GiNaC  1.8.0
power.cpp
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1 
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22 
23 #include "power.h"
24 #include "expairseq.h"
25 #include "add.h"
26 #include "mul.h"
27 #include "ncmul.h"
28 #include "numeric.h"
29 #include "constant.h"
30 #include "operators.h"
31 #include "inifcns.h" // for log() in power::derivative()
32 #include "matrix.h"
33 #include "indexed.h"
34 #include "symbol.h"
35 #include "lst.h"
36 #include "archive.h"
37 #include "utils.h"
38 #include "relational.h"
39 #include "compiler.h"
40 
41 #include <iostream>
42 #include <limits>
43 #include <stdexcept>
44 #include <vector>
45 #include <algorithm>
46 
47 namespace GiNaC {
48 
51  print_func<print_latex>(&power::do_print_latex).
52  print_func<print_csrc>(&power::do_print_csrc).
53  print_func<print_python>(&power::do_print_python).
54  print_func<print_python_repr>(&power::do_print_python_repr).
55  print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
56 
57 // default constructor
60 
61 power::power() { }
62 
64 // other constructors
66 
67 // all inlined
68 
70 // archiving
72 
73 void power::read_archive(const archive_node &n, lst &sym_lst)
74 {
75  inherited::read_archive(n, sym_lst);
76  n.find_ex("basis", basis, sym_lst);
77  n.find_ex("exponent", exponent, sym_lst);
78 }
79 
81 {
82  inherited::archive(n);
83  n.add_ex("basis", basis);
84  n.add_ex("exponent", exponent);
85 }
86 
88 // functions overriding virtual functions from base classes
90 
91 // public
92 
93 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
94 {
95  // Ordinary output of powers using '^' or '**'
96  if (precedence() <= level)
97  c.s << openbrace << '(';
98  basis.print(c, precedence());
99  c.s << powersymbol;
100  c.s << openbrace;
102  c.s << closebrace;
103  if (precedence() <= level)
104  c.s << ')' << closebrace;
105 }
106 
107 void power::do_print_dflt(const print_dflt & c, unsigned level) const
108 {
109  if (exponent.is_equal(_ex1_2)) {
110 
111  // Square roots are printed in a special way
112  c.s << "sqrt(";
113  basis.print(c);
114  c.s << ')';
115 
116  } else
117  print_power(c, "^", "", "", level);
118 }
119 
120 void power::do_print_latex(const print_latex & c, unsigned level) const
121 {
122  if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
123 
124  // Powers with negative numeric exponents are printed as fractions
125  c.s << "\\frac{1}{";
126  power(basis, -exponent).eval().print(c);
127  c.s << '}';
128 
129  } else if (exponent.is_equal(_ex1_2)) {
130 
131  // Square roots are printed in a special way
132  c.s << "\\sqrt{";
133  basis.print(c);
134  c.s << '}';
135 
136  } else
137  print_power(c, "^", "{", "}", level);
138 }
139 
140 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
141 {
142  // Optimal output of integer powers of symbols to aid compiler CSE.
143  // C.f. ISO/IEC 14882:2011, section 1.9 [intro execution], paragraph 15
144  // to learn why such a parenthesation is really necessary.
145  if (exp == 1) {
146  x.print(c);
147  } else if (exp == 2) {
148  x.print(c);
149  c.s << "*";
150  x.print(c);
151  } else if (exp & 1) {
152  x.print(c);
153  c.s << "*";
154  print_sym_pow(c, x, exp-1);
155  } else {
156  c.s << "(";
157  print_sym_pow(c, x, exp >> 1);
158  c.s << ")*(";
159  print_sym_pow(c, x, exp >> 1);
160  c.s << ")";
161  }
162 }
163 
164 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
165 {
166  if (exponent.is_equal(_ex_1)) {
167  c.s << "recip(";
168  basis.print(c);
169  c.s << ')';
170  return;
171  }
172  c.s << "expt(";
173  basis.print(c);
174  c.s << ", ";
175  exponent.print(c);
176  c.s << ')';
177 }
178 
179 void power::do_print_csrc(const print_csrc & c, unsigned level) const
180 {
181  // Integer powers of symbols are printed in a special, optimized way
183  (is_a<symbol>(basis) || is_a<constant>(basis))) {
184  int exp = ex_to<numeric>(exponent).to_int();
185  if (exp > 0)
186  c.s << '(';
187  else {
188  exp = -exp;
189  c.s << "1.0/(";
190  }
191  print_sym_pow(c, ex_to<symbol>(basis), exp);
192  c.s << ')';
193 
194  // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
195  } else if (exponent.is_equal(_ex_1)) {
196  c.s << "1.0/(";
197  basis.print(c);
198  c.s << ')';
199 
200  // Otherwise, use the pow() function
201  } else {
202  c.s << "pow(";
203  basis.print(c);
204  c.s << ',';
