ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include "inifcns.h"
  24 #include "ex.h"
  25 #include "constant.h"
  26 #include "lst.h"
  27 #include "fderivative.h"
  28 #include "matrix.h"
  29 #include "mul.h"
  30 #include "power.h"
  31 #include "operators.h"
  32 #include "relational.h"
  33 #include "pseries.h"
  34 #include "symbol.h"
  35 #include "symmetry.h"
  36 #include "utils.h"
  37
  38 #include <stdexcept>
  39 #include <vector>
  40
  41 namespace GiNaC {
  42
  43 //////////
  44 // complex conjugate
  45 //////////
  46
  47 static ex conjugate_evalf(const ex & arg)
  48 {
  49         if (is_exactly_a<numeric>(arg)) {
  50                 return ex_to<numeric>(arg).conjugate();
  51         }
  52         return conjugate_function(arg).hold();
  53 }
  54
  55 static ex conjugate_eval(const ex & arg)
  56 {
  57         return arg.conjugate();
  58 }
  59
  60 static void conjugate_print_latex(const ex & arg, const print_context & c)
  61 {
  62         c.s << "\\bar{"; arg.print(c); c.s << "}";
  63 }
  64
  65 static ex conjugate_conjugate(const ex & arg)
  66 {
  67         return arg;
  68 }
  69
  70 // If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x)
  71 // and conjugate((U+I*V).diff(x))
  72 static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
  73 {
  74         if (s.info(info_flags::real))
  75                 return conjugate(arg.diff(s));
  76         else {
  77                 exvector vec_arg;
  78                 vec_arg.push_back(arg);
  79                 return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
  80         }
  81 }
  82
  83 static ex conjugate_real_part(const ex & arg)
  84 {
  85         return arg.real_part();
  86 }
  87
  88 static ex conjugate_imag_part(const ex & arg)
  89 {
  90         return -arg.imag_part();
  91 }
  92
  93 static bool func_arg_info(const ex & arg, unsigned inf)
  94 {
  95         // for some functions we can return the info() of its argument
  96         // (think of conjugate())
  97         switch (inf) {
  98                 case info_flags::polynomial:
  99                 case info_flags::integer_polynomial:
 100                 case info_flags::cinteger_polynomial:
 101                 case info_flags::rational_polynomial:
 102                 case info_flags::real:
 103                 case info_flags::rational:
 104                 case info_flags::integer:
 105                 case info_flags::crational:
 106                 case info_flags::cinteger:
 107                 case info_flags::even:
 108                 case info_flags::odd:
 109                 case info_flags::prime:
 110                 case info_flags::crational_polynomial:
 111                 case info_flags::rational_function:
 112                 case info_flags::algebraic:
 113                 case info_flags::positive:
 114                 case info_flags::negative:
 115                 case info_flags::nonnegative:
 116                 case info_flags::posint:
 117                 case info_flags::negint:
 118                 case info_flags::nonnegint:
 119                 case info_flags::has_indices:
 120                         return arg.info(inf);
 121         }
 122         return false;
 123 }
 124
 125 static bool conjugate_info(const ex & arg, unsigned inf)
 126 {
 127         return func_arg_info(arg, inf);
 128 }
 129
 130 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
 131                                       evalf_func(conjugate_evalf).
 132                                       expl_derivative_func(conjugate_expl_derivative).
 133                                       info_func(conjugate_info).
 134                                       print_func<print_latex>(conjugate_print_latex).
 135                                       conjugate_func(conjugate_conjugate).
 136                                       real_part_func(conjugate_real_part).
 137                                       imag_part_func(conjugate_imag_part).
 138                                       set_name("conjugate","conjugate"));
 139
 140 //////////
 141 // real part
 142 //////////
 143
 144 static ex real_part_evalf(const ex & arg)
 145 {
 146         if (is_exactly_a<numeric>(arg)) {
 147                 return ex_to<numeric>(arg).real();
 148         }
 149         return real_part_function(arg).hold();
 150 }
 151
 152 static ex real_part_eval(const ex & arg)
 153 {
 154         return arg.real_part();
 155 }
 156
 157 static void real_part_print_latex(const ex & arg, const print_context & c)
 158 {
 159         c.s << "\\Re"; arg.print(c); c.s << "";
 160 }
 161
 162 static ex real_part_conjugate(const ex & arg)
 163 {
 164         return real_part_function(arg).hold();
 165 }
 166
 167 static ex real_part_real_part(const ex & arg)
 168 {
 169         return real_part_function(arg).hold();
 170 }
 171
 172 static ex real_part_imag_part(const ex & arg)
 173 {
 174         return 0;
 175 }
 176
 177 // If x is real then Re(e).diff(x) is equal to Re(e.diff(x))
 178 static ex real_part_expl_derivative(const ex & arg, const symbol & s)
 179 {
 180         if (s.info(info_flags::real))
 181                 return real_part_function(arg.diff(s));
 182         else {
 183                 exvector vec_arg;
 184                 vec_arg.push_back(arg);
 185                 return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
 186         }
 187 }
 188
 189 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
 190                                       evalf_func(real_part_evalf).
