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 static ex csgn_power(const ex & arg, const ex & exp)
 256 {
 257         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
 258                 if (ex_to<numeric>(exp).is_odd())
 259                         return csgn(arg);
 260                 else
 261                         return power(csgn(arg), _ex2).hold();
 262         } else
 263                 return power(csgn(arg), exp).hold();
 264 }
 265
 266
 267 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 268                         evalf_func(csgn_evalf).
 269                         series_func(csgn_series).
 270                         conjugate_func(csgn_conjugate).
 271                         power_func(csgn_power));
 272
 273
 274 //////////
 275 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 276 // This function is closely related to the unwinding number K, sometimes found
 277 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 278 //////////
 279
 280 static ex eta_evalf(const ex &x, const ex &y)
 281 {
 282         // It seems like we basically have to replicate the eval function here,
 283         // since the expression might not be fully evaluated yet.
 284         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 285                 return _ex0;
 286
 287         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 288                 const numeric nx = ex_to<numeric>(x);
 289                 const numeric ny = ex_to<numeric>(y);
 290                 const numeric nxy = ex_to<numeric>(x*y);
 291                 int cut = 0;
 292                 if (nx.is_real() && nx.is_negative())
 293                         cut -= 4;
 294                 if (ny.is_real() && ny.is_negative())
 295                         cut -= 4;
 296                 if (nxy.is_real() && nxy.is_negative())
 297                         cut += 4;
 298                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 299                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 300         }
 301
 302         return eta(x,y).hold();
 303 }
 304
 305 static ex eta_eval(const ex &x, const ex &y)
 306 {
 307         // trivial:  eta(x,c) -> 0  if c is real and positive
 308         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 309                 return _ex0;
 310
 311         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 312                 // don't call eta_evalf here because it would call Pi.evalf()!
 313                 const numeric nx = ex_to<numeric>(x);
 314                 const numeric ny = ex_to<numeric>(y);
 315                 const numeric nxy = ex_to<numeric>(x*y);
 316                 int cut = 0;
 317                 if (nx.is_real() && nx.is_negative())
 318                         cut -= 4;
 319                 if (ny.is_real() && ny.is_negative())
 320                         cut -= 4;
 321                 if (nxy.is_real() && nxy.is_negative())
 322                         cut += 4;
 323                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 324                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 325         }
 326
 327         return eta(x,y).hold();
 328 }
 329
 330 static ex eta_series(const ex & x, const ex & y,
 331                      const relational & rel,
 332                      int order,
 333                      unsigned options)
 334 {
 335         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 336         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 337         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 338             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 339             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 340                         throw (std::domain_error("eta_series(): on discontinuity"));
 341         epvector seq;
 342         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 343         return pseries(rel,seq);
 344 }
 345
 346 static ex eta_conjugate(const ex & x, const ex & y)
 347 {
 348         return -eta(x,y);
 349 }
 350
 351 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 352                        evalf_func(eta_evalf).
 353                        series_func(eta_series).
 354                        latex_name("\\eta").
 355                        set_symmetry(sy_symm(0, 1)).