205  exponent.print(c);
206  c.s << ')';
207  }
208 }
209 
210 void power::do_print_python(const print_python & c, unsigned level) const
211 {
212  print_power(c, "**", "", "", level);
213 }
214 
215 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
216 {
217  c.s << class_name() << '(';
218  basis.print(c);
219  c.s << ',';
220  exponent.print(c);
221  c.s << ')';
222 }
223 
224 bool power::info(unsigned inf) const
225 {
226  switch (inf) {
232  return basis.info(inf) && exponent.info(info_flags::nonnegint);
234  return basis.info(inf) && exponent.info(info_flags::integer);
235  case info_flags::real:
236  return basis.info(inf) && exponent.info(info_flags::integer);
238  return (flags & status_flags::expanded);
246  return true;
248  return false;
249  else if (basis.info(info_flags::has_indices)) {
252  return true;
253  } else {
256  return false;
257  }
258  }
259  }
260  return inherited::info(inf);
261 }
262 
263 size_t power::nops() const
264 {
265  return 2;
266 }
267 
268 ex power::op(size_t i) const
269 {
270  GINAC_ASSERT(i<2);
271 
272  return i==0 ? basis : exponent;
273 }
274 
276 {
277  const ex &mapped_basis = f(basis);
278  const ex &mapped_exponent = f(exponent);
279 
280  if (!are_ex_trivially_equal(basis, mapped_basis)
281  || !are_ex_trivially_equal(exponent, mapped_exponent))
282  return dynallocate<power>(mapped_basis, mapped_exponent);
283  else
284  return *this;
285 }
286 
287 bool power::is_polynomial(const ex & var) const
288 {
289  if (basis.is_polynomial(var)) {
290  if (basis.has(var))
291  // basis is non-constant polynomial in var
293  else
294  // basis is constant in var
295  return !exponent.has(var);
296  }
297  // basis is a non-polynomial function of var
298  return false;
299 }
300 
301 int power::degree(const ex & s) const
302 {
303  if (is_equal(ex_to<basic>(s)))
304  return 1;
305  else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
306  if (basis.is_equal(s))
307  return ex_to<numeric>(exponent).to_int();
308  else
309  return basis.degree(s) * ex_to<numeric>(exponent).to_int();
310  } else if (basis.has(s))
311  throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
312  else
313  return 0;
314 }
315 
316 int power::ldegree(const ex & s) const
317 {
318  if (is_equal(ex_to<basic>(s)))
319  return 1;
320  else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
321  if (basis.is_equal(s))
322  return ex_to<numeric>(exponent).to_int();
323  else
324  return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
325  } else if (basis.has(s))
326  throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
327  else
328  return 0;
329 }
330 
331 ex power::coeff(const ex & s, int n) const
332 {
333  if (is_equal(ex_to<basic>(s)))
334  return n==1 ? _ex1 : _ex0;
335  else if (!basis.is_equal(s)) {
336  // basis not equal to s
337  if (n == 0)
338  return *this;
339  else
340  return _ex0;
341  } else {
342  // basis equal to s
343  if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
344  // integer exponent
345  int int_exp = ex_to<numeric>(exponent).to_int();
346  if (n == int_exp)
347  return _ex1;
348  else
349  return _ex0;
350  } else {
351  // non-integer exponents are treated as zero
352  if (n == 0)
353  return *this;
354  else
355  return _ex0;
356  }
357  }
358 }
359 
375 {
377  return *this;
378 
379  const numeric *num_basis = nullptr;
380  const numeric *num_exponent = nullptr;
381 
382  if (is_exactly_a<numeric>(basis)) {
383  num_basis = &ex_to<numeric>(basis);
384  }
385  if (is_exactly_a<numeric>(exponent)) {
386  num_exponent = &ex_to<numeric>(exponent);
387  }
388 
389  // ^(x,0) -> 1 (0^0 also handled here)
390  if (exponent.is_zero()) {
391  if (basis.is_zero())
392  throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
393  else
394  return _ex1;
395  }
396 
397  // ^(x,1) -> x
398  if (exponent.is_equal(_ex1))
399  return basis;
400 
401  // ^(0,c1) -> 0 or exception (depending on real value of c1)
402  if (basis.is_zero() && num_exponent) {
403  if ((num_exponent->real()).is_zero())
404  throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
405  else if ((num_exponent->real()).is_negative())
406  throw (pole_error("power::eval(): division by zero",1));
407  else
408  return _ex0;
409  }
410 
411  // ^(1,x) -> 1
412  if (basis.is_equal(_ex1))
413  return _ex1;
414 
415  // power of a function calculated by separate rules defined for this function
