ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include "inifcns.h"
  24 #include "ex.h"
  25 #include "constant.h"
  26 #include "lst.h"
  27 #include "matrix.h"
  28 #include "mul.h"
  29 #include "power.h"
  30 #include "operators.h"
  31 #include "relational.h"
  32 #include "pseries.h"
  33 #include "symbol.h"
  34 #include "symmetry.h"
  35 #include "utils.h"
  36
  37 #include <stdexcept>
  38 #include <vector>
  39
  40 namespace GiNaC {
  41
  42 //////////
  43 // complex conjugate
  44 //////////
  45
  46 static ex conjugate_evalf(const ex & arg)
  47 {
  48         if (is_exactly_a<numeric>(arg)) {
  49                 return ex_to<numeric>(arg).conjugate();
  50         }
  51         return conjugate_function(arg).hold();
  52 }
  53
  54 static ex conjugate_eval(const ex & arg)
  55 {
  56         return arg.conjugate();
  57 }
  58
  59 static void conjugate_print_latex(const ex & arg, const print_context & c)
  60 {
  61         c.s << "\\bar{"; arg.print(c); c.s << "}";
  62 }
  63
  64 static ex conjugate_conjugate(const ex & arg)
  65 {
  66         return arg;
  67 }
  68
  69 static ex conjugate_real_part(const ex & arg)
  70 {
  71         return arg.real_part();
  72 }
  73
  74 static ex conjugate_imag_part(const ex & arg)
  75 {
  76         return -arg.imag_part();
  77 }
  78
  79 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         if (is_ex_the_function(arg, exp))
 201                 return exp(arg.op(0).real_part());
 202
 203         if (is_exactly_a<power>(arg)) {
 204                 const ex& base = arg.op(0);
 205                 const ex& exponent = arg.op(1);
 206                 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
 207                         return pow(abs(base), exponent.real_part());
 208         }
 209
 210         if (is_ex_the_function(arg, conjugate_function))
 211                 return abs(arg.op(0));
 212
 213         if (is_ex_the_function(arg, step))
 214                 return arg;
 215
 216         return abs(arg).hold();
 217 }
 218
 219 static void abs_print_latex(const ex & arg, const print_context & c)
 220 {
 221         c.s << "{|"; arg.print(c); c.s << "|}";
 222 }
 223
 224 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 225 {
 226         c.s << "fabs("; arg.print(c); c.s << ")";
 227 }
 228
 229 static ex abs_conjugate(const ex & arg)
 230 {
 231         return abs(arg).hold();
 232 }
 233
 234 static ex abs_real_part(const ex & arg)
 235 {
 236         return abs(arg).hold();
 237 }
 238
 239 static ex abs_imag_part(const ex& arg)
 240 {
 241         return 0;
 242 }
 243
 244 static ex abs_power(const ex & arg, const ex & exp)
 245 {
 246         if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
 247                                                 || exp.info(info_flags::even)))
 248                 return power(arg, exp);
 249         else
 250                 return power(abs(arg), exp).hold();
 251 }
 252
 253 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 254                        evalf_func(abs_evalf).
 255                        print_func<print_latex>(abs_print_latex).
 256                        print_func<print_csrc_float>(abs_print_csrc_float).
 257                        print_func<print_csrc_double>(abs_print_csrc_float).
 258                        conjugate_func(abs_conjugate).
 259                        real_part_func(abs_real_part).
 260                        imag_part_func(abs_imag_part).
