182 static ex imag_part_eval(const ex & arg)
 183 {
 184         return arg.imag_part();
 185 }
 186
 187 static void imag_part_print_latex(const ex & arg, const print_context & c)
 188 {
 189         c.s << "\\Im"; arg.print(c); c.s << "";
 190 }
 191
 192 static ex imag_part_conjugate(const ex & arg)
 193 {
 194         return imag_part_function(arg).hold();
 195 }
 196
 197 static ex imag_part_real_part(const ex & arg)
 198 {
 199         return imag_part_function(arg).hold();
 200 }
 201
 202 static ex imag_part_imag_part(const ex & arg)
 203 {
 204         return 0;
 205 }
 206
 207 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
 208                                       evalf_func(imag_part_evalf).
 209                                       print_func<print_latex>(imag_part_print_latex).
 210                                       conjugate_func(imag_part_conjugate).
 211                                       real_part_func(imag_part_real_part).
 212                                       imag_part_func(imag_part_imag_part).
 213                                       set_name("imag_part","imag_part"));
 214
 215 //////////
 216 // absolute value
 217 //////////
 218
 219 static ex abs_evalf(const ex & arg)
 220 {
 221         if (is_exactly_a<numeric>(arg))
 222                 return abs(ex_to<numeric>(arg));
 223
 224         return abs(arg).hold();
 225 }
 226
 227 static ex abs_eval(const ex & arg)
 228 {
 229         if (is_exactly_a<numeric>(arg))
 230                 return abs(ex_to<numeric>(arg));
 231
 232         if (arg.info(info_flags::nonnegative))
 233                 return arg;
 234
 235         if (is_ex_the_function(arg, abs))
 236                 return arg;
 237
 238         if (is_ex_the_function(arg, exp))
 239                 return exp(arg.op(0).real_part());
 240
 241         if (is_exactly_a<power>(arg)) {
 242                 const ex& base = arg.op(0);
 243                 const ex& exponent = arg.op(1);
 244                 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
 245                         return pow(abs(base), exponent.real_part());
 246         }
 247
 248         if (is_ex_the_function(arg, conjugate_function))
 249                 return abs(arg.op(0));
 250
 251         if (is_ex_the_function(arg, step))
 252                 return arg;
 253
 254         return abs(arg).hold();
 255 }
 256
 257 static ex abs_expand(const ex & arg, unsigned options)
 258 {
 259         if ((options & expand_options::expand_transcendental)
 260                 && is_exactly_a<mul>(arg)) {
 261                 exvector prodseq;
 262                 prodseq.reserve(arg.nops());
 263                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 264                         if (options & expand_options::expand_function_args)
 265                                 prodseq.push_back(abs(i->expand(options)));
 266                         else
 267                                 prodseq.push_back(abs(*i));
 268                 }
 269                 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
 270         }
 271
 272         if (options & expand_options::expand_function_args)
 273                 return abs(arg.expand(options)).hold();
 274         else
 275                 return abs(arg).hold();
 276 }
 277
 278 static void abs_print_latex(const ex & arg, const print_context & c)
 279 {
 280         c.s << "{|"; arg.print(c); c.s << "|}";
 281 }
 282
 283 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 284 {
 285         c.s << "fabs("; arg.print(c); c.s << ")";
 286 }
 287
 288 static ex abs_conjugate(const ex & arg)
 289 {
 290         return abs(arg).hold();
 291 }
 292
 293 static ex abs_real_part(const ex & arg)
 294 {
 295         return abs(arg).hold();
 296 }
 297
 298 static ex abs_imag_part(const ex& arg)
 299 {
 300         return 0;
 301 }
 302
 303 static ex abs_power(const ex & arg, const ex & exp)
 304 {
 305         if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
 306                 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
 307                         return power(arg, exp);
 308                 else
 309                         return power(arg, exp/2)*power(arg.conjugate(), exp/2);
 310         } else
 311                 return power(abs(arg), exp).hold();
 312 }
 313
 314 bool abs_info(const ex & arg, unsigned inf)
 315 {
 316         switch (inf) {
 317                 case info_flags::integer:
 318                 case info_flags::even:
 319                 case info_flags::odd:
 320                 case info_flags::prime:
 321                         return arg.info(inf);
 322                 case info_flags::nonnegint:
 323                         return arg.info(info_flags::integer);
 324                 case info_flags::nonnegative:
 325                 case info_flags::real:
 326                         return true;
 327                 case info_flags::negative:
 328                         return false;
 329                 case info_flags::positive:
 330                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
 331                 case info_flags::has_indices: {
 332                         if (arg.info(info_flags::has_indices))
 333                                 return true;
 334                         else
 335                                 return false;
 336                 }
 337         }
 338         return false;
 339 }
 340
 341 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 342                        evalf_func(abs_evalf).