416  if (is_exactly_a<function>(basis))
417  return ex_to<function>(basis).power(exponent);
418 
419  // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
420  if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
421  return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
422 
423  if ( num_exponent ) {
424 
425  // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
426  // except if c1,c2 are rational, but c1^c2 is not)
427  if ( num_basis ) {
428  const bool basis_is_crational = num_basis->is_crational();
429  const bool exponent_is_crational = num_exponent->is_crational();
430  if (!basis_is_crational || !exponent_is_crational) {
431  // return a plain float
432  return dynallocate<numeric>(num_basis->power(*num_exponent));
433  }
434 
435  const numeric res = num_basis->power(*num_exponent);
436  if (res.is_crational()) {
437  return res;
438  }
439  GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
440 
441  // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
442  if (basis_is_crational && exponent_is_crational
443  && num_exponent->is_real()
444  && !num_exponent->is_integer()) {
445  const numeric n = num_exponent->numer();
446  const numeric m = num_exponent->denom();
447  numeric r;
448  numeric q = iquo(n, m, r);
449  if (r.is_negative()) {
450  r += m;
451  --q;
452  }
453  if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
454  if (num_basis->is_rational() && !num_basis->is_integer()) {
455  // try it for numerator and denominator separately, in order to
456  // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
457  const numeric bnum = num_basis->numer();
458  const numeric bden = num_basis->denom();
459  const numeric res_bnum = bnum.power(*num_exponent);
460  const numeric res_bden = bden.power(*num_exponent);
461  if (res_bnum.is_integer())
462  return dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
463  if (res_bden.is_integer())
464  return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(status_flags::evaluated);
465  }
466  return this->hold();
467  } else {
468  // assemble resulting product, but allowing for a re-evaluation,
469  // because otherwise we'll end up with something like
470  // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
471  // instead of 7/16*7^(1/3).
472  return pow(basis, r.div(m)) * pow(basis, q);
473  }
474  }
475  }
476 
477  // ^(^(x,c1),c2) -> ^(x,c1*c2)
478  // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
479  // case c1==1 should not happen, see below!)
480  if (is_exactly_a<power>(basis)) {
481  const power & sub_power = ex_to<power>(basis);
482  const ex & sub_basis = sub_power.basis;
483  const ex & sub_exponent = sub_power.exponent;
484  if (is_exactly_a<numeric>(sub_exponent)) {
485  const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
486  GINAC_ASSERT(num_sub_exponent!=numeric(1));
487  if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
488  (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
489  return dynallocate<power>(sub_basis, num_sub_exponent.mul(*num_exponent));
490  }
491  }
492  }
493 
494  // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
495  if (num_exponent->is_integer() && is_exactly_a<mul>(basis)) {
496  return expand_mul(ex_to<mul>(basis), *num_exponent, false);
497  }
498 
499  // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
500  if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
501  numeric icont = basis.integer_content();
502  const numeric lead_coeff =
503  ex_to<numeric>(ex_to<add>(basis).seq.begin()->coeff).div(icont);
504 
505  const bool canonicalizable = lead_coeff.is_integer();
506  const bool unit_normal = lead_coeff.is_pos_integer();
507  if (canonicalizable && (! unit_normal))
508  icont = icont.mul(*_num_1_p);
509 
510  if (canonicalizable && (icont != *_num1_p)) {
511  const add& addref = ex_to<add>(basis);
512  add & addp = dynallocate<add>(addref);
514  addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
515  for (auto & i : addp.seq)
516  i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
517 
518  const numeric c = icont.power(*num_exponent);
519  if (likely(c != *_num1_p))
520  return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
521  else
522  return dynallocate<power>(addp, *num_exponent);
523  }
524  }
525 
526  // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
