ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2011 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 void abs_print_latex(const ex & arg, const print_context & c)
 258 {
 259         c.s << "{|"; arg.print(c); c.s << "|}";
 260 }
 261
 262 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 263 {
 264         c.s << "fabs("; arg.print(c); c.s << ")";
 265 }
 266
 267 static ex abs_conjugate(const ex & arg)
 268 {
 269         return abs(arg).hold();
 270 }
 271
 272 static ex abs_real_part(const ex & arg)
 273 {
 274         return abs(arg).hold();
 275 }
 276
 277 static ex abs_imag_part(const ex& arg)
 278 {
 279         return 0;
 280 }
 281
 282 static ex abs_power(const ex & arg, const ex & exp)
 283 {
 284         if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
 285                                                 || exp.info(info_flags::even)))
 286                 return power(arg, exp);
 287         else
 288                 return power(abs(arg), exp).hold();
 289 }
 290
 291 bool abs_info(const ex & arg, unsigned inf)
 292 {
 293         switch (inf) {
 294                 case info_flags::integer:
 295                 case info_flags::even:
 296                 case info_flags::odd:
 297                 case info_flags::prime:
 298                         return arg.info(inf);
 299                 case info_flags::nonnegint:
 300                         return arg.info(info_flags::integer);
 301                 case info_flags::nonnegative:
 302                 case info_flags::real:
 303                         return true;
 304                 case info_flags::negative:
 305                         return false;
 306                 case info_flags::positive:
 307                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
 308                 case info_flags::has_indices: {
 309                         if (arg.info(info_flags::has_indices))
 310                                 return true;
 311                         else
 312                                 return false;
 313                 }
 314         }
 315         return false;
 316 }
 317
 318 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 319                        evalf_func(abs_evalf).
 320                        info_func(abs_info).
 321                        print_func<print_latex>(abs_print_latex).
 322                        print_func<print_csrc_float>(abs_print_csrc_float).
 323                        print_func<print_csrc_double>(abs_print_csrc_float).
 324                        conjugate_func(abs_conjugate).
 325                        real_part_func(abs_real_part).
 326                        imag_part_func(abs_imag_part).
 327                        power_func(abs_power));
 328
 329 //////////
 330 // Step function
 331 //////////
 332
 333 static ex step_evalf(const ex & arg)
 334 {
 335         if (is_exactly_a<numeric>(arg))
 336                 return step(ex_to<numeric>(arg));
 337
 338         return step(arg).hold();
 339 }
 340
 341 static ex step_eval(const ex & arg)
 342 {
 343         if (is_exactly_a<numeric>(arg))
 344                 return step(ex_to<numeric>(arg));
 345
 346         else if (is_exactly_a<mul>(arg) &&
 347                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 348                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 349                 if (oc.is_real()) {
 350                         if (oc > 0)
 351                                 // step(42*x) -> step(x)
 352                                 return step(arg/oc).hold();
 353                         else
 354                                 // step(-42*x) -> step(-x)
 355                                 return step(-arg/oc).hold();
 356                 }
 357                 if (oc.real().is_zero()) {
 358                         if (oc.imag() > 0)
 359                                 // step(42*I*x) -> step(I*x)
 360                                 return step(I*arg/oc).hold();
 361                         else
 362                                 // step(-42*I*x) -> step(-I*x)
 363                                 return step(-I*arg/oc).hold();
 364                 }
 365         }
 366
 367         return step(arg).hold();
 368 }
 369
 370 static ex step_series(const ex & arg,
 371                       const relational & rel,
 372                       int order,
 373                       unsigned options)
 374 {
 375         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 376         if (arg_pt.info(info_flags::numeric)
 377             && ex_to<numeric>(arg_pt).real().is_zero()
 378             && !(options & series_options::suppress_branchcut))
 379                 throw (std::domain_error("step_series(): on imaginary axis"));
 380
 381         epvector seq;
 382         seq.push_back(expair(step(arg_pt), _ex0));
 383         return pseries(rel,seq);
 384 }
 385
 386 static ex step_conjugate(const ex& arg)
 387 {
 388         return step(arg).hold();
 389 }
 390
 391 static ex step_real_part(const ex& arg)
 392 {
 393         return step(arg).hold();
 394 }
 395
 396 static ex step_imag_part(const ex& arg)
 397 {
 398         return 0;
 399 }
 400
 401 REGISTER_FUNCTION(step, eval_func(step_eval).
 402                         evalf_func(step_evalf).
