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