527  // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
528  if (is_exactly_a<mul>(basis)) {
529  GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
530  const mul & mulref = ex_to<mul>(basis);
531  if (!mulref.overall_coeff.is_equal(_ex1)) {
532  const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
533  if (num_coeff.is_real()) {
534  if (num_coeff.is_positive()) {
535  mul & mulp = dynallocate<mul>(mulref);
536  mulp.overall_coeff = _ex1;
538  return dynallocate<mul>(dynallocate<power>(mulp, exponent),
539  dynallocate<power>(num_coeff, *num_exponent));
540  } else {
541  GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
542  if (!num_coeff.is_equal(*_num_1_p)) {
543  mul & mulp = dynallocate<mul>(mulref);
544  mulp.overall_coeff = _ex_1;
546  return dynallocate<mul>(dynallocate<power>(mulp, exponent),
547  dynallocate<power>(abs(num_coeff), *num_exponent));
548  }
549  }
550  }
551  }
552  }
553 
554  // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
555  if (num_exponent->is_pos_integer() &&
557  !is_a<matrix>(basis)) {
558  return ncmul(exvector(num_exponent->to_int(), basis));
559  }
560  }
561 
562  return this->hold();
563 }
564 
566 {
567  ex ebasis = basis.evalf();
568  ex eexponent;
569 
570  if (!is_exactly_a<numeric>(exponent))
571  eexponent = exponent.evalf();
572  else
573  eexponent = exponent;
574 
575  return dynallocate<power>(ebasis, eexponent);
576 }
577 
579 {
580  const ex ebasis = basis.evalm();
581  const ex eexponent = exponent.evalm();
582  if (is_a<matrix>(ebasis)) {
583  if (is_exactly_a<numeric>(eexponent)) {
584  return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
585  }
586  }
587  return dynallocate<power>(ebasis, eexponent);
588 }
589 
590 bool power::has(const ex & other, unsigned options) const
591 {
593  return basic::has(other, options);
594  if (!is_a<power>(other))
595  return basic::has(other, options);
597  !other.op(1).info(info_flags::integer))
598  return basic::has(other, options);
600  other.op(1).info(info_flags::posint) &&
601  ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
602  basis.match(other.op(0)))
603  return true;
605  other.op(1).info(info_flags::negint) &&
606  ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
607  basis.match(other.op(0)))
608  return true;
609  return basic::has(other, options);
610 }
611 
612 // from mul.cpp
613 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
614 
615 ex power::subs(const exmap & m, unsigned options) const
616 {
617  const ex &subsed_basis = basis.subs(m, options);
618  const ex &subsed_exponent = exponent.subs(m, options);
619 
620  if (!are_ex_trivially_equal(basis, subsed_basis)
621  || !are_ex_trivially_equal(exponent, subsed_exponent))
622  return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
623 
625  return subs_one_level(m, options);
626 
627  for (auto & it : m) {
628  int nummatches = std::numeric_limits<int>::max();
629  exmap repls;
630  if (tryfactsubs(*this, it.first, nummatches, repls)) {
631  ex anum = it.second.subs(repls, subs_options::no_pattern);
632  ex aden = it.first.subs(repls, subs_options::no_pattern);
633  ex result = (*this) * pow(anum/aden, nummatches);
634  return (ex_to<basic>(result)).subs_one_level(m, options);
635  }
636  }
637 
638  return subs_one_level(m, options);
639 }
640 
641 ex power::eval_ncmul(const exvector & v) const
642 {
643  return inherited::eval_ncmul(v);
644 }
645 
647 {
648  // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
649  // branch cut which runs along the negative real axis.
651  ex newexponent = exponent.conjugate();
652  if (are_ex_trivially_equal(exponent, newexponent)) {
653  return *this;
654  }
655  return dynallocate<power>(basis, newexponent);
656  }
658  ex newbasis = basis.conjugate();
659  if (are_ex_trivially_equal(basis, newbasis)) {
660  return *this;
661  }
662  return dynallocate<power>(newbasis, exponent);
663  }
664  return conjugate_function(*this).hold();
665 }
666 
668 {
669  // basis == a+I*b, exponent == c+I*d
670  const ex a = basis.real_part();
671  const ex c = exponent.real_part();
672  if (basis.is_equal(a) && exponent.is_equal(c) &&
674  // Re(a^c)
675  return *this;
676  }
677 
678  const ex b = basis.imag_part();
680  // Re((a+I*b)^c) w/ c ∈ ℤ
681  long N = ex_to<numeric>(c).to_long();
682  // Use real terms in Binomial expansion to construct
683  // Re(expand(pow(a+I*b, N))).