 403                         series_func(step_series).
 404                         conjugate_func(step_conjugate).
 405                         real_part_func(step_real_part).
 406                         imag_part_func(step_imag_part));
 407
 408 //////////
 409 // Complex sign
 410 //////////
 411
 412 static ex csgn_evalf(const ex & arg)
 413 {
 414         if (is_exactly_a<numeric>(arg))
 415                 return csgn(ex_to<numeric>(arg));
 416
 417         return csgn(arg).hold();
 418 }
 419
 420 static ex csgn_eval(const ex & arg)
 421 {
 422         if (is_exactly_a<numeric>(arg))
 423                 return csgn(ex_to<numeric>(arg));
 424
 425         else if (is_exactly_a<mul>(arg) &&
 426                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 427                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 428                 if (oc.is_real()) {
 429                         if (oc > 0)
 430                                 // csgn(42*x) -> csgn(x)
 431                                 return csgn(arg/oc).hold();
 432                         else
 433                                 // csgn(-42*x) -> -csgn(x)
 434                                 return -csgn(arg/oc).hold();
 435                 }
 436                 if (oc.real().is_zero()) {
 437                         if (oc.imag() > 0)
 438                                 // csgn(42*I*x) -> csgn(I*x)
 439                                 return csgn(I*arg/oc).hold();
 440                         else
 441                                 // csgn(-42*I*x) -> -csgn(I*x)
 442                                 return -csgn(I*arg/oc).hold();
 443                 }
 444         }
 445
 446         return csgn(arg).hold();
 447 }
 448
 449 static ex csgn_series(const ex & arg,
 450                       const relational & rel,
 451                       int order,
 452                       unsigned options)
 453 {
 454         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 455         if (arg_pt.info(info_flags::numeric)
 456             && ex_to<numeric>(arg_pt).real().is_zero()
 457             && !(options & series_options::suppress_branchcut))
 458                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 459
 460         epvector seq;
 461         seq.push_back(expair(csgn(arg_pt), _ex0));
 462         return pseries(rel,seq);
 463 }
 464
 465 static ex csgn_conjugate(const ex& arg)
 466 {
 467         return csgn(arg).hold();
 468 }
 469
 470 static ex csgn_real_part(const ex& arg)
 471 {
 472         return csgn(arg).hold();
 473 }
 474
 475 static ex csgn_imag_part(const ex& arg)
 476 {
 477         return 0;
 478 }
 479
 480 static ex csgn_power(const ex & arg, const ex & exp)
 481 {
 482         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 483                 if (ex_to<numeric>(exp).is_odd())
 484                         return csgn(arg).hold();
 485                 else
 486                         return power(csgn(arg), _ex2).hold();
 487         } else
 488                 return power(csgn(arg), exp).hold();
 489 }
 490
 491
 492 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 493                         evalf_func(csgn_evalf).
 494                         series_func(csgn_series).
 495                         conjugate_func(csgn_conjugate).
 496                         real_part_func(csgn_real_part).
 497                         imag_part_func(csgn_imag_part).
 498                         power_func(csgn_power));
 499
 500
 501 //////////
 502 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 503 // This function is closely related to the unwinding number K, sometimes found
 504 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 505 //////////
 506
 507 static ex eta_evalf(const ex &x, const ex &y)
 508 {
 509         // It seems like we basically have to replicate the eval function here,
 510         // since the expression might not be fully evaluated yet.
 511         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 512                 return _ex0;
 513
 514         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 515                 const numeric nx = ex_to<numeric>(x);
 516                 const numeric ny = ex_to<numeric>(y);
 517                 const numeric nxy = ex_to<numeric>(x*y);
 518                 int cut = 0;
 519                 if (nx.is_real() && nx.is_negative())
 520                         cut -= 4;
 521                 if (ny.is_real() && ny.is_negative())
 522                         cut -= 4;
 523                 if (nxy.is_real() && nxy.is_negative())
 524                         cut += 4;
 525                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 526                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 527         }
 528
 529         return eta(x,y).hold();
 530 }
 531
 532 static ex eta_eval(const ex &x, const ex &y)
 533 {
 534         // trivial:  eta(x,c) -> 0  if c is real and positive
 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                 // don't call eta_evalf here because it would call Pi.evalf()!
