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