 191                                       expl_derivative_func(real_part_expl_derivative).
 192                                       print_func<print_latex>(real_part_print_latex).
 193                                       conjugate_func(real_part_conjugate).
 194                                       real_part_func(real_part_real_part).
 195                                       imag_part_func(real_part_imag_part).
 196                                       set_name("real_part","real_part"));
 197
 198 //////////
 199 // imag part
 200 //////////
 201
 202 static ex imag_part_evalf(const ex & arg)
 203 {
 204         if (is_exactly_a<numeric>(arg)) {
 205                 return ex_to<numeric>(arg).imag();
 206         }
 207         return imag_part_function(arg).hold();
 208 }
 209
 210 static ex imag_part_eval(const ex & arg)
 211 {
 212         return arg.imag_part();
 213 }
 214
 215 static void imag_part_print_latex(const ex & arg, const print_context & c)
 216 {
 217         c.s << "\\Im"; arg.print(c); c.s << "";
 218 }
 219
 220 static ex imag_part_conjugate(const ex & arg)
 221 {
 222         return imag_part_function(arg).hold();
 223 }
 224
 225 static ex imag_part_real_part(const ex & arg)
 226 {
 227         return imag_part_function(arg).hold();
 228 }
 229
 230 static ex imag_part_imag_part(const ex & arg)
 231 {
 232         return 0;
 233 }
 234
 235 // If x is real then Im(e).diff(x) is equal to Im(e.diff(x))
 236 static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
 237 {
 238         if (s.info(info_flags::real))
 239                 return imag_part_function(arg.diff(s));
 240         else {
 241                 exvector vec_arg;
 242                 vec_arg.push_back(arg);
 243                 return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
 244         }
 245 }
 246
 247 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
 248                                       evalf_func(imag_part_evalf).
 249                                       expl_derivative_func(imag_part_expl_derivative).
 250                                       print_func<print_latex>(imag_part_print_latex).
 251                                       conjugate_func(imag_part_conjugate).
 252                                       real_part_func(imag_part_real_part).
 253                                       imag_part_func(imag_part_imag_part).
 254                                       set_name("imag_part","imag_part"));
 255
 256 //////////
 257 // absolute value
 258 //////////
 259
 260 static ex abs_evalf(const ex & arg)
 261 {
 262         if (is_exactly_a<numeric>(arg))
 263                 return abs(ex_to<numeric>(arg));
 264
 265         return abs(arg).hold();
 266 }
 267
 268 static ex abs_eval(const ex & arg)
 269 {
 270         if (is_exactly_a<numeric>(arg))
 271                 return abs(ex_to<numeric>(arg));
 272
 273         if (arg.info(info_flags::nonnegative))
 274                 return arg;
 275
 276         if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
 277                 return -arg;
 278
 279         if (is_ex_the_function(arg, abs))
 280                 return arg;
 281
 282         if (is_ex_the_function(arg, exp))
 283                 return exp(arg.op(0).real_part());
 284
 285         if (is_exactly_a<power>(arg)) {
 286                 const ex& base = arg.op(0);
 287                 const ex& exponent = arg.op(1);
 288                 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
 289                         return pow(abs(base), exponent.real_part());
 290         }
 291
 292         if (is_ex_the_function(arg, conjugate_function))
 293                 return abs(arg.op(0));
 294
 295         if (is_ex_the_function(arg, step))
 296                 return arg;
 297
 298         return abs(arg).hold();
 299 }
 300
 301 static ex abs_expand(const ex & arg, unsigned options)
 302 {
 303         if ((options & expand_options::expand_transcendental)
 304                 && is_exactly_a<mul>(arg)) {
 305                 exvector prodseq;
 306                 prodseq.reserve(arg.nops());
 307                 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
 308                         if (options & expand_options::expand_function_args)
 309                                 prodseq.push_back(abs(i->expand(options)));
 310                         else
 311                                 prodseq.push_back(abs(*i));
 312                 }
 313                 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
 314         }
 315
 316         if (options & expand_options::expand_function_args)
 317                 return abs(arg.expand(options)).hold();
 318         else
 319                 return abs(arg).hold();
 320 }
 321
 322 static ex abs_expl_derivative(const ex & arg, const symbol & s)
 323 {
 324         ex diff_arg = arg.diff(s);
 325         return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
 326 }
 327
 328 static void abs_print_latex(const ex & arg, const print_context & c)
 329 {
 330         c.s << "{|"; arg.print(c); c.s << "|}";
 331 }
 332
 333 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 334 {
 335         c.s << "fabs("; arg.print(c); c.s << ")";
 336 }
 337
 338 static ex abs_conjugate(const ex & arg)
 339 {
 340         return abs(arg).hold();
 341 }
 342
 343 static ex abs_real_part(const ex & arg)
 344 {
 345         return abs(arg).hold();
 346 }
 347
 348 static ex abs_imag_part(const ex& arg)
 349 {
 350         return 0;
 351 }
 352
 353 static ex abs_power(const ex & arg, const ex & exp)
 354 {
 355         if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
 356                 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
 357                         return power(arg, exp);
 358                 else
 359                         return power(arg, exp/2)*power(arg.conjugate(), exp/2);
 360         } else
 361                 return power(abs(arg), exp).hold();
 362 }
 363
 364 bool abs_info(const ex & arg, unsigned inf)
 365 {
 366         switch (inf) {
 367                 case info_flags::integer:
 368                 case info_flags::even:
 369                 case info_flags::odd:
 370                 case info_flags::prime:
 371                         return arg.info(inf);
 372                 case info_flags::nonnegint:
 373                         return arg.info(info_flags::integer);
 374                 case info_flags::nonnegative:
 375                 case info_flags::real:
 376                         return true;
 377                 case info_flags::negative:
 378                         return false;
 379                 case info_flags::positive:
 380                         return arg.info(info_flags::positive) || arg.info(info_flags::negative);
 381                 case info_flags::has_indices: {
 382                         if (arg.info(info_flags::has_indices))
 383                                 return true;
 384                         else
 385                                 return false;
 386                 }
 387         }
 388         return false;
 389 }
 390
 391 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 392                        evalf_func(abs_evalf).