 540                 const numeric nx = ex_to<numeric>(x);
 541                 const numeric ny = ex_to<numeric>(y);
 542                 const numeric nxy = ex_to<numeric>(x*y);
 543                 int cut = 0;
 544                 if (nx.is_real() && nx.is_negative())
 545                         cut -= 4;
 546                 if (ny.is_real() && ny.is_negative())
 547                         cut -= 4;
 548                 if (nxy.is_real() && nxy.is_negative())
 549                         cut += 4;
 550                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 551                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 552         }
 553
 554         return eta(x,y).hold();
 555 }
 556
 557 static ex eta_series(const ex & x, const ex & y,
 558                      const relational & rel,
 559                      int order,
 560                      unsigned options)
 561 {
 562         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 563         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 564         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 565             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 566             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 567                         throw (std::domain_error("eta_series(): on discontinuity"));
 568         epvector seq;
 569         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 570         return pseries(rel,seq);
 571 }
 572
 573 static ex eta_conjugate(const ex & x, const ex & y)
 574 {
 575         return -eta(x, y).hold();
 576 }
 577
 578 static ex eta_real_part(const ex & x, const ex & y)
 579 {
 580         return 0;
 581 }
 582
 583 static ex eta_imag_part(const ex & x, const ex & y)
 584 {
 585         return -I*eta(x, y).hold();
 586 }
 587
 588 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 589                        evalf_func(eta_evalf).
 590                        series_func(eta_series).
 591                        latex_name("\\eta").
 592                        set_symmetry(sy_symm(0, 1)).
 593                        conjugate_func(eta_conjugate).
 594                        real_part_func(eta_real_part).
 595                        imag_part_func(eta_imag_part));
 596
 597
 598 //////////
 599 // dilogarithm
 600 //////////
 601
 602 static ex Li2_evalf(const ex & x)
 603 {
 604         if (is_exactly_a<numeric>(x))
 605                 return Li2(ex_to<numeric>(x));
 606
 607         return Li2(x).hold();
 608 }
 609
 610 static ex Li2_eval(const ex & x)
 611 {
 612         if (x.info(info_flags::numeric)) {
 613                 // Li2(0) -> 0
 614                 if (x.is_zero())
 615                         return _ex0;
 616                 // Li2(1) -> Pi^2/6
 617                 if (x.is_equal(_ex1))
 618                         return power(Pi,_ex2)/_ex6;
 619                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 620                 if (x.is_equal(_ex1_2))
 621                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 622                 // Li2(-1) -> -Pi^2/12
 623                 if (x.is_equal(_ex_1))
 624                         return -power(Pi,_ex2)/_ex12;
 625                 // Li2(I) -> -Pi^2/48+Catalan*I
 626                 if (x.is_equal(I))
 627                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 628                 // Li2(-I) -> -Pi^2/48-Catalan*I
 629                 if (x.is_equal(-I))
 630                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 631                 // Li2(float)
 632                 if (!x.info(info_flags::crational))
 633                         return Li2(ex_to<numeric>(x));
 634         }
 635
 636         return Li2(x).hold();
 637 }
 638
 639 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 640 {
 641         GINAC_ASSERT(deriv_param==0);
 642
 643         // d/dx Li2(x) -> -log(1-x)/x
 644         return -log(_ex1-x)/x;
 645 }
 646
 647 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 648 {
 649         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 650         if (x_pt.info(info_flags::numeric)) {
 651                 // First special case: x==0 (derivatives have poles)
 652                 if (x_pt.is_zero()) {
 653                         // method:
 654                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 655                         // simply substitute x==0.  The limit, however, exists: it is 1.
 656                         // We also know all higher derivatives' limits:
 657                         // (d/dx)^n Li2(x) == n!/n^2.
 658                         // So the primitive series expansion is
 659                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 660                         // and so on.
 661                         // We first construct such a primitive series expansion manually in
 662                         // a dummy symbol s and then insert the argument's series expansion
 663                         // for s.  Reexpanding the resulting series returns the desired
 664                         // result.
