ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2005 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 <vector>
  24 #include <stdexcept>
  25
  26 #include "inifcns.h"
  27 #include "ex.h"
  28 #include "constant.h"
  29 #include "lst.h"
  30 #include "matrix.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "operators.h"
  34 #include "relational.h"
  35 #include "pseries.h"
  36 #include "symbol.h"
  37 #include "symmetry.h"
  38 #include "utils.h"
  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 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
  70                                       evalf_func(conjugate_evalf).
  71                                       print_func<print_latex>(conjugate_print_latex).
  72                                       conjugate_func(conjugate_conjugate).
  73                                       set_name("conjugate","conjugate"));
  74
  75 //////////
  76 // absolute value
  77 //////////
  78
  79 static ex abs_evalf(const ex & arg)
  80 {
  81         if (is_exactly_a<numeric>(arg))
  82                 return abs(ex_to<numeric>(arg));
  83
  84         return abs(arg).hold();
  85 }
  86
  87 static ex abs_eval(const ex & arg)
  88 {
  89         if (is_exactly_a<numeric>(arg))
  90                 return abs(ex_to<numeric>(arg));
  91         else
  92                 return abs(arg).hold();
  93 }
  94
  95 static void abs_print_latex(const ex & arg, const print_context & c)
  96 {
  97         c.s << "{|"; arg.print(c); c.s << "|}";
  98 }
  99
 100 static void abs_print_csrc_float(const ex & arg, const print_context & c)
 101 {
 102         c.s << "fabs("; arg.print(c); c.s << ")";
 103 }
 104
 105 static ex abs_conjugate(const ex & arg)
 106 {
 107         return abs(arg);
 108 }
 109
 110 static ex abs_power(const ex & arg, const ex & exp)
 111 {
 112         if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
 113                 return power(arg, exp);
 114         else
 115                 return power(abs(arg), exp).hold();
 116 }
 117
 118 REGISTER_FUNCTION(abs, eval_func(abs_eval).
 119                        evalf_func(abs_evalf).
 120                        print_func<print_latex>(abs_print_latex).
 121                        print_func<print_csrc_float>(abs_print_csrc_float).
 122                        print_func<print_csrc_double>(abs_print_csrc_float).
 123                        conjugate_func(abs_conjugate).
 124                        power_func(abs_power));
 125
 126 //////////
 127 // Step function
 128 //////////
 129
 130 static ex step_evalf(const ex & arg)
 131 {
 132         if (is_exactly_a<numeric>(arg))
 133                 return step(ex_to<numeric>(arg));
 134
 135         return step(arg).hold();
 136 }
 137
 138 static ex step_eval(const ex & arg)
 139 {
 140         if (is_exactly_a<numeric>(arg))
 141                 return step(ex_to<numeric>(arg));
 142
 143         else if (is_exactly_a<mul>(arg) &&
 144                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 145                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 146                 if (oc.is_real()) {
 147                         if (oc > 0)
 148                                 // step(42*x) -> step(x)
 149                                 return step(arg/oc).hold();
 150                         else
 151                                 // step(-42*x) -> step(-x)
 152                                 return step(-arg/oc).hold();
 153                 }
 154                 if (oc.real().is_zero()) {
 155                         if (oc.imag() > 0)
 156                                 // step(42*I*x) -> step(I*x)
 157                                 return step(I*arg/oc).hold();
 158                         else
 159                                 // step(-42*I*x) -> step(-I*x)
 160                                 return step(-I*arg/oc).hold();
 161                 }
 162         }
 163
 164         return step(arg).hold();
 165 }
 166
 167 static ex step_series(const ex & arg,
 168                       const relational & rel,
 169                       int order,
 170                       unsigned options)
 171 {
 172         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 173         if (arg_pt.info(info_flags::numeric)
 174             && ex_to<numeric>(arg_pt).real().is_zero()
 175             && !(options & series_options::suppress_branchcut))
 176                 throw (std::domain_error("step_series(): on imaginary axis"));
 177
 178         epvector seq;
 179         seq.push_back(expair(step(arg_pt), _ex0));
 180         return pseries(rel,seq);
 181 }
 182
 183 static ex step_power(const ex & arg, const ex & exp)
 184 {
 185         if (exp.info(info_flags::positive))
 186                 return step(arg);
 187
 188         return power(step(arg), exp).hold();
 189 }
 190
 191 static ex step_conjugate(const ex& arg)
 192 {
 193         return step(arg);
 194 }
 195
 196 REGISTER_FUNCTION(step, eval_func(step_eval).
 197                         evalf_func(step_evalf).