 343                        expand_func(abs_expand).
 344                        info_func(abs_info).
 345                        print_func<print_latex>(abs_print_latex).
 346                        print_func<print_csrc_float>(abs_print_csrc_float).
 347                        print_func<print_csrc_double>(abs_print_csrc_float).
 348                        conjugate_func(abs_conjugate).
 349                        real_part_func(abs_real_part).
 350                        imag_part_func(abs_imag_part).
 351                        power_func(abs_power));
 352
 353 //////////
 354 // Step function
 355 //////////
 356
 357 static ex step_evalf(const ex & arg)
 358 {
 359         if (is_exactly_a<numeric>(arg))
 360                 return step(ex_to<numeric>(arg));
 361
 362         return step(arg).hold();
 363 }
 364
 365 static ex step_eval(const ex & arg)
 366 {
 367         if (is_exactly_a<numeric>(arg))
 368                 return step(ex_to<numeric>(arg));
 369
 370         else if (is_exactly_a<mul>(arg) &&
 371                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 372                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 373                 if (oc.is_real()) {
 374                         if (oc > 0)
 375                                 // step(42*x) -> step(x)
 376                                 return step(arg/oc).hold();
 377                         else
 378                                 // step(-42*x) -> step(-x)
 379                                 return step(-arg/oc).hold();
 380                 }
 381                 if (oc.real().is_zero()) {
 382                         if (oc.imag() > 0)
 383                                 // step(42*I*x) -> step(I*x)
 384                                 return step(I*arg/oc).hold();
 385                         else
 386                                 // step(-42*I*x) -> step(-I*x)
 387                                 return step(-I*arg/oc).hold();
 388                 }
 389         }
 390
 391         return step(arg).hold();
 392 }
 393
 394 static ex step_series(const ex & arg,
 395                       const relational & rel,
 396                       int order,
 397                       unsigned options)
 398 {
 399         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 400         if (arg_pt.info(info_flags::numeric)
 401             && ex_to<numeric>(arg_pt).real().is_zero()
 402             && !(options & series_options::suppress_branchcut))
 403                 throw (std::domain_error("step_series(): on imaginary axis"));
 404
 405         epvector seq;
 406         seq.push_back(expair(step(arg_pt), _ex0));
 407         return pseries(rel,seq);
 408 }
 409
 410 static ex step_conjugate(const ex& arg)
 411 {
 412         return step(arg).hold();
 413 }
 414
 415 static ex step_real_part(const ex& arg)
 416 {
 417         return step(arg).hold();
 418 }
 419
 420 static ex step_imag_part(const ex& arg)
 421 {
 422         return 0;
 423 }
 424
 425 REGISTER_FUNCTION(step, eval_func(step_eval).
 426                         evalf_func(step_evalf).
 427                         series_func(step_series).
 428                         conjugate_func(step_conjugate).
 429                         real_part_func(step_real_part).