 356                        conjugate_func(eta_conjugate));
 357
 358
 359 //////////
 360 // dilogarithm
 361 //////////
 362
 363 static ex Li2_evalf(const ex & x)
 364 {
 365         if (is_exactly_a<numeric>(x))
 366                 return Li2(ex_to<numeric>(x));
 367
 368         return Li2(x).hold();
 369 }
 370
 371 static ex Li2_eval(const ex & x)
 372 {
 373         if (x.info(info_flags::numeric)) {
 374                 // Li2(0) -> 0
 375                 if (x.is_zero())
 376                         return _ex0;
 377                 // Li2(1) -> Pi^2/6
 378                 if (x.is_equal(_ex1))
 379                         return power(Pi,_ex2)/_ex6;
 380                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 381                 if (x.is_equal(_ex1_2))
 382                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 383                 // Li2(-1) -> -Pi^2/12
 384                 if (x.is_equal(_ex_1))
 385                         return -power(Pi,_ex2)/_ex12;
 386                 // Li2(I) -> -Pi^2/48+Catalan*I
 387                 if (x.is_equal(I))
 388                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 389                 // Li2(-I) -> -Pi^2/48-Catalan*I
 390                 if (x.is_equal(-I))
 391                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 392                 // Li2(float)
 393                 if (!x.info(info_flags::crational))
 394                         return Li2(ex_to<numeric>(x));
 395         }
 396
 397         return Li2(x).hold();
 398 }
 399
 400 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 401 {
 402         GINAC_ASSERT(deriv_param==0);
 403
 404         // d/dx Li2(x) -> -log(1-x)/x
 405         return -log(_ex1-x)/x;
 406 }
 407
 408 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 409 {
 410         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 411         if (x_pt.info(info_flags::numeric)) {
 412                 // First special case: x==0 (derivatives have poles)
 413                 if (x_pt.is_zero()) {
 414                         // method:
 415                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 416                         // simply substitute x==0.  The limit, however, exists: it is 1.
 417                         // We also know all higher derivatives' limits:
 418                         // (d/dx)^n Li2(x) == n!/n^2.
 419                         // So the primitive series expansion is
 420                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 421                         // and so on.
 422                         // We first construct such a primitive series expansion manually in
 423                         // a dummy symbol s and then insert the argument's series expansion
 424                         // for s.  Reexpanding the resulting series returns the desired
 425                         // result.
 426                         const symbol s;
 427                         ex ser;
 428                         // manually construct the primitive expansion
 429                         for (int i=1; i<order; ++i)
 430                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
 431                         // substitute the argument's series expansion
 432                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 433                         // maybe that was terminating, so add a proper order term
 434                         epvector nseq;
 435                         nseq.push_back(expair(Order(_ex1), order));
 436                         ser += pseries(rel, nseq);
 437                         // reexpanding it will collapse the series again
 438                         return ser.series(rel, order);
 439                         // NB: Of course, this still does not allow us to compute anything
 440                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 441                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 442                         // substitution x==0.  Probably limits *are* needed for the general
 443                         // cases.  In case L'Hospital's rule is implemented for limits and
 444                         // basic::series() takes care of this, this whole block is probably
 445                         // obsolete!
 446                 }
 447                 // second special case: x==1 (branch point)
 448                 if (x_pt.is_equal(_ex1)) {
 449                         // method:
 450                         // construct series manually in a dummy symbol s
 451                         const symbol s;
 452                         ex ser = zeta(_ex2);
 453                         // manually construct the primitive expansion
 454                         for (int i=1; i<order; ++i)
 455                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 456                         // substitute the argument's series expansion
 457                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 458                         // maybe that was terminating, so add a proper order term
 459                         epvector nseq;
 460                         nseq.push_back(expair(Order(_ex1), order));
 461                         ser += pseries(rel, nseq);
 462                         // reexpanding it will collapse the series again
 463                         return ser.series(rel, order);
 464                 }
 465                 // third special case: x real, >=1 (branch cut)
 466                 if (!(options & series_options::suppress_branchcut) &&
 467                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 468                         // method:
 469                         // This is the branch cut: assemble the primitive series manually
 470                         // and then add the corresponding complex step function.
 471                         const symbol &s = ex_to<symbol>(rel.lhs());
 472                         const ex point = rel.rhs();
 473                         const symbol foo;
 474                         epvector seq;
 475                         // zeroth order term:
 476                         seq.push_back(expair(Li2(x_pt), _ex0));
 477                         // compute the intermediate terms:
 478                         ex replarg = series(Li2(x), s==foo, order);
 479                         for (size_t i=1; i<replarg.nops()-1; ++i)
 480                                 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));
 481                         // append an order term:
 482                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 483                         return pseries(rel, seq);
 484                 }
 485         }
 486         // all other cases should be safe, by now:
 487         throw do_taylor();  // caught by function::series()
 488 }
 489
 490 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 491                        evalf_func(Li2_evalf).