 198                         series_func(step_series).
 199                         conjugate_func(step_conjugate).
 200                         power_func(step_power));
 201
 202 //////////
 203 // Complex sign
 204 //////////
 205
 206 static ex csgn_evalf(const ex & arg)
 207 {
 208         if (is_exactly_a<numeric>(arg))
 209                 return csgn(ex_to<numeric>(arg));
 210
 211         return csgn(arg).hold();
 212 }
 213
 214 static ex csgn_eval(const ex & arg)
 215 {
 216         if (is_exactly_a<numeric>(arg))
 217                 return csgn(ex_to<numeric>(arg));
 218
 219         else if (is_exactly_a<mul>(arg) &&
 220                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
 221                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
 222                 if (oc.is_real()) {
 223                         if (oc > 0)
 224                                 // csgn(42*x) -> csgn(x)
 225                                 return csgn(arg/oc).hold();
 226                         else
 227                                 // csgn(-42*x) -> -csgn(x)
 228                                 return -csgn(arg/oc).hold();
 229                 }
 230                 if (oc.real().is_zero()) {
 231                         if (oc.imag() > 0)
 232                                 // csgn(42*I*x) -> csgn(I*x)
 233                                 return csgn(I*arg/oc).hold();
 234                         else
 235                                 // csgn(-42*I*x) -> -csgn(I*x)
 236                                 return -csgn(I*arg/oc).hold();
 237                 }
 238         }
 239
 240         return csgn(arg).hold();
 241 }
 242
 243 static ex csgn_series(const ex & arg,
 244                       const relational & rel,
 245                       int order,
 246                       unsigned options)
 247 {
 248         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 249         if (arg_pt.info(info_flags::numeric)
 250             && ex_to<numeric>(arg_pt).real().is_zero()
 251             && !(options & series_options::suppress_branchcut))
 252                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 253
 254         epvector seq;
 255         seq.push_back(expair(csgn(arg_pt), _ex0));
 256         return pseries(rel,seq);
 257 }
 258
 259 static ex csgn_conjugate(const ex& arg)
 260 {
 261         return csgn(arg);
 262 }
 263
 264 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 265                         evalf_func(csgn_evalf).
 266                         series_func(csgn_series).
 267                         conjugate_func(csgn_conjugate));
 268
 269
 270 //////////
 271 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 272 // This function is closely related to the unwinding number K, sometimes found
 273 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 274 //////////
 275
 276 static ex eta_evalf(const ex &x, const ex &y)
 277 {
 278         // It seems like we basically have to replicate the eval function here,
 279         // since the expression might not be fully evaluated yet.
 280         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 281                 return _ex0;
 282
 283         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 284                 const numeric nx = ex_to<numeric>(x);
 285                 const numeric ny = ex_to<numeric>(y);
 286                 const numeric nxy = ex_to<numeric>(x*y);
 287                 int cut = 0;
 288                 if (nx.is_real() && nx.is_negative())
 289                         cut -= 4;
 290                 if (ny.is_real() && ny.is_negative())
 291                         cut -= 4;
 292                 if (nxy.is_real() && nxy.is_negative())
 293                         cut += 4;
 294                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 295                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 296         }
 297
 298         return eta(x,y).hold();
 299 }
 300
 301 static ex eta_eval(const ex &x, const ex &y)
 302 {
 303         // trivial:  eta(x,c) -> 0  if c is real and positive
 304         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 305                 return _ex0;
 306
 307         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 308                 // don't call eta_evalf here because it would call Pi.evalf()!