 261                        power_func(abs_power));
 262
 263 //////////
 264 // Step function
 265 //////////
 266
 267 static ex step_evalf(const ex & arg)
 268 {
 269         if (is_exactly_a<numeric>(arg))
 270                 return step(ex_to<numeric>(arg));
 271
 272         return step(arg).hold();
 273 }
 274
 275 static ex step_eval(const ex & arg)
 276 {
 277         if (is_exactly_a<numeric>(arg))
 278                 return step(ex_to<numeric>(arg));
 279
 280         else if (is_exactly_a<mul>(arg) &&
 281                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 282                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 283                 if (oc.is_real()) {
 284                         if (oc > 0)
 285                                 // step(42*x) -> step(x)
 286                                 return step(arg/oc).hold();
 287                         else
 288                                 // step(-42*x) -> step(-x)
 289                                 return step(-arg/oc).hold();
 290                 }
 291                 if (oc.real().is_zero()) {
 292                         if (oc.imag() > 0)
 293                                 // step(42*I*x) -> step(I*x)
 294                                 return step(I*arg/oc).hold();
 295                         else
 296                                 // step(-42*I*x) -> step(-I*x)
 297                                 return step(-I*arg/oc).hold();
 298                 }
 299         }
 300
 301         return step(arg).hold();
 302 }
 303
 304 static ex step_series(const ex & arg,
 305                       const relational & rel,
 306                       int order,
 307                       unsigned options)
 308 {
 309         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 310         if (arg_pt.info(info_flags::numeric)
 311             && ex_to<numeric>(arg_pt).real().is_zero()
 312             && !(options & series_options::suppress_branchcut))
 313                 throw (std::domain_error("step_series(): on imaginary axis"));
 314
 315         epvector seq;
 316         seq.push_back(expair(step(arg_pt), _ex0));
 317         return pseries(rel,seq);
 318 }
 319
 320 static ex step_conjugate(const ex& arg)
 321 {
 322         return step(arg).hold();
 323 }
 324
 325 static ex step_real_part(const ex& arg)
 326 {
 327         return step(arg).hold();
 328 }
 329
 330 static ex step_imag_part(const ex& arg)
 331 {
 332         return 0;
 333 }
 334
 335 REGISTER_FUNCTION(step, eval_func(step_eval).
 336                         evalf_func(step_evalf).
 337                         series_func(step_series).
 338                         conjugate_func(step_conjugate).
 339                         real_part_func(step_real_part).
 340                         imag_part_func(step_imag_part));
 341
 342 //////////
 343 // Complex sign
 344 //////////
 345
 346 static ex csgn_evalf(const ex & arg)
 347 {
 348         if (is_exactly_a<numeric>(arg))
 349                 return csgn(ex_to<numeric>(arg));
 350
 351         return csgn(arg).hold();
 352 }
 353
 354 static ex csgn_eval(const ex & arg)
 355 {
 356         if (is_exactly_a<numeric>(arg))
 357                 return csgn(ex_to<numeric>(arg));
 358
 359         else if (is_exactly_a<mul>(arg) &&
 360                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 361                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 362                 if (oc.is_real()) {
 363                         if (oc > 0)
 364                                 // csgn(42*x) -> csgn(x)
 365                                 return csgn(arg/oc).hold();
 366                         else
 367                                 // csgn(-42*x) -> -csgn(x)
 368                                 return -csgn(arg/oc).hold();
 369                 }
 370                 if (oc.real().is_zero()) {
 371                         if (oc.imag() > 0)
 372                                 // csgn(42*I*x) -> csgn(I*x)
 373                                 return csgn(I*arg/oc).hold();
 374                         else
 375                                 // csgn(-42*I*x) -> -csgn(I*x)
 376                                 return -csgn(I*arg/oc).hold();
 377                 }
 378         }
 379
 380         return csgn(arg).hold();
 381 }
 382
 383 static ex csgn_series(const ex & arg,
 384                       const relational & rel,
 385                       int order,
 386                       unsigned options)
 387 {
 388         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 389         if (arg_pt.info(info_flags::numeric)
 390             && ex_to<numeric>(arg_pt).real().is_zero()
 391             && !(options & series_options::suppress_branchcut))
 392                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 393
 394         epvector seq;
 395         seq.push_back(expair(csgn(arg_pt), _ex0));
 396         return pseries(rel,seq);
 397 }
 398
 399 static ex csgn_conjugate(const ex& arg)
 400 {
 401         return csgn(arg).hold();
 402 }
 403
 404 static ex csgn_real_part(const ex& arg)
 405 {
 406         return csgn(arg).hold();
 407 }
 408
 409 static ex csgn_imag_part(const ex& arg)
 410 {
 411         return 0;
 412 }
 413
 414 static ex csgn_power(const ex & arg, const ex & exp)
 415 {
 416         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 417                 if (ex_to<numeric>(exp).is_odd())
 418                         return csgn(arg).hold();
 419                 else
 420                         return power(csgn(arg), _ex2).hold();
 421         } else
 422                 return power(csgn(arg), exp).hold();
 423 }
 424
 425
 426 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 427                         evalf_func(csgn_evalf).
 428                         series_func(csgn_series).
 429                         conjugate_func(csgn_conjugate).
 430                         real_part_func(csgn_real_part).
 431                         imag_part_func(csgn_imag_part).
