ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2010 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 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
  80                                       evalf_func(conjugate_evalf).
  81                                       print_func<print_latex>(conjugate_print_latex).
  82                                       conjugate_func(conjugate_conjugate).
  83                                       real_part_func(conjugate_real_part).
  84                                       imag_part_func(conjugate_imag_part).
  85                                       set_name("conjugate","conjugate"));
  86
  87 //////////
  88 // real part
  89 //////////
  90
  91 static ex real_part_evalf(const ex & arg)
  92 {
  93         if (is_exactly_a<numeric>(arg)) {
  94                 return ex_to<numeric>(arg).real();
  95         }
  96         return real_part_function(arg).hold();
  97 }
  98
  99 static ex real_part_eval(const ex & arg)
 100 {
 101         return arg.real_part();
 102 }
 103
 104 static void real_part_print_latex(const ex & arg, const print_context & c)
 105 {
 106         c.s << "\\Re"; arg.print(c); c.s << "";
 107 }
 108
 109 static ex real_part_conjugate(const ex & arg)
 110 {
 111         return real_part_function(arg).hold();
 112 }
 113
 114 static ex real_part_real_part(const ex & arg)
 115 {
 116         return real_part_function(arg).hold();
 117 }
 118
 119 static ex real_part_imag_part(const ex & arg)
 120 {
 121         return 0;
 122 }
 123
 124 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
 125                                       evalf_func(real_part_evalf).
 126                                       print_func<print_latex>(real_part_print_latex).
 127                                       conjugate_func(real_part_conjugate).
 128                                       real_part_func(real_part_real_part).
 129                                       imag_part_func(real_part_imag_part).
 130                                       set_name("real_part","real_part"));
 131
 132 //////////
 133 // imag part
 134 //////////
 135
 136 static ex imag_part_evalf(const ex & arg)
 137 {
 138         if (is_exactly_a<numeric>(arg)) {
 139                 return ex_to<numeric>(arg).imag();
 140         }
 141         return imag_part_function(arg).hold();
 142 }
 143
 144 static ex imag_part_eval(const ex & arg)
 145 {
 146         return arg.imag_part();
 147 }
 148
 149 static void imag_part_print_latex(const ex & arg, const print_context & c)
 150 {
 151         c.s << "\\Im"; arg.print(c); c.s << "";
 152 }
 153
 154 static ex imag_part_conjugate(const ex & arg)
 155 {
 156         return imag_part_function(arg).hold();
 157 }
 158
 159 static ex imag_part_real_part(const ex & arg)
 160 {
 161         return imag_part_function(arg).hold();
 162 }
 163
 164 static ex imag_part_imag_part(const ex & arg)
 165 {
 166         return 0;
 167 }
 168
 169 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
 170                                       evalf_func(imag_part_evalf).
 171                                       print_func<print_latex>(imag_part_print_latex).
 172                                       conjugate_func(imag_part_conjugate).
 173                                       real_part_func(imag_part_real_part).
 174                                       imag_part_func(imag_part_imag_part).
 175                                       set_name("imag_part","imag_part"));
 176
 177 //////////
 178 // absolute value
 179 //////////
 180
 181 static ex abs_evalf(const ex & arg)
 182 {
 183         if (is_exactly_a<numeric>(arg))
 184                 return abs(ex_to<numeric>(arg));
 185
 186         return abs(arg).hold();
 187 }
 188
 189 static ex abs_eval(const ex & arg)
 190 {
 191         if (is_exactly_a<numeric>(arg))
 192                 return abs(ex_to<numeric>(arg));
 193
 194         if (arg.info(info_flags::nonnegative))
 195                 return arg;
 196
 197         if (is_ex_the_function(arg, abs))
 198                 return arg;
 199
 200         return abs(arg).hold();
 201 }
 202
 203 static void abs_print_latex(const ex & arg, const print_context & c)
 204 {
 205         c.s << "{|"; arg.print(c); c.s << "|}";
 206 }
 207
 208 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 209 {
 210         c.s << "fabs("; arg.print(c); c.s << ")";
 211 }
 212
 213 static ex abs_conjugate(const ex & arg)
 214 {
 215         return abs(arg);
 216 }
 217
 218 static ex abs_real_part(const ex & arg)
 219 {
 220         return abs(arg).hold();
 221 }
 222
 223 static ex abs_imag_part(const ex& arg)
 224 {
 225         return 0;
 226 }
 227
 228 static ex abs_power(const ex & arg, const ex & exp)
 229 {
 230         if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
 231                 return power(arg, exp);
 232         else
 233                 return power(abs(arg), exp).hold();
 234 }
 235
 236 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 237                        evalf_func(abs_evalf).