 665                         const symbol s;
 666                         ex ser;
 667                         // manually construct the primitive expansion
 668                         for (int i=1; i<order; ++i)
 669                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 670                         // substitute the argument's series expansion
 671                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 672                         // maybe that was terminating, so add a proper order term
 673                         epvector nseq;
 674                         nseq.push_back(expair(Order(_ex1), order));
 675                         ser += pseries(rel, nseq);
 676                         // reexpanding it will collapse the series again
 677                         return ser.series(rel, order);
 678                         // NB: Of course, this still does not allow us to compute anything
 679                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 680                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 681                         // substitution x==0.  Probably limits *are* needed for the general
 682                         // cases.  In case L'Hospital's rule is implemented for limits and
 683                         // basic::series() takes care of this, this whole block is probably
 684                         // obsolete!
 685                 }
 686                 // second special case: x==1 (branch point)
 687                 if (x_pt.is_equal(_ex1)) {
 688                         // method:
 689                         // construct series manually in a dummy symbol s
 690                         const symbol s;
 691                         ex ser = zeta(_ex2);
 692                         // manually construct the primitive expansion
 693                         for (int i=1; i<order; ++i)
 694                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 695                         // substitute the argument's series expansion
 696                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 697                         // maybe that was terminating, so add a proper order term
 698                         epvector nseq;
 699                         nseq.push_back(expair(Order(_ex1), order));
 700                         ser += pseries(rel, nseq);
 701                         // reexpanding it will collapse the series again
 702                         return ser.series(rel, order);
 703                 }
 704                 // third special case: x real, >=1 (branch cut)
 705                 if (!(options & series_options::suppress_branchcut) &&
 706                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 707                         // method:
 708                         // This is the branch cut: assemble the primitive series manually
 709                         // and then add the corresponding complex step function.
 710                         const symbol &s = ex_to<symbol>(rel.lhs());
 711                         const ex point = rel.rhs();
 712                         const symbol foo;
 713                         epvector seq;
 714                         // zeroth order term:
 715                         seq.push_back(expair(Li2(x_pt), _ex0));
 716                         // compute the intermediate terms:
 717                         ex replarg = series(Li2(x), s==foo, order);
 718                         for (size_t i=1; i<replarg.nops()-1; ++i)
 719                                 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));
 720                         // append an order term:
 721                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 722                         return pseries(rel, seq);
 723                 }
 724         }
 725         // all other cases should be safe, by now:
 726         throw do_taylor();  // caught by function::series()
 727 }
 728
 729 static ex Li2_conjugate(const ex & x)
 730 {
 731         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 732         // run along the positive real axis beginning at 1.
 733         if (x.info(info_flags::negative)) {
 734                 return Li2(x).hold();
 735         }
 736         if (is_exactly_a<numeric>(x) &&
 737             (!x.imag_part().is_zero() || x < *_num1_p)) {
 738                 return Li2(x.conjugate());
 739         }
 740         return conjugate_function(Li2(x)).hold();
 741 }
 742
 743 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 744                        evalf_func(Li2_evalf).
 745                        derivative_func(Li2_deriv).
 746                        series_func(Li2_series).
 747                        conjugate_func(Li2_conjugate).
 748                        latex_name("\\mathrm{Li}_2"));
 749
 750 //////////
 751 // trilogarithm
 752 //////////
 753
 754 static ex Li3_eval(const ex & x)
 755 {
 756         if (x.is_zero())
 757                 return x;
 758         return Li3(x).hold();
 759 }
 760
 761 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 762                        latex_name("\\mathrm{Li}_3"));
 763
 764 //////////
 765 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 766 //////////
 767
 768 static ex zetaderiv_eval(const ex & n, const ex & x)
 769 {
 770         if (n.info(info_flags::numeric)) {
 771                 // zetaderiv(0,x) -> zeta(x)
 772                 if (n.is_zero())
 773                         return zeta(x).hold();
 774         }
 775
 776         return zetaderiv(n, x).hold();
 777 }
 778
 779 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 780 {
 781         GINAC_ASSERT(deriv_param<2);
 782
 783         if (deriv_param==0) {
 784                 // d/dn zeta(n,x)
 785                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 786         }
 787         // d/dx psi(n,x)
 788         return zetaderiv(n+1,x);
 789 }
 790
 791 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 792                                  derivative_func(zetaderiv_deriv).