 432                         power_func(csgn_power));
 433
 434
 435 //////////
 436 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 437 // This function is closely related to the unwinding number K, sometimes found
 438 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 439 //////////
 440
 441 static ex eta_evalf(const ex &x, const ex &y)
 442 {
 443         // It seems like we basically have to replicate the eval function here,
 444         // since the expression might not be fully evaluated yet.
 445         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 446                 return _ex0;
 447
 448         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 449                 const numeric nx = ex_to<numeric>(x);
 450                 const numeric ny = ex_to<numeric>(y);
 451                 const numeric nxy = ex_to<numeric>(x*y);
 452                 int cut = 0;
 453                 if (nx.is_real() && nx.is_negative())
 454                         cut -= 4;
 455                 if (ny.is_real() && ny.is_negative())
 456                         cut -= 4;
 457                 if (nxy.is_real() && nxy.is_negative())
 458                         cut += 4;
 459                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 460                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 461         }
 462
 463         return eta(x,y).hold();
 464 }
 465
 466 static ex eta_eval(const ex &x, const ex &y)
 467 {
 468         // trivial:  eta(x,c) -> 0  if c is real and positive
 469         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 470                 return _ex0;
 471
 472         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 473                 // don't call eta_evalf here because it would call Pi.evalf()!
 474                 const numeric nx = ex_to<numeric>(x);
 475                 const numeric ny = ex_to<numeric>(y);
 476                 const numeric nxy = ex_to<numeric>(x*y);
 477                 int cut = 0;
 478                 if (nx.is_real() && nx.is_negative())
 479                         cut -= 4;
 480                 if (ny.is_real() && ny.is_negative())
 481                         cut -= 4;
 482                 if (nxy.is_real() && nxy.is_negative())
 483                         cut += 4;
 484                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 485                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 486         }
 487
 488         return eta(x,y).hold();
 489 }
 490
 491 static ex eta_series(const ex & x, const ex & y,
 492                      const relational & rel,
 493                      int order,
 494                      unsigned options)
 495 {
 496         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 497         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 498         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 499             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 500             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 501                         throw (std::domain_error("eta_series(): on discontinuity"));
 502         epvector seq;
 503         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 504         return pseries(rel,seq);
 505 }
 506
 507 static ex eta_conjugate(const ex & x, const ex & y)
 508 {
 509         return -eta(x, y).hold();
 510 }
 511
 512 static ex eta_real_part(const ex & x, const ex & y)
 513 {
 514         return 0;
 515 }
 516
 517 static ex eta_imag_part(const ex & x, const ex & y)
 518 {
 519         return -I*eta(x, y).hold();
 520 }
 521
 522 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 523                        evalf_func(eta_evalf).
 524                        series_func(eta_series).
 525                        latex_name("\\eta").
 526                        set_symmetry(sy_symm(0, 1)).
 527                        conjugate_func(eta_conjugate).
 528                        real_part_func(eta_real_part).
 529                        imag_part_func(eta_imag_part));
 530
 531
 532 //////////
 533 // dilogarithm
 534 //////////
 535
 536 static ex Li2_evalf(const ex & x)
 537 {
 538         if (is_exactly_a<numeric>(x))
 539                 return Li2(ex_to<numeric>(x));
 540
 541         return Li2(x).hold();
 542 }
 543
 544 static ex Li2_eval(const ex & x)
 545 {
 546         if (x.info(info_flags::numeric)) {
 547                 // Li2(0) -> 0
 548                 if (x.is_zero())
 549                         return _ex0;
 550                 // Li2(1) -> Pi^2/6
 551                 if (x.is_equal(_ex1))
 552                         return power(Pi,_ex2)/_ex6;
 553                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 554                 if (x.is_equal(_ex1_2))
 555                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 556                 // Li2(-1) -> -Pi^2/12
 557                 if (x.is_equal(_ex_1))
 558                         return -power(Pi,_ex2)/_ex12;
 559                 // Li2(I) -> -Pi^2/48+Catalan*I
 560                 if (x.is_equal(I))
 561                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 562                 // Li2(-I) -> -Pi^2/48-Catalan*I
 563                 if (x.is_equal(-I))
 564                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 565                 // Li2(float)
 566                 if (!x.info(info_flags::crational))
 567                         return Li2(ex_to<numeric>(x));
 568         }
 569
 570         return Li2(x).hold();
 571 }
 572
 573 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 574 {
 575         GINAC_ASSERT(deriv_param==0);
 576
 577         // d/dx Li2(x) -> -log(1-x)/x
 578         return -log(_ex1-x)/x;
 579 }
 580
 581 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 582 {
 583         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 584         if (x_pt.info(info_flags::numeric)) {
 585                 // First special case: x==0 (derivatives have poles)
 586                 if (x_pt.is_zero()) {
 587                         // method:
 588                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 589                         // simply substitute x==0.  The limit, however, exists: it is 1.