 238                        print_func<print_latex>(abs_print_latex).
 239                        print_func<print_csrc_float>(abs_print_csrc_float).
 240                        print_func<print_csrc_double>(abs_print_csrc_float).
 241                        conjugate_func(abs_conjugate).
 242                        real_part_func(abs_real_part).
 243                        imag_part_func(abs_imag_part).
 244                        power_func(abs_power));
 245
 246 //////////
 247 // Step function
 248 //////////
 249
 250 static ex step_evalf(const ex & arg)
 251 {
 252         if (is_exactly_a<numeric>(arg))
 253                 return step(ex_to<numeric>(arg));
 254
 255         return step(arg).hold();
 256 }
 257
 258 static ex step_eval(const ex & arg)
 259 {
 260         if (is_exactly_a<numeric>(arg))
 261                 return step(ex_to<numeric>(arg));
 262
 263         else if (is_exactly_a<mul>(arg) &&
 264                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 265                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 266                 if (oc.is_real()) {
 267                         if (oc > 0)
 268                                 // step(42*x) -> step(x)
 269                                 return step(arg/oc).hold();
 270                         else
 271                                 // step(-42*x) -> step(-x)
 272                                 return step(-arg/oc).hold();
 273                 }
 274                 if (oc.real().is_zero()) {
 275                         if (oc.imag() > 0)
 276                                 // step(42*I*x) -> step(I*x)
 277                                 return step(I*arg/oc).hold();
 278                         else
 279                                 // step(-42*I*x) -> step(-I*x)
 280                                 return step(-I*arg/oc).hold();
 281                 }
 282         }
 283
 284         return step(arg).hold();
 285 }
 286
 287 static ex step_series(const ex & arg,
 288                       const relational & rel,
 289                       int order,
 290                       unsigned options)
 291 {
 292         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 293         if (arg_pt.info(info_flags::numeric)
 294             && ex_to<numeric>(arg_pt).real().is_zero()
 295             && !(options & series_options::suppress_branchcut))
 296                 throw (std::domain_error("step_series(): on imaginary axis"));
 297
 298         epvector seq;
 299         seq.push_back(expair(step(arg_pt), _ex0));
 300         return pseries(rel,seq);
 301 }
 302
 303 static ex step_conjugate(const ex& arg)
 304 {
 305         return step(arg).hold();
 306 }
 307
 308 static ex step_real_part(const ex& arg)
 309 {
 310         return step(arg).hold();
 311 }
 312
 313 static ex step_imag_part(const ex& arg)
 314 {
 315         return 0;
 316 }
 317
 318 REGISTER_FUNCTION(step, eval_func(step_eval).
 319                         evalf_func(step_evalf).
 320                         series_func(step_series).
 321                         conjugate_func(step_conjugate).
 322                         real_part_func(step_real_part).