684  long NN = N > 0 ? N : -N;
685  ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
686  ex result = 0;
687  for (long n = 0; n <= NN; n += 2) {
688  ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
689  if (n % 4 == 0) {
690  result += term; // sign: I^n w/ n == 4*m
691  } else {
692  result -= term; // sign: I^n w/ n == 4*m+2
693  }
694  }
695  return result;
696  }
697 
698  // Re((a+I*b)^(c+I*d))
699  const ex d = exponent.imag_part();
700  return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
701 }
702 
704 {
705  // basis == a+I*b, exponent == c+I*d
706  const ex a = basis.real_part();
707  const ex c = exponent.real_part();
708  if (basis.is_equal(a) && exponent.is_equal(c) &&
710  // Im(a^c)
711  return 0;
712  }
713 
714  const ex b = basis.imag_part();
716  // Im((a+I*b)^c) w/ c ∈ ℤ
717  long N = ex_to<numeric>(c).to_long();
718  // Use imaginary terms in Binomial expansion to construct
719  // Im(expand(pow(a+I*b, N))).
720  long p = N > 0 ? 1 : 3; // modulus for positive sign
721  long NN = N > 0 ? N : -N;
722  ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
723  ex result = 0;
724  for (long n = 1; n <= NN; n += 2) {
725  ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
726  if (n % 4 == p) {
727  result += term; // sign: I^n w/ n == 4*m+p
728  } else {
729  result -= term; // sign: I^n w/ n == 4*m+2+p
730  }
731  }
732  return result;
733  }
734 
735  // Im((a+I*b)^(c+I*d))
736  const ex d = exponent.imag_part();
737  return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
738 }
739 
740 // protected
741 
744 ex power::derivative(const symbol & s) const
745 {
746  if (is_a<numeric>(exponent)) {
747  // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
748  const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
749  return dynallocate<mul>(std::move(newseq), exponent);
750  } else {
751  // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
752  return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
753  }
754 }
755 
756 int power::compare_same_type(const basic & other) const
757 {
758  GINAC_ASSERT(is_exactly_a<power>(other));
759  const power &o = static_cast<const power &>(other);
760 
761  int cmpval = basis.compare(o.basis);
762  if (cmpval)
763  return cmpval;
764  else
765  return exponent.compare(o.exponent);
766 }
767 
768 unsigned power::return_type() const
769 {
770  return basis.return_type();
771 }
772 
774 {
775  return basis.return_type_tinfo();
776 }
777 
778 ex power::expand(unsigned options) const
779 {
780  if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
781  // A special case worth optimizing.
783  return *this;
784  }
785 
786  // (x*p)^c -> x^c * p^c, if p>0
787  // makes sense before expanding the basis
788  if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
789  const mul &m = ex_to<mul>(basis);
790  exvector prodseq;
791  epvector powseq;
792  prodseq.reserve(m.seq.size() + 1);
793  powseq.reserve(m.seq.size() + 1);
794  bool possign = true;
795 
796  // search for positive/negative factors
797  for (auto & cit : m.seq) {
798  ex e=m.recombine_pair_to_ex(cit);
799  if (e.info(info_flags::positive))
800  prodseq.push_back(pow(e, exponent).expand(options));
801  else if (e.info(info_flags::negative)) {
802  prodseq.push_back(pow(-e, exponent).expand(options));
803  possign = !possign;
804  } else
805  powseq.push_back(cit);
806  }
807 
808  // take care on the numeric coefficient
809  ex coeff=(possign? _ex1 : _ex_1);
810  if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
811  prodseq.push_back(pow(m.overall_coeff, exponent));
812  else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1) {
813  prodseq.push_back(pow(-m.overall_coeff, exponent));
814  coeff = -coeff;
815  } else
816  coeff *= m.overall_coeff;
817 
818  // If positive/negative factors are found, then extract them.
819  // In either case we set a flag to avoid the second run on a part
820  // which does not have positive/negative terms.