 393                        expand_func(abs_expand).
 394                        expl_derivative_func(abs_expl_derivative).
 395                        info_func(abs_info).
 396                        print_func<print_latex>(abs_print_latex).
 397                        print_func<print_csrc_float>(abs_print_csrc_float).
 398                        print_func<print_csrc_double>(abs_print_csrc_float).
 399                        conjugate_func(abs_conjugate).
 400                        real_part_func(abs_real_part).
 401                        imag_part_func(abs_imag_part).
 402                        power_func(abs_power));
 403
 404 //////////
 405 // Step function
 406 //////////
 407
 408 static ex step_evalf(const ex & arg)
 409 {
 410         if (is_exactly_a<numeric>(arg))
 411                 return step(ex_to<numeric>(arg));
 412
 413         return step(arg).hold();
 414 }
 415
 416 static ex step_eval(const ex & arg)
 417 {
 418         if (is_exactly_a<numeric>(arg))
 419                 return step(ex_to<numeric>(arg));
 420
 421         else if (is_exactly_a<mul>(arg) &&
 422                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 423                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 424                 if (oc.is_real()) {
 425                         if (oc > 0)
 426                                 // step(42*x) -> step(x)
 427                                 return step(arg/oc).hold();
 428                         else
 429                                 // step(-42*x) -> step(-x)
 430                                 return step(-arg/oc).hold();
 431                 }
 432                 if (oc.real().is_zero()) {
 433                         if (oc.imag() > 0)
 434                                 // step(42*I*x) -> step(I*x)
 435                                 return step(I*arg/oc).hold();
 436                         else
 437                                 // step(-42*I*x) -> step(-I*x)
 438                                 return step(-I*arg/oc).hold();
 439                 }
 440         }
 441
 442         return step(arg).hold();
 443 }
 444
 445 static ex step_series(const ex & arg,
 446                       const relational & rel,
 447                       int order,
 448                       unsigned options)
 449 {
 450         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 451         if (arg_pt.info(info_flags::numeric)
 452             && ex_to<numeric>(arg_pt).real().is_zero()
 453             && !(options & series_options::suppress_branchcut))
 454                 throw (std::domain_error("step_series(): on imaginary axis"));
 455
 456         epvector seq;
 457         seq.push_back(expair(step(arg_pt), _ex0));
 458         return pseries(rel,seq);
 459 }
 460
 461 static ex step_conjugate(const ex& arg)
 462 {
 463         return step(arg).hold();
 464 }
 465
 466 static ex step_real_part(const ex& arg)
 467 {
 468         return step(arg).hold();
 469 }
 470
 471 static ex step_imag_part(const ex& arg)
 472 {
 473         return 0;
 474 }
 475
 476 REGISTER_FUNCTION(step, eval_func(step_eval).
 477                         evalf_func(step_evalf).
 478                         series_func(step_series).
 479                         conjugate_func(step_conjugate).
 480                         real_part_func(step_real_part).