 430                         imag_part_func(step_imag_part));
 431
 432 //////////
 433 // Complex sign
 434 //////////
 435
 436 static ex csgn_evalf(const ex & arg)
 437 {
 438         if (is_exactly_a<numeric>(arg))
 439                 return csgn(ex_to<numeric>(arg));
 440
 441         return csgn(arg).hold();
 442 }
 443
 444 static ex csgn_eval(const ex & arg)
 445 {
 446         if (is_exactly_a<numeric>(arg))
 447                 return csgn(ex_to<numeric>(arg));
 448
 449         else if (is_exactly_a<mul>(arg) &&
 450                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 451                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 452                 if (oc.is_real()) {
 453                         if (oc > 0)
 454                                 // csgn(42*x) -> csgn(x)
 455                                 return csgn(arg/oc).hold();
 456                         else
 457                                 // csgn(-42*x) -> -csgn(x)
 458                                 return -csgn(arg/oc).hold();
 459                 }
 460                 if (oc.real().is_zero()) {
 461                         if (oc.imag() > 0)
 462                                 // csgn(42*I*x) -> csgn(I*x)
 463                                 return csgn(I*arg/oc).hold();
 464                         else
 465                                 // csgn(-42*I*x) -> -csgn(I*x)
 466                                 return -csgn(I*arg/oc).hold();
 467                 }
 468         }
 469
 470         return csgn(arg).hold();
 471 }
 472
 473 static ex csgn_series(const ex & arg,
 474                       const relational & rel,
 475                       int order,
 476                       unsigned options)
 477 {
 478         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 479         if (arg_pt.info(info_flags::numeric)
 480             && ex_to<numeric>(arg_pt).real().is_zero()
 481             && !(options & series_options::suppress_branchcut))
 482                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 483
 484         epvector seq;
 485         seq.push_back(expair(csgn(arg_pt), _ex0));
 486         return pseries(rel,seq);
 487 }
 488
 489 static ex csgn_conjugate(const ex& arg)
 490 {
 491         return csgn(arg).hold();
 492 }
 493
 494 static ex csgn_real_part(const ex& arg)
 495 {
 496         return csgn(arg).hold();
 497 }
 498
 499 static ex csgn_imag_part(const ex& arg)
 500 {
 501         return 0;
 502 }
 503
 504 static ex csgn_power(const ex & arg, const ex & exp)
 505 {
 506         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 507                 if (ex_to<numeric>(exp).is_odd())
 508                         return csgn(arg).hold();
 509                 else
 510                         return power(csgn(arg), _ex2).hold();
 511         } else
 512                 return power(csgn(arg), exp).hold();
 513 }
 514
 515
 516 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 517                         evalf_func(csgn_evalf).
 518                         series_func(csgn_series).
 519                         conjugate_func(csgn_conjugate).
 520                         real_part_func(csgn_real_part).
 521                         imag_part_func(csgn_imag_part).
 522                         power_func(csgn_power));
 523
 524
 525 //////////
 526 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 527 // This function is closely related to the unwinding number K, sometimes found
 528 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 529 //////////
 530
 531 static ex eta_evalf(const ex &x, const ex &y)
 532 {
 533         // It seems like we basically have to replicate the eval function here,
 534         // since the expression might not be fully evaluated yet.
 535         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 536                 return _ex0;
 537
 538         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 539                 const numeric nx = ex_to<numeric>(x);
 540                 const numeric ny = ex_to<numeric>(y);
 541                 const numeric nxy = ex_to<numeric>(x*y);
 542                 int cut = 0;
 543                 if (nx.is_real() && nx.is_negative())
 544                         cut -= 4;
 545                 if (ny.is_real() && ny.is_negative())
 546                         cut -= 4;
 547                 if (nxy.is_real() && nxy.is_negative())
 548                         cut += 4;
 549                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 550                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 551         }
 552
 553         return eta(x,y).hold();
 554 }
 555
 556 static ex eta_eval(const ex &x, const ex &y)
 557 {
 558         // trivial:  eta(x,c) -> 0  if c is real and positive
 559         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 560                 return _ex0;
 561
 562         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 563                 // don't call eta_evalf here because it would call Pi.evalf()!