821  if (prodseq.size() > 0) {
822  ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
823  ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
824  return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
825  } else
826  ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
827  }
828 
829  const ex expanded_basis = basis.expand(options);
830  const ex expanded_exponent = exponent.expand(options);
831 
832  // x^(a+b) -> x^a * x^b
833  if (is_exactly_a<add>(expanded_exponent)) {
834  const add &a = ex_to<add>(expanded_exponent);
835  exvector distrseq;
836  distrseq.reserve(a.seq.size() + 1);
837  for (auto & cit : a.seq) {
838  distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
839  }
840 
841  // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
842  if (ex_to<numeric>(a.overall_coeff).is_integer()) {
843  const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
844  long int_exponent = num_exponent.to_int();
845  if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
846  distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
847  else
848  distrseq.push_back(pow(expanded_basis, a.overall_coeff));
849  } else
850  distrseq.push_back(pow(expanded_basis, a.overall_coeff));
851 
852  // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
853  ex r = dynallocate<mul>(distrseq);
854  return r.expand(options);
855  }
856 
857  if (!is_exactly_a<numeric>(expanded_exponent) ||
858  !ex_to<numeric>(expanded_exponent).is_integer()) {
859  if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
860  return this->hold();
861  } else {
862  return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
863  }
864  }
865 
866  // integer numeric exponent
867  const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
868  long int_exponent = num_exponent.to_long();
869 
870  // (x+y)^n, n>0
871  if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
872  return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
873 
874  // (x*y)^n -> x^n * y^n
875  if (is_exactly_a<mul>(expanded_basis))
876  return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
877 
878  // cannot expand further
879  if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
880  return this->hold();
881  else
882  return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
883 }
884 
886 // new virtual functions which can be overridden by derived classes
888 
889 // none
890 
892 // non-virtual functions in this class
894 
897 ex power::expand_add(const add & a, long n, unsigned options)
898 {
899  // The special case power(+(x,...y;x),2) can be optimized better.
900  if (n==2)
901  return expand_add_2(a, options);
902 
903  // method:
904  //
905  // Consider base as the sum of all symbolic terms and the overall numeric
906  // coefficient and apply the binomial theorem:
907  // S = power(+(x,...,z;c),n)
908  // = power(+(+(x,...,z;0);c),n)
909  // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
910  // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
911  // The multinomial theorem is computed by an outer loop over all
912  // partitions of the exponent and an inner loop over all compositions of
913  // that partition. This method makes the expansion a combinatorial
914  // problem and allows us to directly construct the expanded sum and also
915  // to re-use the multinomial coefficients (since they depend only on the
916  // partition, not on the composition).
917  //
918  // multinomial power(+(x,y,z;0),3) example:
919  // partition : compositions : multinomial coefficient
920  // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
921  // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
922  // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
923  // => (x + y + z)^3 =
924  // x^3 + y^3 + z^3
925  // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
926  // + 6*x*y*z
927  //
928  // multinomial power(+(x,y,z;0),4) example:
929  // partition : compositions : multinomial coefficient
930  // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
931  // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
932  // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
933  // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
934  // (no [1,1,1,1] partition since it has too many parts)
935  // => (x + y + z)^4 =
936  // x^4 + y^4 + z^4
937  // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
938  // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
939  // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
940  //
941  // Summary:
942  // r = 0
943  // for k from 0 to n:
944  // f = c^(n-k)*binomial(n,k)
945  // for p in all partitions of n with m parts (including zero parts):
946  // h = f * multinomial coefficient of p
947  // for c in all compositions of p:
948  // t = 1
949  // for e in all elements of c:
950  // t = t * a[e]^e
951  // r = r + h*t
952  // return r
953 
954  epvector result;
955  // The number of terms will be the number of combinatorial compositions,
956  // i.e. the number of unordered arrangements of m nonnegative integers
957  // which sum up to n. It is frequently written as C_n(m) and directly
958  // related with binomial coefficients: binomial(n+m-1,m-1).
959  size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
960  if (!a.overall_coeff.is_zero()) {
961  // the result's overall_coeff is one of the terms
962  --result_size;
963  }
964  result.reserve(result_size);
965 
966  // Iterate over all terms in binomial expansion of
967  // S = power(+(x,...,z;c),n)
968  // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
969  for (int k = 1; k <= n; ++k) {
970  numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
971  if (a.overall_coeff.is_zero()) {
972  // degenerate case with zero overall_coeff:
973  // apply multinomial theorem directly to power(+(x,...z;0),n)
974  binomial_coefficient = 1;
975  if (k < n) {
976  continue;
977  }
978  } else {
979  binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
980  }
981 
982  // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
983  // Iterate over all partitions of k with exactly as many parts as
984  // there are symbolic terms in the basis (including zero parts).
985  partition_with_zero_parts_generator partitions(k, a.seq.size());
986  do {
987  const std::vector<unsigned>& partition = partitions.get();
988  // All monomials of this partition have the same number of terms and the same coefficient.
989  const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
990  const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
991 
992  // Iterate over all compositions of the current partition.