 323                         imag_part_func(step_imag_part));
 324
 325 //////////
 326 // Complex sign
 327 //////////
 328
 329 static ex csgn_evalf(const ex & arg)
 330 {
 331         if (is_exactly_a<numeric>(arg))
 332                 return csgn(ex_to<numeric>(arg));
 333
 334         return csgn(arg).hold();
 335 }
 336
 337 static ex csgn_eval(const ex & arg)
 338 {
 339         if (is_exactly_a<numeric>(arg))
 340                 return csgn(ex_to<numeric>(arg));
 341
 342         else if (is_exactly_a<mul>(arg) &&
 343                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 344                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 345                 if (oc.is_real()) {
 346                         if (oc > 0)
 347                                 // csgn(42*x) -> csgn(x)
 348                                 return csgn(arg/oc).hold();
 349                         else
 350                                 // csgn(-42*x) -> -csgn(x)
 351                                 return -csgn(arg/oc).hold();
 352                 }
 353                 if (oc.real().is_zero()) {
 354                         if (oc.imag() > 0)
 355                                 // csgn(42*I*x) -> csgn(I*x)
 356                                 return csgn(I*arg/oc).hold();
 357                         else
 358                                 // csgn(-42*I*x) -> -csgn(I*x)
 359                                 return -csgn(I*arg/oc).hold();
 360                 }
 361         }
 362
 363         return csgn(arg).hold();
 364 }
 365
 366 static ex csgn_series(const ex & arg,
 367                       const relational & rel,
 368                       int order,
 369                       unsigned options)
 370 {
 371         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 372         if (arg_pt.info(info_flags::numeric)
 373             && ex_to<numeric>(arg_pt).real().is_zero()
 374             && !(options & series_options::suppress_branchcut))
 375                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 376
 377         epvector seq;
 378         seq.push_back(expair(csgn(arg_pt), _ex0));
 379         return pseries(rel,seq);
 380 }
 381
 382 static ex csgn_conjugate(const ex& arg)
 383 {
 384         return csgn(arg).hold();
 385 }
 386
 387 static ex csgn_real_part(const ex& arg)
 388 {
 389         return csgn(arg).hold();
 390 }
 391
 392 static ex csgn_imag_part(const ex& arg)
 393 {
 394         return 0;
 395 }
 396
 397 static ex csgn_power(const ex & arg, const ex & exp)
 398 {
 399         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 400                 if (ex_to<numeric>(exp).is_odd())
 401                         return csgn(arg);
 402                 else
 403                         return power(csgn(arg), _ex2).hold();
 404         } else
 405                 return power(csgn(arg), exp).hold();
 406 }
 407
 408
 409 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 410                         evalf_func(csgn_evalf).
 411                         series_func(csgn_series).
 412                         conjugate_func(csgn_conjugate).
 413                         real_part_func(csgn_real_part).
 414                         imag_part_func(csgn_imag_part).
 415                         power_func(csgn_power));
 416
 417
 418 //////////
 419 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 420 // This function is closely related to the unwinding number K, sometimes found
 421 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 422 //////////
 423
 424 static ex eta_evalf(const ex &x, const ex &y)
 425 {
 426         // It seems like we basically have to replicate the eval function here,
 427         // since the expression might not be fully evaluated yet.
 428         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 429                 return _ex0;
 430
 431         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 432                 const numeric nx = ex_to<numeric>(x);
 433                 const numeric ny = ex_to<numeric>(y);
 434                 const numeric nxy = ex_to<numeric>(x*y);
 435                 int cut = 0;
 436                 if (nx.is_real() && nx.is_negative())
 437                         cut -= 4;
 438                 if (ny.is_real() && ny.is_negative())
 439                         cut -= 4;
 440                 if (nxy.is_real() && nxy.is_negative())
 441                         cut += 4;
 442                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 443                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 444         }
 445
 446         return eta(x,y).hold();
 447 }
 448
 449 static ex eta_eval(const ex &x, const ex &y)
 450 {
 451         // trivial:  eta(x,c) -> 0  if c is real and positive
 452         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 453                 return _ex0;
 454
 455         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 456                 // don't call eta_evalf here because it would call Pi.evalf()!