 309                 const numeric nx = ex_to<numeric>(x);
 310                 const numeric ny = ex_to<numeric>(y);
 311                 const numeric nxy = ex_to<numeric>(x*y);
 312                 int cut = 0;
 313                 if (nx.is_real() && nx.is_negative())
 314                         cut -= 4;
 315                 if (ny.is_real() && ny.is_negative())
 316                         cut -= 4;
 317                 if (nxy.is_real() && nxy.is_negative())
 318                         cut += 4;
 319                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 320                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 321         }
 322
 323         return eta(x,y).hold();
 324 }
 325
 326 static ex eta_series(const ex & x, const ex & y,
 327                      const relational & rel,
 328                      int order,
 329                      unsigned options)
 330 {
 331         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 332         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 333         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 334             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 335             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 336                         throw (std::domain_error("eta_series(): on discontinuity"));
 337         epvector seq;
 338         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 339         return pseries(rel,seq);
 340 }
 341
 342 static ex eta_conjugate(const ex & x, const ex & y)
 343 {
 344         return -eta(x,y);
 345 }
 346
 347 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 348                        evalf_func(eta_evalf).
 349                        series_func(eta_series).
 350                        latex_name("\\eta").
 351                        set_symmetry(sy_symm(0, 1)).
 352                        conjugate_func(eta_conjugate));
 353
 354
 355 //////////
 356 // dilogarithm
 357 //////////
 358
 359 static ex Li2_evalf(const ex & x)
 360 {
 361         if (is_exactly_a<numeric>(x))
 362                 return Li2(ex_to<numeric>(x));
 363
 364         return Li2(x).hold();
 365 }
 366
 367 static ex Li2_eval(const ex & x)
 368 {
 369         if (x.info(info_flags::numeric)) {
 370                 // Li2(0) -> 0
 371                 if (x.is_zero())
 372                         return _ex0;
 373                 // Li2(1) -> Pi^2/6
 374                 if (x.is_equal(_ex1))
 375                         return power(Pi,_ex2)/_ex6;
 376                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 377                 if (x.is_equal(_ex1_2))
 378                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 379                 // Li2(-1) -> -Pi^2/12
 380                 if (x.is_equal(_ex_1))
 381                         return -power(Pi,_ex2)/_ex12;
 382                 // Li2(I) -> -Pi^2/48+Catalan*I
 383                 if (x.is_equal(I))
 384                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 385                 // Li2(-I) -> -Pi^2/48-Catalan*I
 386                 if (x.is_equal(-I))
 387                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 388                 // Li2(float)
 389                 if (!x.info(info_flags::crational))
 390                         return Li2(ex_to<numeric>(x));
 391         }
 392
 393         return Li2(x).hold();
 394 }
 395
 396 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 397 {
 398         GINAC_ASSERT(deriv_param==0);
 399
 400         // d/dx Li2(x) -> -log(1-x)/x
 401         return -log(_ex1-x)/x;
 402 }
 403
 404 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 405 {
 406         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 407         if (x_pt.info(info_flags::numeric)) {
 408                 // First special case: x==0 (derivatives have poles)
 409                 if (x_pt.is_zero()) {
 410                         // method:
 411                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 412                         // simply substitute x==0.  The limit, however, exists: it is 1.
 413                         // We also know all higher derivatives' limits:
 414                         // (d/dx)^n Li2(x) == n!/n^2.
 415                         // So the primitive series expansion is
 416                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 417                         // and so on.
 418                         // We first construct such a primitive series expansion manually in
 419                         // a dummy symbol s and then insert the argument's series expansion
 420                         // for s.  Reexpanding the resulting series returns the desired
 421                         // result.
 422                         const symbol s;
 423                         ex ser;
 424                         // manually construct the primitive expansion
 425                         for (int i=1; i<order; ++i)
 426                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 427                         // substitute the argument's series expansion
 428                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 429                         // maybe that was terminating, so add a proper order term
 430                         epvector nseq;
 431                         nseq.push_back(expair(Order(_ex1), order));
 432                         ser += pseries(rel, nseq);
 433                         // reexpanding it will collapse the series again
 434                         return ser.series(rel, order);
 435                         // NB: Of course, this still does not allow us to compute anything
 436                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 437                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 438                         // substitution x==0.  Probably limits *are* needed for the general
 439                         // cases.  In case L'Hospital's rule is implemented for limits and
 440                         // basic::series() takes care of this, this whole block is probably
 441                         // obsolete!