993  composition_generator compositions(partition);
994  do {
995  const std::vector<unsigned>& exponent = compositions.get();
996  epvector monomial;
997  monomial.reserve(msize);
998  numeric factor = coeff;
999  for (unsigned i = 0; i < exponent.size(); ++i) {
1000  const ex & r = a.seq[i].rest;
1001  GINAC_ASSERT(!is_exactly_a<add>(r));
1002  GINAC_ASSERT(!is_exactly_a<power>(r) ||
1003  !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1004  !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1005  !is_exactly_a<add>(ex_to<power>(r).basis) ||
1006  !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1007  !is_exactly_a<power>(ex_to<power>(r).basis));
1008  GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1009  const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1010  if (exponent[i] == 0) {
1011  // optimize away
1012  } else if (exponent[i] == 1) {
1013  // optimized
1014  monomial.emplace_back(expair(r, _ex1));
1015  if (c != *_num1_p)
1016  factor = factor.mul(c);
1017  } else { // general case exponent[i] > 1
1018  monomial.emplace_back(expair(r, exponent[i]));
1019  if (c != *_num1_p)
1020  factor = factor.mul(c.power(exponent[i]));
1021  }
1022  }
1023  result.emplace_back(expair(mul(std::move(monomial)).expand(options), factor));
1024  } while (compositions.next());
1025  } while (partitions.next());
1026  }
1027 
1028  GINAC_ASSERT(result.size() == result_size);
1029  if (a.overall_coeff.is_zero()) {
1030  return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1031  } else {
1032  return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
1033  }
1034 }
1035 
1036 
1039 ex power::expand_add_2(const add & a, unsigned options)
1040 {
1041  epvector result;
1042  size_t result_size = (a.nops() * (a.nops()+1)) / 2;
1043  if (!a.overall_coeff.is_zero()) {
1044  // the result's overall_coeff is one of the terms
1045  --result_size;
1046  }
1047  result.reserve(result_size);
1048 
1049  auto last = a.seq.end();
1050 
1051  // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1052  // first part: ignore overall_coeff and expand other terms
1053  for (auto cit0=a.seq.begin(); cit0!=last; ++cit0) {
1054  const ex & r = cit0->rest;
1055  const ex & c = cit0->coeff;
1056 
1057  GINAC_ASSERT(!is_exactly_a<add>(r));
1058  GINAC_ASSERT(!is_exactly_a<power>(r) ||
1059  !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1060  !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1061  !is_exactly_a<add>(ex_to<power>(r).basis) ||
1062  !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1063  !is_exactly_a<power>(ex_to<power>(r).basis));
1064 
1065  if (c.is_equal(_ex1)) {
1066  if (is_exactly_a<mul>(r)) {
1067  result.emplace_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1068  _ex1));
1069  } else {
1070  result.emplace_back(expair(dynallocate<power>(r, _ex2),
1071  _ex1));
1072  }
1073  } else {
1074  if (is_exactly_a<mul>(r)) {
1075  result.emplace_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1076  ex_to<numeric>(c).power_dyn(*_num2_p)));
1077  } else {
1078  result.emplace_back(expair(dynallocate<power>(r, _ex2),
1079  ex_to<numeric>(c).power_dyn(*_num2_p)));
1080  }
1081  }
1082 
1083  for (auto cit1=cit0+1; cit1!=last; ++cit1) {
1084  const ex & r1 = cit1->rest;
1085  const ex & c1 = cit1->coeff;
1086  result.emplace_back(expair(mul(r,r1).expand(options),
1087  _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1088  }
1089  }
1090 
1091  // second part: add terms coming from overall_coeff (if != 0)
1092  if (!a.overall_coeff.is_zero()) {
1093  for (auto & i : a.seq)
1094  result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1095  }
1096 
1097  GINAC_ASSERT(result.size() == result_size);
1098 
1099  if (a.overall_coeff.is_zero()) {
1100  return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1101  } else {
1102  return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
1103  }
1104 }
1105 
1108 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
1109 {
1110  GINAC_ASSERT(n.is_integer());
1111 
1112  if (n.is_zero()) {
1113  return _ex1;
1114  }
1115 
1116  // do not bother to rename indices if there are no any.
1118  m.info(info_flags::has_indices))
1120  // Leave it to multiplication since dummy indices have to be renamed
1122  (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1123  ex result = m;
1125  sort(va.begin(), va.end(), ex_is_less());
1126 
1127  for (int i=1; i < n.to_int(); i++)
1128  result *= rename_dummy_indices_uniquely(va, m);
1129  return result;
1130  }
1131 
1132  epvector distrseq;
1133  distrseq.reserve(m.seq.size());
1134  bool need_reexpand = false;
1135 
1136  for (auto & cit : m.seq) {
1137  expair p = m.combine_pair_with_coeff_to_pair(cit, n);
1138  if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1139  // this happens when e.g. (a+b)^(1/2) gets squared and
1140  // the resulting product needs to be reexpanded
1141  need_reexpand = true;
1142  }
1143  distrseq.push_back(p);
1144  }
1145 
1146  const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
1147  if (need_reexpand)
1148  return ex(result).expand(options);
1149  if (from_expand)
1150  return result.setflag(status_flags::expanded);
1151  return result;
1152 }
1153 
1155 
1156 } // namespace GiNaC
GiNaC::print_dflt
Context for default (ginsh-parsable) output.