 457                 const numeric nx = ex_to<numeric>(x);
 458                 const numeric ny = ex_to<numeric>(y);
 459                 const numeric nxy = ex_to<numeric>(x*y);
 460                 int cut = 0;
 461                 if (nx.is_real() && nx.is_negative())
 462                         cut -= 4;
 463                 if (ny.is_real() && ny.is_negative())
 464                         cut -= 4;
 465                 if (nxy.is_real() && nxy.is_negative())
 466                         cut += 4;
 467                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 468                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 469         }
 470
 471         return eta(x,y).hold();
 472 }
 473
 474 static ex eta_series(const ex & x, const ex & y,
 475                      const relational & rel,
 476                      int order,
 477                      unsigned options)
 478 {
 479         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 480         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 481         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 482             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 483             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 484                         throw (std::domain_error("eta_series(): on discontinuity"));
 485         epvector seq;
 486         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 487         return pseries(rel,seq);
 488 }
 489
 490 static ex eta_conjugate(const ex & x, const ex & y)
 491 {
 492         return -eta(x, y);
 493 }
 494
 495 static ex eta_real_part(const ex & x, const ex & y)
 496 {
 497         return 0;
 498 }
 499
 500 static ex eta_imag_part(const ex & x, const ex & y)
 501 {
 502         return -I*eta(x, y).hold();
 503 }
 504
 505 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 506                        evalf_func(eta_evalf).
 507                        series_func(eta_series).
 508                        latex_name("\\eta").
 509                        set_symmetry(sy_symm(0, 1)).
 510                        conjugate_func(eta_conjugate).
 511                        real_part_func(eta_real_part).
 512                        imag_part_func(eta_imag_part));
 513
 514
 515 //////////
 516 // dilogarithm
 517 //////////
 518
 519 static ex Li2_evalf(const ex & x)
 520 {
 521         if (is_exactly_a<numeric>(x))
 522                 return Li2(ex_to<numeric>(x));
 523
 524         return Li2(x).hold();
 525 }
 526
 527 static ex Li2_eval(const ex & x)
 528 {
 529         if (x.info(info_flags::numeric)) {
 530                 // Li2(0) -> 0
 531                 if (x.is_zero())
 532                         return _ex0;
 533                 // Li2(1) -> Pi^2/6
 534                 if (x.is_equal(_ex1))
 535                         return power(Pi,_ex2)/_ex6;
 536                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 537                 if (x.is_equal(_ex1_2))
 538                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 539                 // Li2(-1) -> -Pi^2/12
 540                 if (x.is_equal(_ex_1))
 541                         return -power(Pi,_ex2)/_ex12;
 542                 // Li2(I) -> -Pi^2/48+Catalan*I
 543                 if (x.is_equal(I))
 544                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 545                 // Li2(-I) -> -Pi^2/48-Catalan*I
 546                 if (x.is_equal(-I))
 547                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 548                 // Li2(float)
 549                 if (!x.info(info_flags::crational))
 550                         return Li2(ex_to<numeric>(x));
 551         }
 552
 553         return Li2(x).hold();
 554 }
 555
 556 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 557 {
 558         GINAC_ASSERT(deriv_param==0);
 559
 560         // d/dx Li2(x) -> -log(1-x)/x
 561         return -log(_ex1-x)/x;
 562 }
 563
 564 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 565 {
 566         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 567         if (x_pt.info(info_flags::numeric)) {
 568                 // First special case: x==0 (derivatives have poles)
 569                 if (x_pt.is_zero()) {
 570                         // method:
 571                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 572                         // simply substitute x==0.  The limit, however, exists: it is 1.
 573                         // We also know all higher derivatives' limits:
 574                         // (d/dx)^n Li2(x) == n!/n^2.
 575                         // So the primitive series expansion is
 576                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 577                         // and so on.
 578                         // We first construct such a primitive series expansion manually in
 579                         // a dummy symbol s and then insert the argument's series expansion
 580                         // for s.  Reexpanding the resulting series returns the desired
 581                         // result.
 582                         const symbol s;
 583                         ex ser;
 584                         // manually construct the primitive expansion
 585                         for (int i=1; i<order; ++i)
 586                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 587                         // substitute the argument's series expansion
 588                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 589                         // maybe that was terminating, so add a proper order term
 590                         epvector nseq;
 591                         nseq.push_back(expair(Order(_ex1), order));
 592                         ser += pseries(rel, nseq);
 593                         // reexpanding it will collapse the series again
 594                         return ser.series(rel, order);
 595                         // NB: Of course, this still does not allow us to compute anything
 596                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 597                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 598                         // substitution x==0.  Probably limits *are* needed for the general
 599                         // cases.  In case L'Hospital's rule is implemented for limits and
 600                         // basic::series() takes care of this, this whole block is probably
 601                         // obsolete!