 564                 const numeric nx = ex_to<numeric>(x);
 565                 const numeric ny = ex_to<numeric>(y);
 566                 const numeric nxy = ex_to<numeric>(x*y);
 567                 int cut = 0;
 568                 if (nx.is_real() && nx.is_negative())
 569                         cut -= 4;
 570                 if (ny.is_real() && ny.is_negative())
 571                         cut -= 4;
 572                 if (nxy.is_real() && nxy.is_negative())
 573                         cut += 4;
 574                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 575                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 576         }
 577
 578         return eta(x,y).hold();
 579 }
 580
 581 static ex eta_series(const ex & x, const ex & y,
 582                      const relational & rel,
 583                      int order,
 584                      unsigned options)
 585 {
 586         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 587         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 588         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 589             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 590             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 591                         throw (std::domain_error("eta_series(): on discontinuity"));
 592         epvector seq;
 593         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 594         return pseries(rel,seq);
 595 }
 596
 597 static ex eta_conjugate(const ex & x, const ex & y)
 598 {
 599         return -eta(x, y).hold();
 600 }
 601
 602 static ex eta_real_part(const ex & x, const ex & y)
 603 {
 604         return 0;
 605 }
 606
 607 static ex eta_imag_part(const ex & x, const ex & y)
 608 {
 609         return -I*eta(x, y).hold();
 610 }
 611
 612 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 613                        evalf_func(eta_evalf).
 614                        series_func(eta_series).
 615                        latex_name("\\eta").
 616                        set_symmetry(sy_symm(0, 1)).
 617                        conjugate_func(eta_conjugate).
 618                        real_part_func(eta_real_part).
 619                        imag_part_func(eta_imag_part));
 620
 621
 622 //////////
 623 // dilogarithm
 624 //////////
 625
 626 static ex Li2_evalf(const ex & x)
 627 {
 628         if (is_exactly_a<numeric>(x))
 629                 return Li2(ex_to<numeric>(x));
 630
 631         return Li2(x).hold();
 632 }
 633
 634 static ex Li2_eval(const ex & x)
 635 {
 636         if (x.info(info_flags::numeric)) {
 637                 // Li2(0) -> 0
 638                 if (x.is_zero())
 639                         return _ex0;
 640                 // Li2(1) -> Pi^2/6
 641                 if (x.is_equal(_ex1))
 642                         return power(Pi,_ex2)/_ex6;
 643                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 644                 if (x.is_equal(_ex1_2))
 645                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 646                 // Li2(-1) -> -Pi^2/12
 647                 if (x.is_equal(_ex_1))
 648                         return -power(Pi,_ex2)/_ex12;
 649                 // Li2(I) -> -Pi^2/48+Catalan*I
 650                 if (x.is_equal(I))
 651                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 652                 // Li2(-I) -> -Pi^2/48-Catalan*I
 653                 if (x.is_equal(-I))
 654                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 655                 // Li2(float)
 656                 if (!x.info(info_flags::crational))
 657                         return Li2(ex_to<numeric>(x));
 658         }
 659
 660         return Li2(x).hold();
 661 }
 662
 663 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 664 {
 665         GINAC_ASSERT(deriv_param==0);
 666
 667         // d/dx Li2(x) -> -log(1-x)/x
 668         return -log(_ex1-x)/x;
 669 }
 670
 671 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 672 {
 673         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 674         if (x_pt.info(info_flags::numeric)) {
 675                 // First special case: x==0 (derivatives have poles)
 676                 if (x_pt.is_zero()) {
 677                         // method:
 678                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 679                         // simply substitute x==0.  The limit, however, exists: it is 1.
 680                         // We also know all higher derivatives' limits:
 681                         // (d/dx)^n Li2(x) == n!/n^2.
 682                         // So the primitive series expansion is
 683                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 684                         // and so on.
 685                         // We first construct such a primitive series expansion manually in
 686                         // a dummy symbol s and then insert the argument's series expansion
 687                         // for s.  Reexpanding the resulting series returns the desired
 688                         // result.
 689                         const symbol s;
 690                         ex ser;
 691                         // manually construct the primitive expansion
 692                         for (int i=1; i<order; ++i)
 693                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 694                         // substitute the argument's series expansion
 695                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 696                         // maybe that was terminating, so add a proper order term
 697                         epvector nseq;
 698                         nseq.push_back(expair(Order(_ex1), order));
 699                         ser += pseries(rel, nseq);
 700                         // reexpanding it will collapse the series again
 701                         return ser.series(rel, order);
 702                         // NB: Of course, this still does not allow us to compute anything
 703                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 704                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 705                         // substitution x==0.  Probably limits *are* needed for the general
 706                         // cases.  In case L'Hospital's rule is implemented for limits and
 707                         // basic::series() takes care of this, this whole block is probably
 708                         // obsolete!