 492                        derivative_func(Li2_deriv).
 493                        series_func(Li2_series).
 494                        latex_name("\\mbox{Li}_2"));
 495
 496 //////////
 497 // trilogarithm
 498 //////////
 499
 500 static ex Li3_eval(const ex & x)
 501 {
 502         if (x.is_zero())
 503                 return x;
 504         return Li3(x).hold();
 505 }
 506
 507 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 508                        latex_name("\\mbox{Li}_3"));
 509
 510 //////////
 511 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
 512 //////////
 513
 514 static ex zetaderiv_eval(const ex & n, const ex & x)
 515 {
 516         if (n.info(info_flags::numeric)) {
 517                 // zetaderiv(0,x) -> zeta(x)
 518                 if (n.is_zero())
 519                         return zeta(x);
 520         }
 521
 522         return zetaderiv(n, x).hold();
 523 }
 524
 525 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
 526 {
 527         GINAC_ASSERT(deriv_param<2);
 528
 529         if (deriv_param==0) {
 530                 // d/dn zeta(n,x)
 531                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
 532         }
 533         // d/dx psi(n,x)
 534         return zetaderiv(n+1,x);
 535 }
 536
 537 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
 538                                  derivative_func(zetaderiv_deriv).
 539                                  latex_name("\\zeta^\\prime"));
 540
 541 //////////
 542 // factorial
 543 //////////
 544
 545 static ex factorial_evalf(const ex & x)
 546 {
 547         return factorial(x).hold();
 548 }
 549
 550 static ex factorial_eval(const ex & x)
 551 {
 552         if (is_exactly_a<numeric>(x))
 553                 return factorial(ex_to<numeric>(x));
 554         else
 555                 return factorial(x).hold();
 556 }
 557
 558 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 559 {
 560         if (is_exactly_a<symbol>(x) ||
 561             is_exactly_a<constant>(x) ||
 562                 is_exactly_a<function>(x)) {
 563                 x.print(c); c.s << "!";
 564         } else {
 565                 c.s << "("; x.print(c); c.s << ")!";
 566         }
 567 }
 568
 569 static ex factorial_conjugate(const ex & x)
 570 {
 571         return factorial(x);
 572 }
 573
 574 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 575                              evalf_func(factorial_evalf).
 576                              print_func<print_dflt>(factorial_print_dflt_latex).
 577                              print_func<print_latex>(factorial_print_dflt_latex).
 578                              conjugate_func(factorial_conjugate));
 579
 580 //////////
 581 // binomial
 582 //////////
 583
 584 static ex binomial_evalf(const ex & x, const ex & y)
 585 {
 586         return binomial(x, y).hold();
 587 }
 588
 589 static ex binomial_sym(const ex & x, const numeric & y)
 590 {
 591         if (y.is_integer()) {
 592                 if (y.is_nonneg_integer()) {
 593                         const unsigned N = y.to_int();
 594                         if (N == 0) return _ex0;
 595                         if (N == 1) return x;
 596                         ex t = x.expand();
 597                         for (unsigned i = 2; i <= N; ++i)
 598                                 t = (t * (x + i - y - 1)).expand() / i;
 599                         return t;
 600                 } else
 601                         return _ex0;
 602         }
 603
 604         return binomial(x, y).hold();
 605 }
 606
 607 static ex binomial_eval(const ex & x, const ex &y)
 608 {
 609         if (is_exactly_a<numeric>(y)) {
 610                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
 611                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 612                 else
 613                         return binomial_sym(x, ex_to<numeric>(y));
 614         } else
 615                 return binomial(x, y).hold();
 616 }
 617
 618 // At the moment the numeric evaluation of a binomail function always
 619 // gives a real number, but if this would be implemented using the gamma
 620 // function, also complex conjugation should be changed (or rather, deleted).