 793                                  latex_name("\\zeta^\\prime"));
 794
 795 //////////
 796 // factorial
 797 //////////
 798
 799 static ex factorial_evalf(const ex & x)
 800 {
 801         return factorial(x).hold();
 802 }
 803
 804 static ex factorial_eval(const ex & x)
 805 {
 806         if (is_exactly_a<numeric>(x))
 807                 return factorial(ex_to<numeric>(x));
 808         else
 809                 return factorial(x).hold();
 810 }
 811
 812 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 813 {
 814         if (is_exactly_a<symbol>(x) ||
 815             is_exactly_a<constant>(x) ||
 816                 is_exactly_a<function>(x)) {
 817                 x.print(c); c.s << "!";
 818         } else {
 819                 c.s << "("; x.print(c); c.s << ")!";
 820         }
 821 }
 822
 823 static ex factorial_conjugate(const ex & x)
 824 {
 825         return factorial(x).hold();
 826 }
 827
 828 static ex factorial_real_part(const ex & x)
 829 {
 830         return factorial(x).hold();
 831 }
 832
 833 static ex factorial_imag_part(const ex & x)
 834 {
 835         return 0;
 836 }
 837
 838 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 839                              evalf_func(factorial_evalf).
 840                              print_func<print_dflt>(factorial_print_dflt_latex).
 841                              print_func<print_latex>(factorial_print_dflt_latex).
 842                              conjugate_func(factorial_conjugate).
 843                              real_part_func(factorial_real_part).
 844                              imag_part_func(factorial_imag_part));
 845
 846 //////////
 847 // binomial
 848 //////////
 849
 850 static ex binomial_evalf(const ex & x, const ex & y)
 851 {
 852         return binomial(x, y).hold();
 853 }
 854
 855 static ex binomial_sym(const ex & x, const numeric & y)
 856 {
 857         if (y.is_integer()) {
 858                 if (y.is_nonneg_integer()) {
 859                         const unsigned N = y.to_int();
 860                         if (N == 0) return _ex1;
 861                         if (N == 1) return x;
 862                         ex t = x.expand();
 863                         for (unsigned i = 2; i <= N; ++i)
 864                                 t = (t * (x + i - y - 1)).expand() / i;
 865                         return t;
 866                 } else
 867                         return _ex0;
 868         }
 869
 870         return binomial(x, y).hold();
 871 }
 872
 873 static ex binomial_eval(const ex & x, const ex &y)
 874 {
 875         if (is_exactly_a<numeric>(y)) {
 876                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 877                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 878                 else
 879                         return binomial_sym(x, ex_to<numeric>(y));
 880         } else
 881                 return binomial(x, y).hold();
 882 }
 883
 884 // At the moment the numeric evaluation of a binomail function always
 885 // gives a real number, but if this would be implemented using the gamma
 886 // function, also complex conjugation should be changed (or rather, deleted).
 887 static ex binomial_conjugate(const ex & x, const ex & y)
 888 {
 889         return binomial(x,y).hold();
 890 }
 891
 892 static ex binomial_real_part(const ex & x, const ex & y)
 893 {
 894         return binomial(x,y).hold();
 895 }
 896
 897 static ex binomial_imag_part(const ex & x, const ex & y)
 898 {
 899         return 0;
 900 }
 901
 902 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 903                             evalf_func(binomial_evalf).
 904                             conjugate_func(binomial_conjugate).
 905                             real_part_func(binomial_real_part).