 442                 }
 443                 // second special case: x==1 (branch point)
 444                 if (x_pt.is_equal(_ex1)) {
 445                         // method:
 446                         // construct series manually in a dummy symbol s
 447                         const symbol s;
 448                         ex ser = zeta(_ex2);
 449                         // manually construct the primitive expansion
 450                         for (int i=1; i<order; ++i)
 451                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 452                         // substitute the argument's series expansion
 453                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 454                         // maybe that was terminating, so add a proper order term
 455                         epvector nseq;
 456                         nseq.push_back(expair(Order(_ex1), order));
 457                         ser += pseries(rel, nseq);
 458                         // reexpanding it will collapse the series again
 459                         return ser.series(rel, order);
 460                 }
 461                 // third special case: x real, >=1 (branch cut)
 462                 if (!(options & series_options::suppress_branchcut) &&
 463                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 464                         // method:
 465                         // This is the branch cut: assemble the primitive series manually
 466                         // and then add the corresponding complex step function.
 467                         const symbol &s = ex_to<symbol>(rel.lhs());
 468                         const ex point = rel.rhs();
 469                         const symbol foo;
 470                         epvector seq;
 471                         // zeroth order term:
 472                         seq.push_back(expair(Li2(x_pt), _ex0));
 473                         // compute the intermediate terms:
 474                         ex replarg = series(Li2(x), s==foo, order);
 475                         for (size_t i=1; i<replarg.nops()-1; ++i)
 476                                 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));
 477                         // append an order term:
 478                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 479                         return pseries(rel, seq);
 480                 }
 481         }
 482         // all other cases should be safe, by now:
 483         throw do_taylor();  // caught by function::series()
 484 }
 485
 486 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 487                        evalf_func(Li2_evalf).
 488                        derivative_func(Li2_deriv).
 489                        series_func(Li2_series).
 490                        latex_name("\\mbox{Li}_2"));
 491
 492 //////////
 493 // trilogarithm
 494 //////////
 495
 496 static ex Li3_eval(const ex & x)
 497 {
 498         if (x.is_zero())
 499                 return x;
 500         return Li3(x).hold();
 501 }
 502
 503 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 504                        latex_name("\\mbox{Li}_3"));
 505
 506 //////////
 507 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 508 //////////
 509
 510 static ex zetaderiv_eval(const ex & n, const ex & x)
 511 {
 512         if (n.info(info_flags::numeric)) {
 513                 // zetaderiv(0,x) -> zeta(x)
 514                 if (n.is_zero())
 515                         return zeta(x);
 516         }
 517
 518         return zetaderiv(n, x).hold();
 519 }
 520
 521 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 522 {
 523         GINAC_ASSERT(deriv_param<2);
 524
 525         if (deriv_param==0) {
 526                 // d/dn zeta(n,x)
 527                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 528         }
 529         // d/dx psi(n,x)
 530         return zetaderiv(n+1,x);
 531 }
 532
 533 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 534                                  derivative_func(zetaderiv_deriv).
 535                                  latex_name("\\zeta^\\prime"));
 536
 537 //////////
 538 // factorial
 539 //////////
 540
 541 static ex factorial_evalf(const ex & x)
 542 {
 543         return factorial(x).hold();
 544 }
 545
 546 static ex factorial_eval(const ex & x)
 547 {
 548         if (is_exactly_a<numeric>(x))
 549                 return factorial(ex_to<numeric>(x));
 550         else
 551                 return factorial(x).hold();
 552 }
 553
 554 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 555 {
 556         if (is_exactly_a<symbol>(x) ||
 557             is_exactly_a<constant>(x) ||
 558                 is_exactly_a<function>(x)) {
 559                 x.print(c); c.s << "!";
 560         } else {
 561                 c.s << "("; x.print(c); c.s << ")!";
 562         }
 563 }
 564
 565 static ex factorial_conjugate(const ex & x)
 566 {
 567         return factorial(x);
 568 }
 569
 570 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 571                              evalf_func(factorial_evalf).
 572                              print_func<print_dflt>(factorial_print_dflt_latex).
 573                              print_func<print_latex>(factorial_print_dflt_latex).