 621 static ex binomial_conjugate(const ex & x, const ex & y)
 622 {
 623         return binomial(x,y);
 624 }
 625
 626 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 627                             evalf_func(binomial_evalf).
 628                             conjugate_func(binomial_conjugate));
 629
 630 //////////
 631 // Order term function (for truncated power series)
 632 //////////
 633
 634 static ex Order_eval(const ex & x)
 635 {
 636         if (is_exactly_a<numeric>(x)) {
 637                 // O(c) -> O(1) or 0
 638                 if (!x.is_zero())
 639                         return Order(_ex1).hold();
 640                 else
 641                         return _ex0;
 642         } else if (is_exactly_a<mul>(x)) {
 643                 const mul &m = ex_to<mul>(x);
 644                 // O(c*expr) -> O(expr)
 645                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 646                         return Order(x / m.op(m.nops() - 1)).hold();
 647         }
 648         return Order(x).hold();
 649 }
 650
 651 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 652 {
 653         // Just wrap the function into a pseries object
 654         epvector new_seq;
 655         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 656         const symbol &s = ex_to<symbol>(r.lhs());
 657         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 658         return pseries(r, new_seq);
 659 }
 660
 661 static ex Order_conjugate(const ex & x)
 662 {
 663         return Order(x);
 664 }
 665
 666 // Differentiation is handled in function::derivative because of its special requirements
 667
 668 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 669                          series_func(Order_series).
 670                          latex_name("\\mathcal{O}").
 671                          conjugate_func(Order_conjugate));
 672
 673 //////////
 674 // Solve linear system
 675 //////////
 676
 677 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 678 {
 679         // solve a system of linear equations
 680         if (eqns.info(info_flags::relation_equal)) {
 681                 if (!symbols.info(info_flags::symbol))
 682                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 683                 const ex sol = lsolve(lst(eqns),lst(symbols));
 684
 685                 GINAC_ASSERT(sol.nops()==1);
 686                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 687
 688                 return sol.op(0).op(1); // return rhs of first solution
 689         }
 690
 691         // syntax checks
 692         if (!eqns.info(info_flags::list)) {
 693                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 694         }
 695         for (size_t i=0; i<eqns.nops(); i++) {
 696                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 697                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 698                 }
 699         }
 700         if (!symbols.info(info_flags::list)) {
 701                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 702         }
 703         for (size_t i=0; i<symbols.nops(); i++) {
 704                 if (!symbols.op(i).info(info_flags::symbol)) {
 705                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 706                 }
 707         }
 708
 709         // build matrix from equation system
 710         matrix sys(eqns.nops(),symbols.nops());
 711         matrix rhs(eqns.nops(),1);
 712         matrix vars(symbols.nops(),1);
 713
 714         for (size_t r=0; r<eqns.nops(); r++) {
 715                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 716                 ex linpart = eq;
 717                 for (size_t c=0; c<symbols.nops(); c++) {
 718                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 719                         linpart -= co*symbols.op(c);
 720                         sys(r,c) = co;
 721                 }
 722                 linpart = linpart.expand();
 723                 rhs(r,0) = -linpart;
 724         }
 725
 726         // test if system is linear and fill vars matrix
 727         for (size_t i=0; i<symbols.nops(); i++) {
 728                 vars(i,0) = symbols.op(i);
 729                 if (sys.has(symbols.op(i)))
 730                         throw(std::logic_error("lsolve: system is not linear"));
 731                 if (rhs.has(symbols.op(i)))
 732                         throw(std::logic_error("lsolve: system is not linear"));
 733         }
 734
 735         matrix solution;
 736         try {
 737                 solution = sys.solve(vars,rhs,options);
 738         } catch (const std::runtime_error & e) {
 739                 // Probably singular matrix or otherwise overdetermined system:
 740                 // It is consistent to return an empty list
 741                 return lst();
 742         }
 743         GINAC_ASSERT(solution.cols()==1);
 744         GINAC_ASSERT(solution.rows()==symbols.nops());
 745
 746         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 747         lst sollist;
 748         for (size_t i=0; i<symbols.nops(); i++)
 749                 sollist.append(symbols.op(i)==solution(i,0));
 750
 751         return sollist;
 752 }
 753
 754 //////////
 755 // Find real root of f(x) numerically
 756 //////////
 757
 758 const numeric
 759 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
 760 {
 761         if (!x1.is_real() || !x2.is_real()) {
 762                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
 763         }
 764         if (x1==x2) {
 765                 throw std::runtime_error("fsolve(): vanishing interval");
 766         }
 767         // xx[0] == left interval limit, xx[1] == right interval limit.