 709                 }
 710                 // second special case: x==1 (branch point)
 711                 if (x_pt.is_equal(_ex1)) {
 712                         // method:
 713                         // construct series manually in a dummy symbol s
 714                         const symbol s;
 715                         ex ser = zeta(_ex2);
 716                         // manually construct the primitive expansion
 717                         for (int i=1; i<order; ++i)
 718                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 719                         // substitute the argument's series expansion
 720                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 721                         // maybe that was terminating, so add a proper order term
 722                         epvector nseq;
 723                         nseq.push_back(expair(Order(_ex1), order));
 724                         ser += pseries(rel, nseq);
 725                         // reexpanding it will collapse the series again
 726                         return ser.series(rel, order);
 727                 }
 728                 // third special case: x real, >=1 (branch cut)
 729                 if (!(options & series_options::suppress_branchcut) &&
 730                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 731                         // method:
 732                         // This is the branch cut: assemble the primitive series manually
 733                         // and then add the corresponding complex step function.
 734                         const symbol &s = ex_to<symbol>(rel.lhs());
 735                         const ex point = rel.rhs();
 736                         const symbol foo;
 737                         epvector seq;
 738                         // zeroth order term:
 739                         seq.push_back(expair(Li2(x_pt), _ex0));
 740                         // compute the intermediate terms:
 741                         ex replarg = series(Li2(x), s==foo, order);
 742                         for (size_t i=1; i<replarg.nops()-1; ++i)
 743                                 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));
 744                         // append an order term:
 745                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 746                         return pseries(rel, seq);
 747                 }
 748         }
 749         // all other cases should be safe, by now:
 750         throw do_taylor();  // caught by function::series()
 751 }
 752
 753 static ex Li2_conjugate(const ex & x)
 754 {
 755         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 756         // run along the positive real axis beginning at 1.
 757         if (x.info(info_flags::negative)) {
 758                 return Li2(x).hold();
 759         }
 760         if (is_exactly_a<numeric>(x) &&
 761             (!x.imag_part().is_zero() || x < *_num1_p)) {
 762                 return Li2(x.conjugate());
 763         }
 764         return conjugate_function(Li2(x)).hold();
 765 }
 766
 767 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 768                        evalf_func(Li2_evalf).
 769                        derivative_func(Li2_deriv).
 770                        series_func(Li2_series).
 771                        conjugate_func(Li2_conjugate).
 772                        latex_name("\\mathrm{Li}_2"));
 773
 774 //////////
 775 // trilogarithm
 776 //////////
 777
 778 static ex Li3_eval(const ex & x)
 779 {
 780         if (x.is_zero())
 781                 return x;
 782         return Li3(x).hold();
 783 }
 784
 785 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 786                        latex_name("\\mathrm{Li}_3"));
 787
 788 //////////
 789 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 790 //////////
 791
 792 static ex zetaderiv_eval(const ex & n, const ex & x)
 793 {
 794         if (n.info(info_flags::numeric)) {
 795                 // zetaderiv(0,x) -> zeta(x)
 796                 if (n.is_zero())
 797                         return zeta(x).hold();
 798         }
 799
 800         return zetaderiv(n, x).hold();
 801 }
 802
 803 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 804 {
 805         GINAC_ASSERT(deriv_param<2);
 806
 807         if (deriv_param==0) {
 808                 // d/dn zeta(n,x)
 809                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 810         }
 811         // d/dx psi(n,x)
 812         return zetaderiv(n+1,x);
 813 }
 814
 815 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 816                                  derivative_func(zetaderiv_deriv).