 590                         // We also know all higher derivatives' limits:
 591                         // (d/dx)^n Li2(x) == n!/n^2.
 592                         // So the primitive series expansion is
 593                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 594                         // and so on.
 595                         // We first construct such a primitive series expansion manually in
 596                         // a dummy symbol s and then insert the argument's series expansion
 597                         // for s.  Reexpanding the resulting series returns the desired
 598                         // result.
 599                         const symbol s;
 600                         ex ser;
 601                         // manually construct the primitive expansion
 602                         for (int i=1; i<order; ++i)
 603                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 604                         // substitute the argument's series expansion
 605                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 606                         // maybe that was terminating, so add a proper order term
 607                         epvector nseq;
 608                         nseq.push_back(expair(Order(_ex1), order));
 609                         ser += pseries(rel, nseq);
 610                         // reexpanding it will collapse the series again
 611                         return ser.series(rel, order);
 612                         // NB: Of course, this still does not allow us to compute anything
 613                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 614                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 615                         // substitution x==0.  Probably limits *are* needed for the general
 616                         // cases.  In case L'Hospital's rule is implemented for limits and
 617                         // basic::series() takes care of this, this whole block is probably
 618                         // obsolete!
 619                 }
 620                 // second special case: x==1 (branch point)
 621                 if (x_pt.is_equal(_ex1)) {
 622                         // method:
 623                         // construct series manually in a dummy symbol s
 624                         const symbol s;
 625                         ex ser = zeta(_ex2);
 626                         // manually construct the primitive expansion
 627                         for (int i=1; i<order; ++i)
 628                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 629                         // substitute the argument's series expansion
 630                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 631                         // maybe that was terminating, so add a proper order term
 632                         epvector nseq;
 633                         nseq.push_back(expair(Order(_ex1), order));
 634                         ser += pseries(rel, nseq);
 635                         // reexpanding it will collapse the series again
 636                         return ser.series(rel, order);
 637                 }
 638                 // third special case: x real, >=1 (branch cut)
 639                 if (!(options & series_options::suppress_branchcut) &&
 640                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 641                         // method:
 642                         // This is the branch cut: assemble the primitive series manually
 643                         // and then add the corresponding complex step function.
 644                         const symbol &s = ex_to<symbol>(rel.lhs());
 645                         const ex point = rel.rhs();
 646                         const symbol foo;
 647                         epvector seq;
 648                         // zeroth order term:
 649                         seq.push_back(expair(Li2(x_pt), _ex0));
 650                         // compute the intermediate terms:
 651                         ex replarg = series(Li2(x), s==foo, order);
 652                         for (size_t i=1; i<replarg.nops()-1; ++i)
 653                                 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));
 654                         // append an order term:
 655                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 656                         return pseries(rel, seq);
 657                 }
 658         }
 659         // all other cases should be safe, by now:
 660         throw do_taylor();  // caught by function::series()
 661 }
 662
 663 static ex Li2_conjugate(const ex & x)
 664 {
 665         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
 666         // run along the positive real axis beginning at 1.
 667         if (x.info(info_flags::negative)) {
 668                 return Li2(x).hold();
 669         }
 670         if (is_exactly_a<numeric>(x) &&
 671             (!x.imag_part().is_zero() || x < *_num1_p)) {
 672                 return Li2(x.conjugate());
 673         }
 674         return conjugate_function(Li2(x)).hold();
 675 }
 676
 677 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 678                        evalf_func(Li2_evalf).
 679                        derivative_func(Li2_deriv).
 680                        series_func(Li2_series).
 681                        conjugate_func(Li2_conjugate).