 602                 }
 603                 // second special case: x==1 (branch point)
 604                 if (x_pt.is_equal(_ex1)) {
 605                         // method:
 606                         // construct series manually in a dummy symbol s
 607                         const symbol s;
 608                         ex ser = zeta(_ex2);
 609                         // manually construct the primitive expansion
 610                         for (int i=1; i<order; ++i)
 611                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 612                         // substitute the argument's series expansion
 613                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 614                         // maybe that was terminating, so add a proper order term
 615                         epvector nseq;
 616                         nseq.push_back(expair(Order(_ex1), order));
 617                         ser += pseries(rel, nseq);
 618                         // reexpanding it will collapse the series again
 619                         return ser.series(rel, order);
 620                 }
 621                 // third special case: x real, >=1 (branch cut)
 622                 if (!(options & series_options::suppress_branchcut) &&
 623                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 624                         // method:
 625                         // This is the branch cut: assemble the primitive series manually
 626                         // and then add the corresponding complex step function.
 627                         const symbol &s = ex_to<symbol>(rel.lhs());
 628                         const ex point = rel.rhs();
 629                         const symbol foo;
 630                         epvector seq;
 631                         // zeroth order term:
 632                         seq.push_back(expair(Li2(x_pt), _ex0));
 633                         // compute the intermediate terms:
 634                         ex replarg = series(Li2(x), s==foo, order);
 635                         for (size_t i=1; i<replarg.nops()-1; ++i)
 636                                 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));
 637                         // append an order term:
 638                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 639                         return pseries(rel, seq);
 640                 }
 641         }
 642         // all other cases should be safe, by now:
 643         throw do_taylor();  // caught by function::series()
 644 }
 645
 646 static ex Li2_conjugate(const ex & x)
 647 {
 648         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 649         // run along the positive real axis beginning at 1.
 650         if (x.info(info_flags::negative)) {
 651                 return Li2(x);
 652         }
 653         if (is_exactly_a<numeric>(x) &&
 654             (!x.imag_part().is_zero() || x < *_num1_p)) {
 655                 return Li2(x.conjugate());
 656         }
 657         return conjugate_function(Li2(x)).hold();
 658 }
 659
 660 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 661                        evalf_func(Li2_evalf).
 662                        derivative_func(Li2_deriv).
 663                        series_func(Li2_series).
 664                        conjugate_func(Li2_conjugate).
 665                        latex_name("\\mathrm{Li}_2"));
 666
 667 //////////
 668 // trilogarithm
 669 //////////
 670
 671 static ex Li3_eval(const ex & x)
 672 {
 673         if (x.is_zero())
 674                 return x;
 675         return Li3(x).hold();
 676 }
 677
 678 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 679                        latex_name("\\mathrm{Li}_3"));
 680
 681 //////////
 682 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 683 //////////
 684
 685 static ex zetaderiv_eval(const ex & n, const ex & x)
 686 {
 687         if (n.info(info_flags::numeric)) {
 688                 // zetaderiv(0,x) -> zeta(x)
 689                 if (n.is_zero())
 690                         return zeta(x);
 691         }
 692
 693         return zetaderiv(n, x).hold();
 694 }
 695
 696 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 697 {
 698         GINAC_ASSERT(deriv_param<2);
 699
 700         if (deriv_param==0) {
 701                 // d/dn zeta(n,x)
 702                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 703         }
 704         // d/dx psi(n,x)
 705         return zetaderiv(n+1,x);
 706 }
 707
 708 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 709                                  derivative_func(zetaderiv_deriv).