 817                                  latex_name("\\zeta^\\prime"));
 818
 819 //////////
 820 // factorial
 821 //////////
 822
 823 static ex factorial_evalf(const ex & x)
 824 {
 825         return factorial(x).hold();
 826 }
 827
 828 static ex factorial_eval(const ex & x)
 829 {
 830         if (is_exactly_a<numeric>(x))
 831                 return factorial(ex_to<numeric>(x));
 832         else
 833                 return factorial(x).hold();
 834 }
 835
 836 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 837 {
 838         if (is_exactly_a<symbol>(x) ||
 839             is_exactly_a<constant>(x) ||
 840                 is_exactly_a<function>(x)) {
 841                 x.print(c); c.s << "!";
 842         } else {
 843                 c.s << "("; x.print(c); c.s << ")!";
 844         }
 845 }
 846
 847 static ex factorial_conjugate(const ex & x)
 848 {
 849         return factorial(x).hold();
 850 }
 851
 852 static ex factorial_real_part(const ex & x)
 853 {
 854         return factorial(x).hold();
 855 }
 856
 857 static ex factorial_imag_part(const ex & x)
 858 {
 859         return 0;
 860 }
 861
 862 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 863                              evalf_func(factorial_evalf).
 864                              print_func<print_dflt>(factorial_print_dflt_latex).
 865                              print_func<print_latex>(factorial_print_dflt_latex).
 866                              conjugate_func(factorial_conjugate).
 867                              real_part_func(factorial_real_part).
 868                              imag_part_func(factorial_imag_part));
 869
 870 //////////
 871 // binomial
 872 //////////
 873
 874 static ex binomial_evalf(const ex & x, const ex & y)
 875 {
 876         return binomial(x, y).hold();
 877 }
 878
 879 static ex binomial_sym(const ex & x, const numeric & y)
 880 {
 881         if (y.is_integer()) {
 882                 if (y.is_nonneg_integer()) {
 883                         const unsigned N = y.to_int();
 884                         if (N == 0) return _ex1;
 885                         if (N == 1) return x;
 886                         ex t = x.expand();
 887                         for (unsigned i = 2; i <= N; ++i)
 888                                 t = (t * (x + i - y - 1)).expand() / i;
 889                         return t;
 890                 } else
 891                         return _ex0;
 892         }
 893
 894         return binomial(x, y).hold();
 895 }
 896
 897 static ex binomial_eval(const ex & x, const ex &y)
 898 {
 899         if (is_exactly_a<numeric>(y)) {
 900                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 901                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 902                 else
 903                         return binomial_sym(x, ex_to<numeric>(y));
 904         } else
 905                 return binomial(x, y).hold();
 906 }
 907
 908 // At the moment the numeric evaluation of a binomail function always
 909 // gives a real number, but if this would be implemented using the gamma
 910 // function, also complex conjugation should be changed (or rather, deleted).
 911 static ex binomial_conjugate(const ex & x, const ex & y)
 912 {
 913         return binomial(x,y).hold();
 914 }
 915
 916 static ex binomial_real_part(const ex & x, const ex & y)
 917 {
 918         return binomial(x,y).hold();
 919 }
 920
 921 static ex binomial_imag_part(const ex & x, const ex & y)
 922 {
 923         return 0;
 924 }
 925
 926 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 927                             evalf_func(binomial_evalf).
 928                             conjugate_func(binomial_conjugate).
 929                             real_part_func(binomial_real_part).
 930                             imag_part_func(binomial_imag_part));
 931
 932 //////////
 933 // Order term function (for truncated power series)
 934 //////////
 935
 936 static ex Order_eval(const ex & x)
 937 {
 938         if (is_exactly_a<numeric>(x)) {
 939                 // O(c) -> O(1) or 0
 940                 if (!x.is_zero())
 941                         return Order(_ex1).hold();
 942                 else
 943                         return _ex0;
 944         } else if (is_exactly_a<mul>(x)) {
 945                 const mul &m = ex_to<mul>(x);
 946                 // O(c*expr) -> O(expr)
 947                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 948                         return Order(x / m.op(m.nops() - 1)).hold();
 949         }
 950         return Order(x).hold();
 951 }
 952
 953 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 954 {
 955         // Just wrap the function into a pseries object
 956         epvector new_seq;
 957         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 958         const symbol &s = ex_to<symbol>(r.lhs());
 959         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 960         return pseries(r, new_seq);
 961 }
 962
 963 static ex Order_conjugate(const ex & x)
 964 {
 965         return Order(x).hold();
 966 }
 967
 968 static ex Order_real_part(const ex & x)
 969 {
 970         return Order(x).hold();
 971 }
 972
 973 static ex Order_imag_part(const ex & x)
 974 {
 975         if(x.info(info_flags::real))
 976                 return 0;
 977         return Order(x).hold();
 978 }
 979
 980 // Differentiation is handled in function::derivative because of its special requirements
 981
 982 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 983                          series_func(Order_series).