 906                             imag_part_func(binomial_imag_part));
 907
 908 //////////
 909 // Order term function (for truncated power series)
 910 //////////
 911
 912 static ex Order_eval(const ex & x)
 913 {
 914         if (is_exactly_a<numeric>(x)) {
 915                 // O(c) -> O(1) or 0
 916                 if (!x.is_zero())
 917                         return Order(_ex1).hold();
 918                 else
 919                         return _ex0;
 920         } else if (is_exactly_a<mul>(x)) {
 921                 const mul &m = ex_to<mul>(x);
 922                 // O(c*expr) -> O(expr)
 923                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 924                         return Order(x / m.op(m.nops() - 1)).hold();
 925         }
 926         return Order(x).hold();
 927 }
 928
 929 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 930 {
 931         // Just wrap the function into a pseries object
 932         epvector new_seq;
 933         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 934         const symbol &s = ex_to<symbol>(r.lhs());
 935         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 936         return pseries(r, new_seq);
 937 }
 938
 939 static ex Order_conjugate(const ex & x)
 940 {
 941         return Order(x).hold();
 942 }
 943
 944 static ex Order_real_part(const ex & x)
 945 {
 946         return Order(x).hold();
 947 }
 948
 949 static ex Order_imag_part(const ex & x)
 950 {
 951         if(x.info(info_flags::real))
 952                 return 0;
 953         return Order(x).hold();
 954 }
 955
 956 // Differentiation is handled in function::derivative because of its special requirements
 957
 958 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 959                          series_func(Order_series).
 960                          latex_name("\\mathcal{O}").
 961                          conjugate_func(Order_conjugate).
 962                          real_part_func(Order_real_part).
 963                          imag_part_func(Order_imag_part));
 964
 965 //////////
 966 // Solve linear system
 967 //////////
 968
 969 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 970 {
 971         // solve a system of linear equations
 972         if (eqns.info(info_flags::relation_equal)) {
 973                 if (!symbols.info(info_flags::symbol))
 974                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 975                 const ex sol = lsolve(lst(eqns),lst(symbols));
 976
 977                 GINAC_ASSERT(sol.nops()==1);
 978                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 979
 980                 return sol.op(0).op(1); // return rhs of first solution
 981         }
 982
 983         // syntax checks
 984         if (!eqns.info(info_flags::list)) {
 985                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
 986         }
 987         for (size_t i=0; i<eqns.nops(); i++) {
 988                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 989                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 990                 }
 991         }
 992         if (!symbols.info(info_flags::list)) {
 993                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
 994         }
 995         for (size_t i=0; i<symbols.nops(); i++) {
 996                 if (!symbols.op(i).info(info_flags::symbol)) {
 997                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 998                 }
 999         }
1000
1001         // build matrix from equation system
1002         matrix sys(eqns.nops(),symbols.nops());
1003         matrix rhs(eqns.nops(),1);
1004         matrix vars(symbols.nops(),1);
1005
1006         for (size_t r=0; r<eqns.nops(); r++) {
1007                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1008                 ex linpart = eq;
1009                 for (size_t c=0; c<symbols.nops(); c++) {
1010                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1011                         linpart -= co*symbols.op(c);
1012                         sys(r,c) = co;
1013                 }
1014                 linpart = linpart.expand();
1015                 rhs(r,0) = -linpart;
1016         }
1017
1018         // test if system is linear and fill vars matrix
1019         for (size_t i=0; i<symbols.nops(); i++) {
1020                 vars(i,0) = symbols.op(i);
1021                 if (sys.has(symbols.op(i)))
1022                         throw(std::logic_error("lsolve: system is not linear"));
1023                 if (rhs.has(symbols.op(i)))
1024                         throw(std::logic_error("lsolve: system is not linear"));
1025         }
1026
1027         matrix solution;
1028         try {
1029                 solution = sys.solve(vars,rhs,options);
1030         } catch (const std::runtime_error & e) {
1031                 // Probably singular matrix or otherwise overdetermined system:
1032                 // It is consistent to return an empty list
1033                 return lst();
1034         }
1035         GINAC_ASSERT(solution.cols()==1);
1036         GINAC_ASSERT(solution.rows()==symbols.nops());
1037
1038         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1039         lst sollist;
1040         for (size_t i=0; i<symbols.nops(); i++)
1041                 sollist.append(symbols.op(i)==solution(i,0));
1042
1043         return sollist;
1044 }
1045
1046 //////////
1047 // Find real root of f(x) numerically
1048 //////////
1049
1050 const numeric
1051 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1052 {
1053         if (!x1.is_real() || !x2.is_real()) {
1054                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1055         }
1056         if (x1==x2) {
1057                 throw std::runtime_error("fsolve(): vanishing interval");
1058         }
1059         // xx[0] == left interval limit, xx[1] == right interval limit.