 682                        latex_name("\\mathrm{Li}_2"));
 683
 684 //////////
 685 // trilogarithm
 686 //////////
 687
 688 static ex Li3_eval(const ex & x)
 689 {
 690         if (x.is_zero())
 691                 return x;
 692         return Li3(x).hold();
 693 }
 694
 695 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 696                        latex_name("\\mathrm{Li}_3"));
 697
 698 //////////
 699 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 700 //////////
 701
 702 static ex zetaderiv_eval(const ex & n, const ex & x)
 703 {
 704         if (n.info(info_flags::numeric)) {
 705                 // zetaderiv(0,x) -> zeta(x)
 706                 if (n.is_zero())
 707                         return zeta(x).hold();
 708         }
 709
 710         return zetaderiv(n, x).hold();
 711 }
 712
 713 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 714 {
 715         GINAC_ASSERT(deriv_param<2);
 716
 717         if (deriv_param==0) {
 718                 // d/dn zeta(n,x)
 719                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 720         }
 721         // d/dx psi(n,x)
 722         return zetaderiv(n+1,x);
 723 }
 724
 725 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 726                                  derivative_func(zetaderiv_deriv).
 727                                  latex_name("\\zeta^\\prime"));
 728
 729 //////////
 730 // factorial
 731 //////////
 732
 733 static ex factorial_evalf(const ex & x)
 734 {
 735         return factorial(x).hold();
 736 }
 737
 738 static ex factorial_eval(const ex & x)
 739 {
 740         if (is_exactly_a<numeric>(x))
 741                 return factorial(ex_to<numeric>(x));
 742         else
 743                 return factorial(x).hold();
 744 }
 745
 746 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 747 {
 748         if (is_exactly_a<symbol>(x) ||
 749             is_exactly_a<constant>(x) ||
 750                 is_exactly_a<function>(x)) {
 751                 x.print(c); c.s << "!";
 752         } else {
 753                 c.s << "("; x.print(c); c.s << ")!";
 754         }
 755 }
 756
 757 static ex factorial_conjugate(const ex & x)
 758 {
 759         return factorial(x).hold();
 760 }
 761
 762 static ex factorial_real_part(const ex & x)
 763 {
 764         return factorial(x).hold();
 765 }
 766
 767 static ex factorial_imag_part(const ex & x)
 768 {
 769         return 0;
 770 }
 771
 772 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 773                              evalf_func(factorial_evalf).
 774                              print_func<print_dflt>(factorial_print_dflt_latex).
 775                              print_func<print_latex>(factorial_print_dflt_latex).
 776                              conjugate_func(factorial_conjugate).
 777                              real_part_func(factorial_real_part).
 778                              imag_part_func(factorial_imag_part));
 779
 780 //////////
 781 // binomial
 782 //////////
 783
 784 static ex binomial_evalf(const ex & x, const ex & y)
 785 {
 786         return binomial(x, y).hold();
 787 }
 788
 789 static ex binomial_sym(const ex & x, const numeric & y)
 790 {
 791         if (y.is_integer()) {
 792                 if (y.is_nonneg_integer()) {
 793                         const unsigned N = y.to_int();
 794                         if (N == 0) return _ex1;
 795                         if (N == 1) return x;
 796                         ex t = x.expand();
 797                         for (unsigned i = 2; i <= N; ++i)
 798                                 t = (t * (x + i - y - 1)).expand() / i;
 799                         return t;
 800                 } else
 801                         return _ex0;
 802         }
 803
 804         return binomial(x, y).hold();
 805 }
 806
 807 static ex binomial_eval(const ex & x, const ex &y)
 808 {
 809         if (is_exactly_a<numeric>(y)) {
 810                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 811                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 812                 else
 813                         return binomial_sym(x, ex_to<numeric>(y));
 814         } else
 815                 return binomial(x, y).hold();
 816 }
 817
 818 // At the moment the numeric evaluation of a binomail function always
 819 // gives a real number, but if this would be implemented using the gamma
 820 // function, also complex conjugation should be changed (or rather, deleted).
 821 static ex binomial_conjugate(const ex & x, const ex & y)
 822 {
 823         return binomial(x,y).hold();
 824 }
 825
 826 static ex binomial_real_part(const ex & x, const ex & y)
 827 {
 828         return binomial(x,y).hold();
 829 }
 830
 831 static ex binomial_imag_part(const ex & x, const ex & y)
 832 {
 833         return 0;
 834 }
 835
 836 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 837                             evalf_func(binomial_evalf).
 838                             conjugate_func(binomial_conjugate).
 839                             real_part_func(binomial_real_part).