 710                                  latex_name("\\zeta^\\prime"));
 711
 712 //////////
 713 // factorial
 714 //////////
 715
 716 static ex factorial_evalf(const ex & x)
 717 {
 718         return factorial(x).hold();
 719 }
 720
 721 static ex factorial_eval(const ex & x)
 722 {
 723         if (is_exactly_a<numeric>(x))
 724                 return factorial(ex_to<numeric>(x));
 725         else
 726                 return factorial(x).hold();
 727 }
 728
 729 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 730 {
 731         if (is_exactly_a<symbol>(x) ||
 732             is_exactly_a<constant>(x) ||
 733                 is_exactly_a<function>(x)) {
 734                 x.print(c); c.s << "!";
 735         } else {
 736                 c.s << "("; x.print(c); c.s << ")!";
 737         }
 738 }
 739
 740 static ex factorial_conjugate(const ex & x)
 741 {
 742         return factorial(x).hold();
 743 }
 744
 745 static ex factorial_real_part(const ex & x)
 746 {
 747         return factorial(x).hold();
 748 }
 749
 750 static ex factorial_imag_part(const ex & x)
 751 {
 752         return 0;
 753 }
 754
 755 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 756                              evalf_func(factorial_evalf).
 757                              print_func<print_dflt>(factorial_print_dflt_latex).
 758                              print_func<print_latex>(factorial_print_dflt_latex).
 759                              conjugate_func(factorial_conjugate).
 760                              real_part_func(factorial_real_part).
 761                              imag_part_func(factorial_imag_part));
 762
 763 //////////
 764 // binomial
 765 //////////
 766
 767 static ex binomial_evalf(const ex & x, const ex & y)
 768 {
 769         return binomial(x, y).hold();
 770 }
 771
 772 static ex binomial_sym(const ex & x, const numeric & y)
 773 {
 774         if (y.is_integer()) {
 775                 if (y.is_nonneg_integer()) {
 776                         const unsigned N = y.to_int();
 777                         if (N == 0) return _ex1;
 778                         if (N == 1) return x;
 779                         ex t = x.expand();
 780                         for (unsigned i = 2; i <= N; ++i)
 781                                 t = (t * (x + i - y - 1)).expand() / i;
 782                         return t;
 783                 } else
 784                         return _ex0;
 785         }
 786
 787         return binomial(x, y).hold();
 788 }
 789
 790 static ex binomial_eval(const ex & x, const ex &y)
 791 {
 792         if (is_exactly_a<numeric>(y)) {
 793                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 794                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 795                 else
 796                         return binomial_sym(x, ex_to<numeric>(y));
 797         } else
 798                 return binomial(x, y).hold();
 799 }
 800
 801 // At the moment the numeric evaluation of a binomail function always
 802 // gives a real number, but if this would be implemented using the gamma
 803 // function, also complex conjugation should be changed (or rather, deleted).
 804 static ex binomial_conjugate(const ex & x, const ex & y)
 805 {
 806         return binomial(x,y).hold();
 807 }
 808
 809 static ex binomial_real_part(const ex & x, const ex & y)
 810 {
 811         return binomial(x,y).hold();
 812 }
 813
 814 static ex binomial_imag_part(const ex & x, const ex & y)
 815 {
 816         return 0;
 817 }
 818
 819 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 820                             evalf_func(binomial_evalf).
 821                             conjugate_func(binomial_conjugate).
 822                             real_part_func(binomial_real_part).
 823                             imag_part_func(binomial_imag_part));
 824
 825 //////////
 826 // Order term function (for truncated power series)
 827 //////////
 828
 829 static ex Order_eval(const ex & x)
 830 {
 831         if (is_exactly_a<numeric>(x)) {
 832                 // O(c) -> O(1) or 0
 833                 if (!x.is_zero())
 834                         return Order(_ex1).hold();
 835                 else
 836                         return _ex0;
 837         } else if (is_exactly_a<mul>(x)) {
 838                 const mul &m = ex_to<mul>(x);
 839                 // O(c*expr) -> O(expr)
 840                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 841                         return Order(x / m.op(m.nops() - 1)).hold();
 842         }
 843         return Order(x).hold();
 844 }
 845
 846 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 847 {
 848         // Just wrap the function into a pseries object
 849         epvector new_seq;
 850         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 851         const symbol &s = ex_to<symbol>(r.lhs());
 852         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 853         return pseries(r, new_seq);
 854 }
 855
 856 static ex Order_conjugate(const ex & x)
 857 {
 858         return Order(x).hold();
 859 }
 860
 861 static ex Order_real_part(const ex & x)
 862 {
 863         return Order(x).hold();
 864 }
 865
 866 static ex Order_imag_part(const ex & x)
 867 {
 868         if(x.info(info_flags::real))
 869                 return 0;
 870         return Order(x).hold();
 871 }
 872
 873 // Differentiation is handled in function::derivative because of its special requirements
 874
 875 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 876                          series_func(Order_series).