 984                          latex_name("\\mathcal{O}").
 985                          conjugate_func(Order_conjugate).
 986                          real_part_func(Order_real_part).
 987                          imag_part_func(Order_imag_part));
 988
 989 //////////
 990 // Solve linear system
 991 //////////
 992
 993 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 994 {
 995         // solve a system of linear equations
 996         if (eqns.info(info_flags::relation_equal)) {
 997                 if (!symbols.info(info_flags::symbol))
 998                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 999                 const ex sol = lsolve(lst(eqns),lst(symbols));
1000
1001                 GINAC_ASSERT(sol.nops()==1);
1002                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1003
1004                 return sol.op(0).op(1); // return rhs of first solution
1005         }
1006
1007         // syntax checks
1008         if (!eqns.info(info_flags::list)) {
1009                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1010         }
1011         for (size_t i=0; i<eqns.nops(); i++) {
1012                 if (!eqns.op(i).info(info_flags::relation_equal)) {
1013                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1014                 }
1015         }
1016         if (!symbols.info(info_flags::list)) {
1017                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1018         }
1019         for (size_t i=0; i<symbols.nops(); i++) {
1020                 if (!symbols.op(i).info(info_flags::symbol)) {
1021                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1022                 }
1023         }
1024
1025         // build matrix from equation system
1026         matrix sys(eqns.nops(),symbols.nops());
1027         matrix rhs(eqns.nops(),1);
1028         matrix vars(symbols.nops(),1);
1029
1030         for (size_t r=0; r<eqns.nops(); r++) {
1031                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1032                 ex linpart = eq;
1033                 for (size_t c=0; c<symbols.nops(); c++) {
1034                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1035                         linpart -= co*symbols.op(c);
1036                         sys(r,c) = co;
1037                 }
1038                 linpart = linpart.expand();
1039                 rhs(r,0) = -linpart;
1040         }
1041
1042         // test if system is linear and fill vars matrix
1043         for (size_t i=0; i<symbols.nops(); i++) {
1044                 vars(i,0) = symbols.op(i);
1045                 if (sys.has(symbols.op(i)))
1046                         throw(std::logic_error("lsolve: system is not linear"));
1047                 if (rhs.has(symbols.op(i)))
1048                         throw(std::logic_error("lsolve: system is not linear"));
1049         }
1050
1051         matrix solution;
1052         try {
1053                 solution = sys.solve(vars,rhs,options);
1054         } catch (const std::runtime_error & e) {
1055                 // Probably singular matrix or otherwise overdetermined system:
1056                 // It is consistent to return an empty list
1057                 return lst();
1058         }
1059         GINAC_ASSERT(solution.cols()==1);
1060         GINAC_ASSERT(solution.rows()==symbols.nops());
1061
1062         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1063         lst sollist;
1064         for (size_t i=0; i<symbols.nops(); i++)
1065                 sollist.append(symbols.op(i)==solution(i,0));
1066
1067         return sollist;
1068 }
1069
1070 //////////
1071 // Find real root of f(x) numerically
1072 //////////
1073
1074 const numeric
1075 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1076 {
1077         if (!x1.is_real() || !x2.is_real()) {
1078                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1079         }
1080         if (x1==x2) {
1081                 throw std::runtime_error("fsolve(): vanishing interval");
1082         }
1083         // xx[0] == left interval limit, xx[1] == right interval limit.