 574                              conjugate_func(factorial_conjugate));
 575
 576 //////////
 577 // binomial
 578 //////////
 579
 580 static ex binomial_evalf(const ex & x, const ex & y)
 581 {
 582         return binomial(x, y).hold();
 583 }
 584
 585 static ex binomial_sym(const ex & x, const numeric & y)
 586 {
 587         if (y.is_integer()) {
 588                 if (y.is_nonneg_integer()) {
 589                         const unsigned N = y.to_int();
 590                         if (N == 0) return _ex0;
 591                         if (N == 1) return x;
 592                         ex t = x.expand();
 593                         for (unsigned i = 2; i <= N; ++i)
 594                                 t = (t * (x + i - y - 1)).expand() / i;
 595                         return t;
 596                 } else
 597                         return _ex0;
 598         }
 599
 600         return binomial(x, y).hold();
 601 }
 602
 603 static ex binomial_eval(const ex & x, const ex &y)
 604 {
 605         if (is_exactly_a<numeric>(y)) {
 606                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 607                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 608                 else
 609                         return binomial_sym(x, ex_to<numeric>(y));
 610         } else
 611                 return binomial(x, y).hold();
 612 }
 613
 614 // At the moment the numeric evaluation of a binomail function always
 615 // gives a real number, but if this would be implemented using the gamma
 616 // function, also complex conjugation should be changed (or rather, deleted).
 617 static ex binomial_conjugate(const ex & x, const ex & y)
 618 {
 619         return binomial(x,y);
 620 }
 621
 622 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 623                             evalf_func(binomial_evalf).
 624                             conjugate_func(binomial_conjugate));
 625
 626 //////////
 627 // Order term function (for truncated power series)
 628 //////////
 629
 630 static ex Order_eval(const ex & x)
 631 {
 632         if (is_exactly_a<numeric>(x)) {
 633                 // O(c) -> O(1) or 0
 634                 if (!x.is_zero())
 635                         return Order(_ex1).hold();
 636                 else
 637                         return _ex0;
 638         } else if (is_exactly_a<mul>(x)) {
 639                 const mul &m = ex_to<mul>(x);
 640                 // O(c*expr) -> O(expr)
 641                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 642                         return Order(x / m.op(m.nops() - 1)).hold();
 643         }
 644         return Order(x).hold();
 645 }
 646
 647 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 648 {
 649         // Just wrap the function into a pseries object
 650         epvector new_seq;
 651         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 652         const symbol &s = ex_to<symbol>(r.lhs());
 653         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 654         return pseries(r, new_seq);
 655 }
 656
 657 static ex Order_conjugate(const ex & x)
 658 {
 659         return Order(x);
 660 }
 661
 662 // Differentiation is handled in function::derivative because of its special requirements
 663
 664 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 665                          series_func(Order_series).
 666                          latex_name("\\mathcal{O}").
 667                          conjugate_func(Order_conjugate));
 668
 669 //////////
 670 // Solve linear system
 671 //////////
 672
 673 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 674 {
 675         // solve a system of linear equations
 676         if (eqns.info(info_flags::relation_equal)) {
 677                 if (!symbols.info(info_flags::symbol))
 678                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 679                 const ex sol = lsolve(lst(eqns),lst(symbols));
 680
 681                 GINAC_ASSERT(sol.nops()==1);
 682                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 683
 684                 return sol.op(0).op(1); // return rhs of first solution
 685         }
 686
 687         // syntax checks
 688         if (!eqns.info(info_flags::list)) {
 689                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 690         }
 691         for (size_t i=0; i<eqns.nops(); i++) {
 692                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 693                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 694                 }
 695         }
 696         if (!symbols.info(info_flags::list)) {
 697                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 698         }
 699         for (size_t i=0; i<symbols.nops(); i++) {
 700                 if (!symbols.op(i).info(info_flags::symbol)) {
 701                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 702                 }
 703         }
 704
 705         // build matrix from equation system
 706         matrix sys(eqns.nops(),symbols.nops());
 707         matrix rhs(eqns.nops(),1);
 708         matrix vars(symbols.nops(),1);
 709
 710         for (size_t r=0; r<eqns.nops(); r++) {
 711                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 712                 ex linpart = eq;
 713                 for (size_t c=0; c<symbols.nops(); c++) {
 714                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 715                         linpart -= co*symbols.op(c);
 716                         sys(r,c) = co;
 717                 }
 718                 linpart = linpart.expand();
 719                 rhs(r,0) = -linpart;
 720         }
 721
 722         // test if system is linear and fill vars matrix
 723         for (size_t i=0; i<symbols.nops(); i++) {
 724                 vars(i,0) = symbols.op(i);
 725                 if (sys.has(symbols.op(i)))
 726                         throw(std::logic_error("lsolve: system is not linear"));
 727                 if (rhs.has(symbols.op(i)))
 728                         throw(std::logic_error("lsolve: system is not linear"));
 729         }
 730
 731         matrix solution;
 732         try {
 733                 solution = sys.solve(vars,rhs,options);
 734         } catch (const std::runtime_error & e) {
 735                 // Probably singular matrix or otherwise overdetermined system:
 736                 // It is consistent to return an empty list
 737                 return lst();
 738         }
 739         GINAC_ASSERT(solution.cols()==1);
 740         GINAC_ASSERT(solution.rows()==symbols.nops());
 741
 742         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 743         lst sollist;
 744         for (size_t i=0; i<symbols.nops(); i++)
 745                 sollist.append(symbols.op(i)==solution(i,0));
 746
 747         return sollist;
 748 }
 749
 750 //////////
 751 // Find real root of f(x) numerically
 752 //////////
 753
 754 const numeric
 755 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
 756 {
 757         if (!x1.is_real() || !x2.is_real()) {
 758                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
 759         }
 760         if (x1==x2) {
 761                 throw std::runtime_error("fsolve(): vanishing interval");
 762         }
 763         // xx[0] == left interval limit, xx[1] == right interval limit.