 877                          latex_name("\\mathcal{O}").
 878                          conjugate_func(Order_conjugate).
 879                          real_part_func(Order_real_part).
 880                          imag_part_func(Order_imag_part));
 881
 882 //////////
 883 // Solve linear system
 884 //////////
 885
 886 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 887 {
 888         // solve a system of linear equations
 889         if (eqns.info(info_flags::relation_equal)) {
 890                 if (!symbols.info(info_flags::symbol))
 891                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 892                 const ex sol = lsolve(lst(eqns),lst(symbols));
 893
 894                 GINAC_ASSERT(sol.nops()==1);
 895                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 896
 897                 return sol.op(0).op(1); // return rhs of first solution
 898         }
 899
 900         // syntax checks
 901         if (!eqns.info(info_flags::list)) {
 902                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
 903         }
 904         for (size_t i=0; i<eqns.nops(); i++) {
 905                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 906                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 907                 }
 908         }
 909         if (!symbols.info(info_flags::list)) {
 910                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
 911         }
 912         for (size_t i=0; i<symbols.nops(); i++) {
 913                 if (!symbols.op(i).info(info_flags::symbol)) {
 914                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 915                 }
 916         }
 917
 918         // build matrix from equation system
 919         matrix sys(eqns.nops(),symbols.nops());
 920         matrix rhs(eqns.nops(),1);
 921         matrix vars(symbols.nops(),1);
 922
 923         for (size_t r=0; r<eqns.nops(); r++) {
 924                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 925                 ex linpart = eq;
 926                 for (size_t c=0; c<symbols.nops(); c++) {
 927                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 928                         linpart -= co*symbols.op(c);
 929                         sys(r,c) = co;
 930                 }
 931                 linpart = linpart.expand();
 932                 rhs(r,0) = -linpart;
 933         }
 934
 935         // test if system is linear and fill vars matrix
 936         for (size_t i=0; i<symbols.nops(); i++) {
 937                 vars(i,0) = symbols.op(i);
 938                 if (sys.has(symbols.op(i)))
 939                         throw(std::logic_error("lsolve: system is not linear"));
 940                 if (rhs.has(symbols.op(i)))
 941                         throw(std::logic_error("lsolve: system is not linear"));
 942         }
 943
 944         matrix solution;
 945         try {
 946                 solution = sys.solve(vars,rhs,options);
 947         } catch (const std::runtime_error & e) {
 948                 // Probably singular matrix or otherwise overdetermined system:
 949                 // It is consistent to return an empty list
 950                 return lst();
 951         }
 952         GINAC_ASSERT(solution.cols()==1);
 953         GINAC_ASSERT(solution.rows()==symbols.nops());
 954
 955         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 956         lst sollist;
 957         for (size_t i=0; i<symbols.nops(); i++)
 958                 sollist.append(symbols.op(i)==solution(i,0));
 959
 960         return sollist;
 961 }
 962
 963 //////////
 964 // Find real root of f(x) numerically
 965 //////////
 966
 967 const numeric
 968 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
 969 {
 970         if (!x1.is_real() || !x2.is_real()) {
 971                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
 972         }
 973         if (x1==x2) {
 974                 throw std::runtime_error("fsolve(): vanishing interval");
 975         }
 976         // xx[0] == left interval limit, xx[1] == right interval limit.