 481                         imag_part_func(step_imag_part));
 482
 483 //////////
 484 // Complex sign
 485 //////////
 486
 487 static ex csgn_evalf(const ex & arg)
 488 {
 489         if (is_exactly_a<numeric>(arg))
 490                 return csgn(ex_to<numeric>(arg));
 491
 492         return csgn(arg).hold();
 493 }
 494
 495 static ex csgn_eval(const ex & arg)
 496 {
 497         if (is_exactly_a<numeric>(arg))
 498                 return csgn(ex_to<numeric>(arg));
 499
 500         else if (is_exactly_a<mul>(arg) &&
 501                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 502                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 503                 if (oc.is_real()) {
 504                         if (oc > 0)
 505                                 // csgn(42*x) -> csgn(x)
 506                                 return csgn(arg/oc).hold();
 507                         else
 508                                 // csgn(-42*x) -> -csgn(x)
 509                                 return -csgn(arg/oc).hold();
 510                 }
 511                 if (oc.real().is_zero()) {
 512                         if (oc.imag() > 0)
 513                                 // csgn(42*I*x) -> csgn(I*x)
 514                                 return csgn(I*arg/oc).hold();
 515                         else
 516                                 // csgn(-42*I*x) -> -csgn(I*x)
 517                                 return -csgn(I*arg/oc).hold();
 518                 }
 519         }
 520
 521         return csgn(arg).hold();
 522 }
 523
 524 static ex csgn_series(const ex & arg,
 525                       const relational & rel,
 526                       int order,
 527                       unsigned options)
 528 {
 529         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 530         if (arg_pt.info(info_flags::numeric)
 531             && ex_to<numeric>(arg_pt).real().is_zero()
 532             && !(options & series_options::suppress_branchcut))
 533                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 534
 535         epvector seq;
 536         seq.push_back(expair(csgn(arg_pt), _ex0));
 537         return pseries(rel,seq);
 538 }
 539
 540 static ex csgn_conjugate(const ex& arg)
 541 {
 542         return csgn(arg).hold();
 543 }
 544
 545 static ex csgn_real_part(const ex& arg)
 546 {
 547         return csgn(arg).hold();
 548 }
 549
 550 static ex csgn_imag_part(const ex& arg)
 551 {
 552         return 0;
 553 }
 554
 555 static ex csgn_power(const ex & arg, const ex & exp)
 556 {
 557         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 558                 if (ex_to<numeric>(exp).is_odd())
 559                         return csgn(arg).hold();
 560                 else
 561                         return power(csgn(arg), _ex2).hold();
 562         } else
 563                 return power(csgn(arg), exp).hold();
 564 }
 565
 566
 567 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 568                         evalf_func(csgn_evalf).
 569                         series_func(csgn_series).
 570                         conjugate_func(csgn_conjugate).
 571                         real_part_func(csgn_real_part).
 572                         imag_part_func(csgn_imag_part).
 573                         power_func(csgn_power));
 574
 575
 576 //////////
 577 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 578 // This function is closely related to the unwinding number K, sometimes found
 579 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 580 //////////
 581
 582 static ex eta_evalf(const ex &x, const ex &y)
 583 {
 584         // It seems like we basically have to replicate the eval function here,
 585         // since the expression might not be fully evaluated yet.
 586         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 587                 return _ex0;
 588
 589         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 590                 const numeric nx = ex_to<numeric>(x);
 591                 const numeric ny = ex_to<numeric>(y);
 592                 const numeric nxy = ex_to<numeric>(x*y);
 593                 int cut = 0;
 594                 if (nx.is_real() && nx.is_negative())
 595                         cut -= 4;
 596                 if (ny.is_real() && ny.is_negative())
 597                         cut -= 4;
 598                 if (nxy.is_real() && nxy.is_negative())
 599                         cut += 4;
 600                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 601                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 602         }
 603
 604         return eta(x,y).hold();
 605 }
 606
 607 static ex eta_eval(const ex &x, const ex &y)
 608 {
 609         // trivial:  eta(x,c) -> 0  if c is real and positive
 610         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 611                 return _ex0;
 612
 613         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 614                 // don't call eta_evalf here because it would call Pi.evalf()!
 615                 const numeric nx = ex_to<numeric>(x);
 616                 const numeric ny = ex_to<numeric>(y);
 617                 const numeric nxy = ex_to<numeric>(x*y);
 618                 int cut = 0;
 619                 if (nx.is_real() && nx.is_negative())
 620                         cut -= 4;
 621                 if (ny.is_real() && ny.is_negative())
 622                         cut -= 4;
 623                 if (nxy.is_real() && nxy.is_negative())
 624                         cut += 4;
 625                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 626                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 627         }
 628
 629         return eta(x,y).hold();
 630 }
 631
 632 static ex eta_series(const ex & x, const ex & y,
 633                      const relational & rel,
 634                      int order,
 635                      unsigned options)
 636 {
 637         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 638         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 639         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 640             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 641             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 642                         throw (std::domain_error("eta_series(): on discontinuity"));
 643         epvector seq;
 644         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 645         return pseries(rel,seq);
 646 }
 647
 648 static ex eta_conjugate(const ex & x, const ex & y)
 649 {
 650         return -eta(x, y).hold();
 651 }
 652
 653 static ex eta_real_part(const ex & x, const ex & y)
 654 {
 655         return 0;
 656 }
 657
 658 static ex eta_imag_part(const ex & x, const ex & y)
 659 {
 660         return -I*eta(x, y).hold();
 661 }
 662
 663 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 664                        evalf_func(eta_evalf).