 768         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
 769         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
 770         numeric xx[2] = { x1<x2 ? x1 : x2,
 771                           x1<x2 ? x2 : x1 };
 772         ex f;
 773         if (is_a<relational>(f_in)) {
 774                 f = f_in.lhs()-f_in.rhs();
 775         } else {
 776                 f = f_in;
 777         }
 778         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
 779                             f.subs(x==xx[1]).evalf() };
 780         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
 781                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
 782         }
 783         numeric fx[2] = { ex_to<numeric>(fx_[0]),
 784                           ex_to<numeric>(fx_[1]) };
 785         if (!fx[0].is_real() || !fx[1].is_real()) {
 786                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
 787         }
 788         if (fx[0]*fx[1]>=0) {
 789                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
 790         }
 791
 792         // The Newton-Raphson method has quadratic convergence!  Simply put, it
 793         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
 794         const ex ff = normal(-f/f.diff(x));
 795         int side = 0;  // Start at left interval limit.
 796         numeric xxprev;
 797         numeric fxprev;
 798         do {
 799                 xxprev = xx[side];
 800                 fxprev = fx[side];
 801                 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
 802                 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
 803                 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
 804                         // Oops, Newton-Raphson method shot out of the interval.
 805                         // Restore, and try again with the other side instead!
 806                         xx[side] = xxprev;
 807                         fx[side] = fxprev;
 808                         side = !side;
 809                         xxprev = xx[side];
 810                         fxprev = fx[side];
 811                         xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
 812                         fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
 813                 }
 814                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
 815                         // Oops, the root isn't bracketed any more.
 816                         // Restore, and perform a bisection!
 817                         xx[side] = xxprev;
 818                         fx[side] = fxprev;
 819
 820                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
 821                         // they occur pretty often when Newton-Raphson arrives at an x too
 822                         // close to the result on one side of the interval and
 823                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
 824                         // precision errors! Recall that this function does not have a
 825                         // precision goal as one of its arguments but instead relies on
 826                         // x converging to a fixed point. We speed up the (safe but slow)
 827                         // bisection method by mixing in a dash of the (unsafer but faster)
 828                         // secant method: Instead of splitting the interval at the
 829                         // arithmetic mean (bisection), we split it nearer to the root as
 830                         // determined by the secant between the values xx[0] and xx[1].
 831                         // Don't set the secant_weight to one because that could disturb
 832                         // the convergence in some corner cases!
 833                         static const double secant_weight = 0.984375;  // == 63/64 < 1
 834                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
 835                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
 836                         numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
 837                         if (fxmid.is_zero()) {
 838                                 // Luck strikes...
 839                                 return xxmid;
 840                         }
 841                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
 842                                 side = !side;
 843                         }
 844                         xxprev = xx[side];
 845                         fxprev = fx[side];
 846                         xx[side] = xxmid;
 847                         fx[side] = fxmid;
 848                 }
 849         } while (xxprev!=xx[side]);
 850         return xxprev;
 851 }
 852
 853
 854 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 855  * for static lib (so ginsh will see them). */
 856 unsigned force_include_tgamma = tgamma_SERIAL::serial;
 857 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 858
 859 } // namespace GiNaC