 977         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
 978         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
 979         numeric xx[2] = { x1<x2 ? x1 : x2,
 980                           x1<x2 ? x2 : x1 };
 981         ex f;
 982         if (is_a<relational>(f_in)) {
 983                 f = f_in.lhs()-f_in.rhs();
 984         } else {
 985                 f = f_in;
 986         }
 987         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
 988                             f.subs(x==xx[1]).evalf() };
 989         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
 990                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
 991         }
 992         numeric fx[2] = { ex_to<numeric>(fx_[0]),
 993                           ex_to<numeric>(fx_[1]) };
 994         if (!fx[0].is_real() || !fx[1].is_real()) {
 995                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
 996         }
 997         if (fx[0]*fx[1]>=0) {
 998                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
 999         }
1000
1001         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1002         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1003         const ex ff = normal(-f/f.diff(x));
1004         int side = 0;  // Start at left interval limit.
1005         numeric xxprev;
1006         numeric fxprev;
1007         do {
1008                 xxprev = xx[side];
1009                 fxprev = fx[side];
1010                 ex dx_ = ff.subs(x == xx[side]).evalf();
1011                 if (!is_a<numeric>(dx_))
1012                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1013                 xx[side] += ex_to<numeric>(dx_);
1014                 // Now check if Newton-Raphson method shot out of the interval
1015                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1016                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1017                 if (!bad_shot) {
1018                         // Compute f(x) only if new x is inside the interval.
1019                         // The function might be difficult to compute numerically
1020                         // or even ill defined outside the interval. Also it's
1021                         // a small optimization.
1022                         ex f_x = f.subs(x == xx[side]).evalf();
1023                         if (!is_a<numeric>(f_x))
1024                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1025                         fx[side] = ex_to<numeric>(f_x);
1026                 }
1027                 if (bad_shot) {
1028                         // Oops, Newton-Raphson method shot out of the interval.
1029                         // Restore, and try again with the other side instead!
1030                         xx[side] = xxprev;
1031                         fx[side] = fxprev;
1032                         side = !side;
1033                         xxprev = xx[side];
1034                         fxprev = fx[side];
1035
1036                         ex dx_ = ff.subs(x == xx[side]).evalf();
1037                         if (!is_a<numeric>(dx_))
1038                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1039                         xx[side] += ex_to<numeric>(dx_);
1040
1041                         ex f_x = f.subs(x==xx[side]).evalf();
1042                         if (!is_a<numeric>(f_x))
1043                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1044                         fx[side] = ex_to<numeric>(f_x);
1045                 }
1046                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1047                         // Oops, the root isn't bracketed any more.
1048                         // Restore, and perform a bisection!
1049                         xx[side] = xxprev;
1050                         fx[side] = fxprev;
1051
1052                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1053                         // they occur pretty often when Newton-Raphson arrives at an x too
1054                         // close to the result on one side of the interval and
1055                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1056                         // precision errors! Recall that this function does not have a
1057                         // precision goal as one of its arguments but instead relies on
1058                         // x converging to a fixed point. We speed up the (safe but slow)
1059                         // bisection method by mixing in a dash of the (unsafer but faster)
1060                         // secant method: Instead of splitting the interval at the
1061                         // arithmetic mean (bisection), we split it nearer to the root as
1062                         // determined by the secant between the values xx[0] and xx[1].
1063                         // Don't set the secant_weight to one because that could disturb
1064                         // the convergence in some corner cases!
1065                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1066                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1067                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1068                         ex fxmid_ = f.subs(x == xxmid).evalf();
1069                         if (!is_a<numeric>(fxmid_))
1070                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1071                         numeric fxmid = ex_to<numeric>(fxmid_);
1072                         if (fxmid.is_zero()) {
1073                                 // Luck strikes...
1074                                 return xxmid;
1075                         }
1076                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1077                                 side = !side;
1078                         }
1079                         xxprev = xx[side];
1080                         fxprev = fx[side];
1081                         xx[side] = xxmid;
1082                         fx[side] = fxmid;
1083                 }
1084         } while (xxprev!=xx[side]);
1085         return xxprev;
1086 }
1087
1088
1089 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1090  * for static lib (so ginsh will see them). */
1091 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1092 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1093
1094 } // namespace GiNaC