 764         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
 765         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
 766         numeric xx[2] = { x1<x2 ? x1 : x2,
 767                           x1<x2 ? x2 : x1 };
 768         ex f;
 769         if (is_a<relational>(f_in)) {
 770                 f = f_in.lhs()-f_in.rhs();
 771         } else {
 772                 f = f_in;
 773         }
 774         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
 775                             f.subs(x==xx[1]).evalf() };
 776         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
 777                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
 778         }
 779         numeric fx[2] = { ex_to<numeric>(fx_[0]),
 780                           ex_to<numeric>(fx_[1]) };
 781         if (!fx[0].is_real() || !fx[1].is_real()) {
 782                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
 783         }
 784         if (fx[0]*fx[1]>=0) {
 785                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
 786         }
 787
 788         // The Newton-Raphson method has quadratic convergence!  Simply put, it
 789         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
 790         const ex ff = normal(-f/f.diff(x));
 791         int side = 0;  // Start at left interval limit.
 792         numeric xxprev;
 793         numeric fxprev;
 794         do {
 795                 xxprev = xx[side];
 796                 fxprev = fx[side];
 797                 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
 798                 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
 799                 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
 800                         // Oops, Newton-Raphson method shot out of the interval.
 801                         // Restore, and try again with the other side instead!
 802                         xx[side] = xxprev;
 803                         fx[side] = fxprev;
 804                         side = !side;
 805                         xxprev = xx[side];
 806                         fxprev = fx[side];
 807                         xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
 808                         fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
 809                 }
 810                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
 811                         // Oops, the root isn't bracketed any more.
 812                         // Restore, and perform a bisection!
 813                         xx[side] = xxprev;
 814                         fx[side] = fxprev;
 815
 816                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
 817                         // they occur pretty often when Newton-Raphson arrives at an x too
 818                         // close to the result on one side of the interval and
 819                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
 820                         // precision errors! Recall that this function does not have a
 821                         // precision goal as one of its arguments but instead relies on
 822                         // x converging to a fixed point. We speed up the (safe but slow)
 823                         // bisection method by mixing in a dash of the (unsafer but faster)
 824                         // secant method: Instead of splitting the interval at the
 825                         // arithmetic mean (bisection), we split it nearer to the root as
 826                         // determined by the secant between the values xx[0] and xx[1].
 827                         // Don't set the secant_weight to one because that could disturb
 828                         // the convergence in some corner cases!
 829                         static const double secant_weight = 0.984375;  // == 63/64 < 1
 830                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
 831                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
 832                         numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
 833                         if (fxmid.is_zero()) {
 834                                 // Luck strikes...
 835                                 return xxmid;
 836                         }
 837                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
 838                                 side = !side;
 839                         }
 840                         xxprev = xx[side];
 841                         fxprev = fx[side];
 842                         xx[side] = xxmid;
 843                         fx[side] = fxmid;
 844                 }
 845         } while (xxprev!=xx[side]);
 846         return xxprev;
 847 }
 848
 849
 850 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 851  * for static lib (so ginsh will see them). */
 852 unsigned force_include_tgamma = tgamma_SERIAL::serial;
 853 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 854
 855 } // namespace GiNaC