 665                        series_func(eta_series).
 666                        latex_name("\\eta").
 667                        set_symmetry(sy_symm(0, 1)).
 668                        conjugate_func(eta_conjugate).
 669                        real_part_func(eta_real_part).
 670                        imag_part_func(eta_imag_part));
 671
 672
 673 //////////
 674 // dilogarithm
 675 //////////
 676
 677 static ex Li2_evalf(const ex & x)
 678 {
 679         if (is_exactly_a<numeric>(x))
 680                 return Li2(ex_to<numeric>(x));
 681
 682         return Li2(x).hold();
 683 }
 684
 685 static ex Li2_eval(const ex & x)
 686 {
 687         if (x.info(info_flags::numeric)) {
 688                 // Li2(0) -> 0
 689                 if (x.is_zero())
 690                         return _ex0;
 691                 // Li2(1) -> Pi^2/6
 692                 if (x.is_equal(_ex1))
 693                         return power(Pi,_ex2)/_ex6;
 694                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 695                 if (x.is_equal(_ex1_2))
 696                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 697                 // Li2(-1) -> -Pi^2/12
 698                 if (x.is_equal(_ex_1))
 699                         return -power(Pi,_ex2)/_ex12;
 700                 // Li2(I) -> -Pi^2/48+Catalan*I
 701                 if (x.is_equal(I))
 702                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 703                 // Li2(-I) -> -Pi^2/48-Catalan*I
 704                 if (x.is_equal(-I))
 705                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 706                 // Li2(float)
 707                 if (!x.info(info_flags::crational))
 708                         return Li2(ex_to<numeric>(x));
 709         }
 710
 711         return Li2(x).hold();
 712 }
 713
 714 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 715 {
 716         GINAC_ASSERT(deriv_param==0);
 717
 718         // d/dx Li2(x) -> -log(1-x)/x
 719         return -log(_ex1-x)/x;
 720 }
 721
 722 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 723 {
 724         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 725         if (x_pt.info(info_flags::numeric)) {
 726                 // First special case: x==0 (derivatives have poles)
 727                 if (x_pt.is_zero()) {
 728                         // method:
 729                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 730                         // simply substitute x==0.  The limit, however, exists: it is 1.
 731                         // We also know all higher derivatives' limits:
 732                         // (d/dx)^n Li2(x) == n!/n^2.
 733                         // So the primitive series expansion is
 734                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 735                         // and so on.
 736                         // We first construct such a primitive series expansion manually in
 737                         // a dummy symbol s and then insert the argument's series expansion
 738                         // for s.  Reexpanding the resulting series returns the desired
 739                         // result.
 740                         const symbol s;
 741                         ex ser;
 742                         // manually construct the primitive expansion
 743                         for (int i=1; i<order; ++i)
 744                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 745                         // substitute the argument's series expansion
 746                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 747                         // maybe that was terminating, so add a proper order term
 748                         epvector nseq;
 749                         nseq.push_back(expair(Order(_ex1), order));
 750                         ser += pseries(rel, nseq);
 751                         // reexpanding it will collapse the series again
 752                         return ser.series(rel, order);
 753                         // NB: Of course, this still does not allow us to compute anything
 754                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 755                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 756                         // substitution x==0.  Probably limits *are* needed for the general
 757                         // cases.  In case L'Hospital's rule is implemented for limits and
 758                         // basic::series() takes care of this, this whole block is probably
 759                         // obsolete!
 760                 }
 761                 // second special case: x==1 (branch point)
 762                 if (x_pt.is_equal(_ex1)) {
 763                         // method:
 764                         // construct series manually in a dummy symbol s
 765                         const symbol s;
 766                         ex ser = zeta(_ex2);
 767                         // manually construct the primitive expansion
 768                         for (int i=1; i<order; ++i)
 769                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 770                         // substitute the argument's series expansion
 771                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 772                         // maybe that was terminating, so add a proper order term
 773                         epvector nseq;
 774                         nseq.push_back(expair(Order(_ex1), order));
 775                         ser += pseries(rel, nseq);
 776                         // reexpanding it will collapse the series again
 777                         return ser.series(rel, order);
 778                 }
 779                 // third special case: x real, >=1 (branch cut)
 780                 if (!(options & series_options::suppress_branchcut) &&
 781                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 782                         // method:
 783                         // This is the branch cut: assemble the primitive series manually
 784                         // and then add the corresponding complex step function.
 785                         const symbol &s = ex_to<symbol>(rel.lhs());
 786                         const ex point = rel.rhs();
 787                         const symbol foo;
 788                         epvector seq;
 789                         // zeroth order term:
 790                         seq.push_back(expair(Li2(x_pt), _ex0));
 791                         // compute the intermediate terms:
 792                         ex replarg = series(Li2(x), s==foo, order);
 793                         for (size_t i=1; i<replarg.nops()-1; ++i)
 794                                 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));
 795                         // append an order term:
 796                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 797                         return pseries(rel, seq);
 798                 }
 799         }
 800         // all other cases should be safe, by now:
 801         throw do_taylor();  // caught by function::series()
 802 }
 803
 804 static ex Li2_conjugate(const ex & x)
 805 {
 806         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 807         // run along the positive real axis beginning at 1.