1084         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1085         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1086         numeric xx[2] = { x1<x2 ? x1 : x2,
1087                           x1<x2 ? x2 : x1 };
1088         ex f;
1089         if (is_a<relational>(f_in)) {
1090                 f = f_in.lhs()-f_in.rhs();
1091         } else {
1092                 f = f_in;
1093         }
1094         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1095                             f.subs(x==xx[1]).evalf() };
1096         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1097                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1098         }
1099         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1100                           ex_to<numeric>(fx_[1]) };
1101         if (!fx[0].is_real() || !fx[1].is_real()) {
1102                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1103         }
1104         if (fx[0]*fx[1]>=0) {
1105                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1106         }
1107
1108         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1109         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1110         const ex ff = normal(-f/f.diff(x));
1111         int side = 0;  // Start at left interval limit.
1112         numeric xxprev;
1113         numeric fxprev;
1114         do {
1115                 xxprev = xx[side];
1116                 fxprev = fx[side];
1117                 ex dx_ = ff.subs(x == xx[side]).evalf();
1118                 if (!is_a<numeric>(dx_))
1119                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1120                 xx[side] += ex_to<numeric>(dx_);
1121                 // Now check if Newton-Raphson method shot out of the interval
1122                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1123                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1124                 if (!bad_shot) {
1125                         // Compute f(x) only if new x is inside the interval.
1126                         // The function might be difficult to compute numerically
1127                         // or even ill defined outside the interval. Also it's
1128                         // a small optimization.
1129                         ex f_x = f.subs(x == xx[side]).evalf();
1130                         if (!is_a<numeric>(f_x))
1131                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1132                         fx[side] = ex_to<numeric>(f_x);
1133                 }
1134                 if (bad_shot) {
1135                         // Oops, Newton-Raphson method shot out of the interval.
1136                         // Restore, and try again with the other side instead!
1137                         xx[side] = xxprev;
1138                         fx[side] = fxprev;
1139                         side = !side;
1140                         xxprev = xx[side];
1141                         fxprev = fx[side];
1142
1143                         ex dx_ = ff.subs(x == xx[side]).evalf();
1144                         if (!is_a<numeric>(dx_))
1145                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1146                         xx[side] += ex_to<numeric>(dx_);
1147
1148                         ex f_x = f.subs(x==xx[side]).evalf();
1149                         if (!is_a<numeric>(f_x))
1150                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1151                         fx[side] = ex_to<numeric>(f_x);
1152                 }
1153                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1154                         // Oops, the root isn't bracketed any more.
1155                         // Restore, and perform a bisection!
1156                         xx[side] = xxprev;
1157                         fx[side] = fxprev;
1158
1159                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1160                         // they occur pretty often when Newton-Raphson arrives at an x too
1161                         // close to the result on one side of the interval and
1162                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1163                         // precision errors! Recall that this function does not have a
1164                         // precision goal as one of its arguments but instead relies on
1165                         // x converging to a fixed point. We speed up the (safe but slow)
1166                         // bisection method by mixing in a dash of the (unsafer but faster)
1167                         // secant method: Instead of splitting the interval at the
1168                         // arithmetic mean (bisection), we split it nearer to the root as
1169                         // determined by the secant between the values xx[0] and xx[1].
1170                         // Don't set the secant_weight to one because that could disturb
1171                         // the convergence in some corner cases!
1172                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1173                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1174                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1175                         ex fxmid_ = f.subs(x == xxmid).evalf();
1176                         if (!is_a<numeric>(fxmid_))
1177                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1178                         numeric fxmid = ex_to<numeric>(fxmid_);
1179                         if (fxmid.is_zero()) {
1180                                 // Luck strikes...
1181                                 return xxmid;
1182                         }
1183                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1184                                 side = !side;
1185                         }
1186                         xxprev = xx[side];
1187                         fxprev = fx[side];
1188                         xx[side] = xxmid;
1189                         fx[side] = fxmid;
1190                 }
1191         } while (xxprev!=xx[side]);
1192         return xxprev;
1193 }
1194
1195
1196 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1197  * for static lib (so ginsh will see them). */
1198 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1199 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1200
1201 } // namespace GiNaC