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