1060         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1061         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1062         numeric xx[2] = { x1<x2 ? x1 : x2,
1063                           x1<x2 ? x2 : x1 };
1064         ex f;
1065         if (is_a<relational>(f_in)) {
1066                 f = f_in.lhs()-f_in.rhs();
1067         } else {
1068                 f = f_in;
1069         }
1070         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1071                             f.subs(x==xx[1]).evalf() };
1072         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1073                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1074         }
1075         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1076                           ex_to<numeric>(fx_[1]) };
1077         if (!fx[0].is_real() || !fx[1].is_real()) {
1078                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1079         }
1080         if (fx[0]*fx[1]>=0) {
1081                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1082         }
1083
1084         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1085         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1086         const ex ff = normal(-f/f.diff(x));
1087         int side = 0;  // Start at left interval limit.
1088         numeric xxprev;
1089         numeric fxprev;
1090         do {
1091                 xxprev = xx[side];
1092                 fxprev = fx[side];
1093                 ex dx_ = ff.subs(x == xx[side]).evalf();
1094                 if (!is_a<numeric>(dx_))
1095                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1096                 xx[side] += ex_to<numeric>(dx_);
1097                 // Now check if Newton-Raphson method shot out of the interval
1098                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1099                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1100                 if (!bad_shot) {
1101                         // Compute f(x) only if new x is inside the interval.
1102                         // The function might be difficult to compute numerically
1103                         // or even ill defined outside the interval. Also it's
1104                         // a small optimization.
1105                         ex f_x = f.subs(x == xx[side]).evalf();
1106                         if (!is_a<numeric>(f_x))
1107                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1108                         fx[side] = ex_to<numeric>(f_x);
1109                 }
1110                 if (bad_shot) {
1111                         // Oops, Newton-Raphson method shot out of the interval.
1112                         // Restore, and try again with the other side instead!
1113                         xx[side] = xxprev;
1114                         fx[side] = fxprev;
1115                         side = !side;
1116                         xxprev = xx[side];
1117                         fxprev = fx[side];
1118
1119                         ex dx_ = ff.subs(x == xx[side]).evalf();
1120                         if (!is_a<numeric>(dx_))
1121                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1122                         xx[side] += ex_to<numeric>(dx_);
1123
1124                         ex f_x = f.subs(x==xx[side]).evalf();
1125                         if (!is_a<numeric>(f_x))
1126                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1127                         fx[side] = ex_to<numeric>(f_x);
1128                 }
1129                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1130                         // Oops, the root isn't bracketed any more.
1131                         // Restore, and perform a bisection!
1132                         xx[side] = xxprev;
1133                         fx[side] = fxprev;
1134
1135                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1136                         // they occur pretty often when Newton-Raphson arrives at an x too
1137                         // close to the result on one side of the interval and
1138                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1139                         // precision errors! Recall that this function does not have a
1140                         // precision goal as one of its arguments but instead relies on
1141                         // x converging to a fixed point. We speed up the (safe but slow)
1142                         // bisection method by mixing in a dash of the (unsafer but faster)
1143                         // secant method: Instead of splitting the interval at the
1144                         // arithmetic mean (bisection), we split it nearer to the root as
1145                         // determined by the secant between the values xx[0] and xx[1].
1146                         // Don't set the secant_weight to one because that could disturb
1147                         // the convergence in some corner cases!
1148                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1149                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1150                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1151                         ex fxmid_ = f.subs(x == xxmid).evalf();
1152                         if (!is_a<numeric>(fxmid_))
1153                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1154                         numeric fxmid = ex_to<numeric>(fxmid_);
1155                         if (fxmid.is_zero()) {
1156                                 // Luck strikes...
1157                                 return xxmid;
1158                         }
1159                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1160                                 side = !side;
1161                         }
1162                         xxprev = xx[side];
1163                         fxprev = fx[side];
1164                         xx[side] = xxmid;
1165                         fx[side] = fxmid;
1166                 }
1167         } while (xxprev!=xx[side]);
1168         return xxprev;
1169 }
1170
1171
1172 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1173  * for static lib (so ginsh will see them). */
1174 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1175 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1176
1177 } // namespace GiNaC