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