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