 808         if (x.info(info_flags::negative)) {
 809                 return Li2(x).hold();
 810         }
 811         if (is_exactly_a<numeric>(x) &&
 812             (!x.imag_part().is_zero() || x < *_num1_p)) {
 813                 return Li2(x.conjugate());
 814         }
 815         return conjugate_function(Li2(x)).hold();
 816 }
 817
 818 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 819                        evalf_func(Li2_evalf).
 820                        derivative_func(Li2_deriv).
 821                        series_func(Li2_series).
 822                        conjugate_func(Li2_conjugate).
 823                        latex_name("\\mathrm{Li}_2"));
 824
 825 //////////
 826 // trilogarithm
 827 //////////
 828
 829 static ex Li3_eval(const ex & x)
 830 {
 831         if (x.is_zero())
 832                 return x;
 833         return Li3(x).hold();
 834 }
 835
 836 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 837                        latex_name("\\mathrm{Li}_3"));
 838
 839 //////////
 840 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 841 //////////
 842
 843 static ex zetaderiv_eval(const ex & n, const ex & x)
 844 {
 845         if (n.info(info_flags::numeric)) {
 846                 // zetaderiv(0,x) -> zeta(x)
 847                 if (n.is_zero())
 848                         return zeta(x).hold();
 849         }
 850
 851         return zetaderiv(n, x).hold();
 852 }
 853
 854 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 855 {
 856         GINAC_ASSERT(deriv_param<2);
 857
 858         if (deriv_param==0) {
 859                 // d/dn zeta(n,x)
 860                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 861         }
 862         // d/dx psi(n,x)
 863         return zetaderiv(n+1,x);
 864 }
 865
 866 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 867                                  derivative_func(zetaderiv_deriv).
 868                                  latex_name("\\zeta^\\prime"));
 869
 870 //////////
 871 // factorial
 872 //////////
 873
 874 static ex factorial_evalf(const ex & x)
 875 {
 876         return factorial(x).hold();
 877 }
 878
 879 static ex factorial_eval(const ex & x)
 880 {
 881         if (is_exactly_a<numeric>(x))
 882                 return factorial(ex_to<numeric>(x));
 883         else
 884                 return factorial(x).hold();
 885 }
 886
 887 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 888 {
 889         if (is_exactly_a<symbol>(x) ||
 890             is_exactly_a<constant>(x) ||
 891                 is_exactly_a<function>(x)) {
 892                 x.print(c); c.s << "!";
 893         } else {
 894                 c.s << "("; x.print(c); c.s << ")!";
 895         }
 896 }
 897
 898 static ex factorial_conjugate(const ex & x)
 899 {
 900         return factorial(x).hold();
 901 }
 902
 903 static ex factorial_real_part(const ex & x)
 904 {
 905         return factorial(x).hold();
 906 }
 907
 908 static ex factorial_imag_part(const ex & x)
 909 {
 910         return 0;
 911 }
 912
 913 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 914                              evalf_func(factorial_evalf).
 915                              print_func<print_dflt>(factorial_print_dflt_latex).
 916                              print_func<print_latex>(factorial_print_dflt_latex).
 917                              conjugate_func(factorial_conjugate).
 918                              real_part_func(factorial_real_part).
 919                              imag_part_func(factorial_imag_part));
 920
 921 //////////
 922 // binomial
 923 //////////
 924
 925 static ex binomial_evalf(const ex & x, const ex & y)
 926 {
 927         return binomial(x, y).hold();
 928 }
 929
 930 static ex binomial_sym(const ex & x, const numeric & y)
 931 {
 932         if (y.is_integer()) {
 933                 if (y.is_nonneg_integer()) {
 934                         const unsigned N = y.to_int();
 935                         if (N == 0) return _ex1;
 936                         if (N == 1) return x;
 937                         ex t = x.expand();
 938                         for (unsigned i = 2; i <= N; ++i)
 939                                 t = (t * (x + i - y - 1)).expand() / i;
 940                         return t;
 941                 } else
 942                         return _ex0;
 943         }
 944
 945         return binomial(x, y).hold();
 946 }
 947
 948 static ex binomial_eval(const ex & x, const ex &y)
 949 {
 950         if (is_exactly_a<numeric>(y)) {
 951                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 952                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 953                 else
 954                         return binomial_sym(x, ex_to<numeric>(y));
 955         } else
 956                 return binomial(x, y).hold();
 957 }
 958
 959 // At the moment the numeric evaluation of a binomial function always
 960 // gives a real number, but if this would be implemented using the gamma
 961 // function, also complex conjugation should be changed (or rather, deleted).