 840                             imag_part_func(binomial_imag_part));
 841
 842 //////////
 843 // Order term function (for truncated power series)
 844 //////////
 845
 846 static ex Order_eval(const ex & x)
 847 {
 848         if (is_exactly_a<numeric>(x)) {
 849                 // O(c) -> O(1) or 0
 850                 if (!x.is_zero())
 851                         return Order(_ex1).hold();
 852                 else
 853                         return _ex0;
 854         } else if (is_exactly_a<mul>(x)) {
 855                 const mul &m = ex_to<mul>(x);
 856                 // O(c*expr) -> O(expr)
 857                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 858                         return Order(x / m.op(m.nops() - 1)).hold();
 859         }
 860         return Order(x).hold();
 861 }
 862
 863 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 864 {
 865         // Just wrap the function into a pseries object
 866         epvector new_seq;
 867         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 868         const symbol &s = ex_to<symbol>(r.lhs());
 869         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 870         return pseries(r, new_seq);
 871 }
 872
 873 static ex Order_conjugate(const ex & x)
 874 {
 875         return Order(x).hold();
 876 }
 877
 878 static ex Order_real_part(const ex & x)
 879 {
 880         return Order(x).hold();
 881 }
 882
 883 static ex Order_imag_part(const ex & x)
 884 {
 885         if(x.info(info_flags::real))
 886                 return 0;
 887         return Order(x).hold();
 888 }
 889
 890 // Differentiation is handled in function::derivative because of its special requirements
 891
 892 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 893                          series_func(Order_series).
 894                          latex_name("\\mathcal{O}").
 895                          conjugate_func(Order_conjugate).
 896                          real_part_func(Order_real_part).
 897                          imag_part_func(Order_imag_part));
 898
 899 //////////
 900 // Solve linear system
 901 //////////
 902
 903 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 904 {
 905         // solve a system of linear equations
 906         if (eqns.info(info_flags::relation_equal)) {
 907                 if (!symbols.info(info_flags::symbol))
 908                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 909                 const ex sol = lsolve(lst(eqns),lst(symbols));
 910
 911                 GINAC_ASSERT(sol.nops()==1);
 912                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 913
 914                 return sol.op(0).op(1); // return rhs of first solution
 915         }
 916
 917         // syntax checks
 918         if (!eqns.info(info_flags::list)) {
 919                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
 920         }
 921         for (size_t i=0; i<eqns.nops(); i++) {
 922                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 923                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 924                 }
 925         }
 926         if (!symbols.info(info_flags::list)) {
 927                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
 928         }
 929         for (size_t i=0; i<symbols.nops(); i++) {
 930                 if (!symbols.op(i).info(info_flags::symbol)) {
 931                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 932                 }
 933         }
 934
 935         // build matrix from equation system
 936         matrix sys(eqns.nops(),symbols.nops());
 937         matrix rhs(eqns.nops(),1);
 938         matrix vars(symbols.nops(),1);
 939
 940         for (size_t r=0; r<eqns.nops(); r++) {
 941                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 942                 ex linpart = eq;
 943                 for (size_t c=0; c<symbols.nops(); c++) {
 944                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 945                         linpart -= co*symbols.op(c);
 946                         sys(r,c) = co;
 947                 }
 948                 linpart = linpart.expand();
 949                 rhs(r,0) = -linpart;
 950         }
 951
 952         // test if system is linear and fill vars matrix
 953         for (size_t i=0; i<symbols.nops(); i++) {
 954                 vars(i,0) = symbols.op(i);
 955                 if (sys.has(symbols.op(i)))
 956                         throw(std::logic_error("lsolve: system is not linear"));
 957                 if (rhs.has(symbols.op(i)))
 958                         throw(std::logic_error("lsolve: system is not linear"));
 959         }
 960
 961         matrix solution;
 962         try {
 963                 solution = sys.solve(vars,rhs,options);
 964         } catch (const std::runtime_error & e) {
 965                 // Probably singular matrix or otherwise overdetermined system:
 966                 // It is consistent to return an empty list
 967                 return lst();
 968         }
 969         GINAC_ASSERT(solution.cols()==1);
 970         GINAC_ASSERT(solution.rows()==symbols.nops());
 971
 972         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 973         lst sollist;
 974         for (size_t i=0; i<symbols.nops(); i++)
 975                 sollist.append(symbols.op(i)==solution(i,0));
 976
 977         return sollist;
 978 }
 979
 980 //////////
 981 // Find real root of f(x) numerically
 982 //////////
 983
 984 const numeric
 985 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
 986 {
 987         if (!x1.is_real() || !x2.is_real()) {
 988                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
 989         }
 990         if (x1==x2) {
 991                 throw std::runtime_error("fsolve(): vanishing interval");
 992         }
 993         // xx[0] == left interval limit, xx[1] == right interval limit.