 962 static ex binomial_conjugate(const ex & x, const ex & y)
 963 {
 964         return binomial(x,y).hold();
 965 }
 966
 967 static ex binomial_real_part(const ex & x, const ex & y)
 968 {
 969         return binomial(x,y).hold();
 970 }
 971
 972 static ex binomial_imag_part(const ex & x, const ex & y)
 973 {
 974         return 0;
 975 }
 976
 977 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 978                             evalf_func(binomial_evalf).
 979                             conjugate_func(binomial_conjugate).
 980                             real_part_func(binomial_real_part).
 981                             imag_part_func(binomial_imag_part));
 982
 983 //////////
 984 // Order term function (for truncated power series)
 985 //////////
 986
 987 static ex Order_eval(const ex & x)
 988 {
 989         if (is_exactly_a<numeric>(x)) {
 990                 // O(c) -> O(1) or 0
 991                 if (!x.is_zero())
 992                         return Order(_ex1).hold();
 993                 else
 994                         return _ex0;
 995         } else if (is_exactly_a<mul>(x)) {
 996                 const mul &m = ex_to<mul>(x);
 997                 // O(c*expr) -> O(expr)
 998                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 999                         return Order(x / m.op(m.nops() - 1)).hold();
1000         }
1001         return Order(x).hold();
1002 }
1003
1004 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
1005 {
1006         // Just wrap the function into a pseries object
1007         epvector new_seq;
1008         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1009         const symbol &s = ex_to<symbol>(r.lhs());
1010         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
1011         return pseries(r, new_seq);
1012 }
1013
1014 static ex Order_conjugate(const ex & x)
1015 {
1016         return Order(x).hold();
1017 }
1018
1019 static ex Order_real_part(const ex & x)
1020 {
1021         return Order(x).hold();
1022 }
1023
1024 static ex Order_imag_part(const ex & x)
1025 {
1026         if(x.info(info_flags::real))
1027                 return 0;
1028         return Order(x).hold();
1029 }
1030
1031 static ex Order_expl_derivative(const ex & arg, const symbol & s)
1032 {
1033         return Order(arg.diff(s));
1034 }
1035
1036 REGISTER_FUNCTION(Order, eval_func(Order_eval).
1037                          series_func(Order_series).
1038                          latex_name("\\mathcal{O}").
1039                          expl_derivative_func(Order_expl_derivative).
1040                          conjugate_func(Order_conjugate).
1041                          real_part_func(Order_real_part).
1042                          imag_part_func(Order_imag_part));
1043
1044 //////////
1045 // Solve linear system
1046 //////////
1047
1048 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1049 {
1050         // solve a system of linear equations
1051         if (eqns.info(info_flags::relation_equal)) {
1052                 if (!symbols.info(info_flags::symbol))
1053                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1054                 const ex sol = lsolve(lst(eqns),lst(symbols));
1055
1056                 GINAC_ASSERT(sol.nops()==1);
1057                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1058
1059                 return sol.op(0).op(1); // return rhs of first solution
1060         }
1061
1062         // syntax checks
1063         if (!eqns.info(info_flags::list)) {
1064                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1065         }
1066         for (size_t i=0; i<eqns.nops(); i++) {
1067                 if (!eqns.op(i).info(info_flags::relation_equal)) {
1068                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1069                 }
1070         }
1071         if (!symbols.info(info_flags::list)) {
1072                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1073         }
1074         for (size_t i=0; i<symbols.nops(); i++) {
1075                 if (!symbols.op(i).info(info_flags::symbol)) {
1076                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1077                 }
1078         }
1079
1080         // build matrix from equation system
1081         matrix sys(eqns.nops(),symbols.nops());
1082         matrix rhs(eqns.nops(),1);
1083         matrix vars(symbols.nops(),1);
1084
1085         for (size_t r=0; r<eqns.nops(); r++) {
1086                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1087                 ex linpart = eq;
1088                 for (size_t c=0; c<symbols.nops(); c++) {
1089                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1090                         linpart -= co*symbols.op(c);
1091                         sys(r,c) = co;
1092                 }
1093                 linpart = linpart.expand();
1094                 rhs(r,0) = -linpart;
1095         }
1096
1097         // test if system is linear and fill vars matrix
1098         for (size_t i=0; i<symbols.nops(); i++) {
1099                 vars(i,0) = symbols.op(i);
1100                 if (sys.has(symbols.op(i)))
1101                         throw(std::logic_error("lsolve: system is not linear"));
1102                 if (rhs.has(symbols.op(i)))
1103                         throw(std::logic_error("lsolve: system is not linear"));
1104         }
1105
1106         matrix solution;
1107         try {
1108                 solution = sys.solve(vars,rhs,options);
1109         } catch (const std::runtime_error & e) {
1110                 // Probably singular matrix or otherwise overdetermined system:
1111                 // It is consistent to return an empty list
1112                 return lst();
1113         }
1114         GINAC_ASSERT(solution.cols()==1);
1115         GINAC_ASSERT(solution.rows()==symbols.nops());
1116
1117         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1118         lst sollist;
1119         for (size_t i=0; i<symbols.nops(); i++)
1120                 sollist.append(symbols.op(i)==solution(i,0));
1121
1122         return sollist;
1123 }
1124
1125 //////////
1126 // Find real root of f(x) numerically
1127 //////////
1128
1129 const numeric
1130 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1131 {
1132         if (!x1.is_real() || !x2.is_real()) {
1133                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1134         }
1135         if (x1==x2) {
1136                 throw std::runtime_error("fsolve(): vanishing interval");
1137         }
1138         // xx[0] == left interval limit, xx[1] == right interval limit.
1139         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1140         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1141         numeric xx[2] = { x1<x2 ? x1 : x2,
1142                           x1<x2 ? x2 : x1 };
1143         ex f;
1144         if (is_a<relational>(f_in)) {
1145                 f = f_in.lhs()-f_in.rhs();
1146         } else {
1147                 f = f_in;
1148         }
1149         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1150                             f.subs(x==xx[1]).evalf() };
1151         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1152                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1153         }
1154         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1155                           ex_to<numeric>(fx_[1]) };
1156         if (!fx[0].is_real() || !fx[1].is_real()) {
1157                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1158         }
1159         if (fx[0]*fx[1]>=0) {
1160                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1161         }
1162
1163         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1164         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1165         const ex ff = normal(-f/f.diff(x));
1166         int side = 0;  // Start at left interval limit.
1167         numeric xxprev;
1168         numeric fxprev;
1169         do {
1170                 xxprev = xx[side];
1171                 fxprev = fx[side];
1172                 ex dx_ = ff.subs(x == xx[side]).evalf();
1173                 if (!is_a<numeric>(dx_))
1174                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1175                 xx[side] += ex_to<numeric>(dx_);
1176                 // Now check if Newton-Raphson method shot out of the interval
1177                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1178                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1179                 if (!bad_shot) {
1180                         // Compute f(x) only if new x is inside the interval.
1181                         // The function might be difficult to compute numerically
1182                         // or even ill defined outside the interval. Also it's
1183                         // a small optimization.
1184                         ex f_x = f.subs(x == xx[side]).evalf();
1185                         if (!is_a<numeric>(f_x))
1186                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1187                         fx[side] = ex_to<numeric>(f_x);
1188                 }
1189                 if (bad_shot) {
1190                         // Oops, Newton-Raphson method shot out of the interval.
1191                         // Restore, and try again with the other side instead!
1192                         xx[side] = xxprev;
1193                         fx[side] = fxprev;
1194                         side = !side;
1195                         xxprev = xx[side];
1196                         fxprev = fx[side];
1197
1198                         ex dx_ = ff.subs(x == xx[side]).evalf();
1199                         if (!is_a<numeric>(dx_))
1200                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1201                         xx[side] += ex_to<numeric>(dx_);
1202
1203                         ex f_x = f.subs(x==xx[side]).evalf();
1204                         if (!is_a<numeric>(f_x))
1205                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1206                         fx[side] = ex_to<numeric>(f_x);
1207                 }
1208                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1209                         // Oops, the root isn't bracketed any more.
1210                         // Restore, and perform a bisection!
1211                         xx[side] = xxprev;
1212                         fx[side] = fxprev;
1213
1214                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1215                         // they occur pretty often when Newton-Raphson arrives at an x too
1216                         // close to the result on one side of the interval and
1217                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1218                         // precision errors! Recall that this function does not have a
1219                         // precision goal as one of its arguments but instead relies on
1220                         // x converging to a fixed point. We speed up the (safe but slow)
1221                         // bisection method by mixing in a dash of the (unsafer but faster)
1222                         // secant method: Instead of splitting the interval at the
1223                         // arithmetic mean (bisection), we split it nearer to the root as
1224                         // determined by the secant between the values xx[0] and xx[1].
1225                         // Don't set the secant_weight to one because that could disturb
1226                         // the convergence in some corner cases!
1227                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1228                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1229                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1230                         ex fxmid_ = f.subs(x == xxmid).evalf();
1231                         if (!is_a<numeric>(fxmid_))
1232                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1233                         numeric fxmid = ex_to<numeric>(fxmid_);
1234                         if (fxmid.is_zero()) {
1235                                 // Luck strikes...
1236                                 return xxmid;
1237                         }
1238                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1239                                 side = !side;
1240                         }
1241                         xxprev = xx[side];
1242                         fxprev = fx[side];
1243                         xx[side] = xxmid;
1244                         fx[side] = fxmid;
1245                 }
1246         } while (xxprev!=xx[side]);
1247         return xxprev;
1248 }
1249
1250
1251 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1252  * for static lib (so ginsh will see them). */
1253 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1254 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1255
1256 } // namespace GiNaC