 994         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
 995         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
 996         numeric xx[2] = { x1<x2 ? x1 : x2,
 997                           x1<x2 ? x2 : x1 };
 998         ex f;
 999         if (is_a<relational>(f_in)) {
1000                 f = f_in.lhs()-f_in.rhs();
1001         } else {
1002                 f = f_in;
1003         }
1004         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1005                             f.subs(x==xx[1]).evalf() };
1006         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1007                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1008         }
1009         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1010                           ex_to<numeric>(fx_[1]) };
1011         if (!fx[0].is_real() || !fx[1].is_real()) {
1012                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1013         }
1014         if (fx[0]*fx[1]>=0) {
1015                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1016         }
1017
1018         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1019         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1020         const ex ff = normal(-f/f.diff(x));
1021         int side = 0;  // Start at left interval limit.
1022         numeric xxprev;
1023         numeric fxprev;
1024         do {
1025                 xxprev = xx[side];
1026                 fxprev = fx[side];
1027                 ex dx_ = ff.subs(x == xx[side]).evalf();
1028                 if (!is_a<numeric>(dx_))
1029                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1030                 xx[side] += ex_to<numeric>(dx_);
1031                 // Now check if Newton-Raphson method shot out of the interval
1032                 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1033                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1034                 if (!bad_shot) {
1035                         // Compute f(x) only if new x is inside the interval.
1036                         // The function might be difficult to compute numerically
1037                         // or even ill defined outside the interval. Also it's
1038                         // a small optimization.
1039                         ex f_x = f.subs(x == xx[side]).evalf();
1040                         if (!is_a<numeric>(f_x))
1041                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1042                         fx[side] = ex_to<numeric>(f_x);
1043                 }
1044                 if (bad_shot) {
1045                         // Oops, Newton-Raphson method shot out of the interval.
1046                         // Restore, and try again with the other side instead!
1047                         xx[side] = xxprev;
1048                         fx[side] = fxprev;
1049                         side = !side;
1050                         xxprev = xx[side];
1051                         fxprev = fx[side];
1052
1053                         ex dx_ = ff.subs(x == xx[side]).evalf();
1054                         if (!is_a<numeric>(dx_))
1055                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1056                         xx[side] += ex_to<numeric>(dx_);
1057
1058                         ex f_x = f.subs(x==xx[side]).evalf();
1059                         if (!is_a<numeric>(f_x))
1060                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1061                         fx[side] = ex_to<numeric>(f_x);
1062                 }
1063                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1064                         // Oops, the root isn't bracketed any more.
1065                         // Restore, and perform a bisection!
1066                         xx[side] = xxprev;
1067                         fx[side] = fxprev;
1068
1069                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1070                         // they occur pretty often when Newton-Raphson arrives at an x too
1071                         // close to the result on one side of the interval and
1072                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1073                         // precision errors! Recall that this function does not have a
1074                         // precision goal as one of its arguments but instead relies on
1075                         // x converging to a fixed point. We speed up the (safe but slow)
1076                         // bisection method by mixing in a dash of the (unsafer but faster)
1077                         // secant method: Instead of splitting the interval at the
1078                         // arithmetic mean (bisection), we split it nearer to the root as
1079                         // determined by the secant between the values xx[0] and xx[1].
1080                         // Don't set the secant_weight to one because that could disturb
1081                         // the convergence in some corner cases!
1082                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1083                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1084                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1085                         ex fxmid_ = f.subs(x == xxmid).evalf();
1086                         if (!is_a<numeric>(fxmid_))
1087                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1088                         numeric fxmid = ex_to<numeric>(fxmid_);
1089                         if (fxmid.is_zero()) {
1090                                 // Luck strikes...
1091                                 return xxmid;
1092                         }
1093                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1094                                 side = !side;
1095                         }
1096                         xxprev = xx[side];
1097                         fxprev = fx[side];
1098                         xx[side] = xxmid;
1099                         fx[side] = fxmid;
1100                 }
1101         } while (xxprev!=xx[side]);
1102         return xxprev;
1103 }
1104
1105
1106 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1107  * for static lib (so ginsh will see them). */
1108 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1109 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1110
1111 } // namespace GiNaC