Definition: print.h:115
inifcns.h
Interface to GiNaC's initially known functions.
GiNaC::power::do_print_python_repr
void do_print_python_repr(const print_python_repr &c, unsigned level) const
Definition: power.cpp:215
GiNaC::print_python_repr
Context for python-parsable output.
Definition: print.h:139
GiNaC::epvector
std::vector< expair > epvector
expair-vector
Definition: expairseq.h:33
GiNaC::ex::expand
ex expand(unsigned options=0) const
Definition: ex.cpp:73
GiNaC::info_flags::real
@ real
Definition: flags.h:221
GiNaC::numeric::to_int
int to_int() const
Converts numeric types to machine's int.
Definition: numeric.cpp:1303
GiNaC::power::return_type_tinfo
return_type_t return_type_tinfo() const override
Definition: power.cpp:773
constant.h
Interface to GiNaC's constant types and some special constants.
GiNaC::power::imag_part
ex imag_part() const override
Definition: power.cpp:703
GiNaC::power::mul
friend class mul
Definition: power.h:42
GiNaC::partition_with_zero_parts_generator::next
bool next()
Definition: utils.h:350
GiNaC::power::has
bool has(const ex &other, unsigned options=0) const override
Test for occurrence of a pattern.
Definition: power.cpp:590
GiNaC::GINAC_IMPLEMENT_REGISTERED_CLASS_OPT
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(add, expairseq, print_func< print_context >(&add::do_print). print_func< print_latex >(&add::do_print_latex). print_func< print_csrc >(&add::do_print_csrc). print_func< print_tree >(&add::do_print_tree). print_func< print_python_repr >(&add::do_print_python_repr)) add
Definition: add.cpp:40
GiNaC::info_flags::integer
@ integer
Definition: flags.h:223
GiNaC::numeric::is_integer
bool is_integer() const
True if object is a non-complex integer.
Definition: numeric.cpp:1154
GiNaC::ex::evalf
ex evalf() const
Definition: ex.h:121
GiNaC::power::conjugate
ex conjugate() const override
Definition: power.cpp:646
GiNaC::power::subs
ex subs(const exmap &m, unsigned options=0) const override
Substitute a set of objects by arbitrary expressions.
Definition: power.cpp:615
GiNaC::power::degree
int degree(const ex &s) const override
Return degree of highest power in object s.
Definition: power.cpp:301
GiNaC::numeric::mul
const numeric mul(const numeric &other) const
Numerical multiplication method.
Definition: numeric.cpp:880
GiNaC::add::combine_pair_with_coeff_to_pair
expair combine_pair_with_coeff_to_pair(const expair &p, const ex &c) const override
Definition: add.cpp:550
r
size_t r
Definition: factor.cpp:770
GiNaC::info_flags::negative
@ negative
Definition: flags.h:227
mul.h
Interface to GiNaC's products of expressions.
GiNaC::power::print_power
void print_power(const print_context &c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
Definition: power.cpp:93
GiNaC::expairseq::nops
size_t nops() const override
Number of operands/members.
GiNaC::ex::coeff
ex coeff(const ex &s, int n=1) const
Definition: ex.h:175
GiNaC::status_flags::expanded
@ expanded
.expand(0) has already done its job (other expand() options ignore this flag)
Definition: flags.h:204
GiNaC::_ex0
const ex _ex0
Definition: utils.cpp:177
GiNaC::iquo
const numeric iquo(const numeric &a, const numeric &b)
Numeric integer quotient.
Definition: numeric.cpp:2404
GiNaC::rename_dummy_indices_uniquely
lst rename_dummy_indices_uniquely(const exvector &va, const exvector &vb)
Similar to above, where va and vb are the same and the return value is a list of two lists for substi...
Definition: indexed.cpp:1460
GiNaC::ex::has
bool has(const ex &pattern, unsigned options=0) const
Definition: ex.h:151
numeric.h
Makes the interface to the underlying bignum package available.
GiNaC::map_function
Function object for map().
Definition: basic.h:85
GiNaC::print_sym_pow
static void print_sym_pow(const print_context &c, const symbol &x, int exp)
Definition: power.cpp:140
GiNaC::ex::compare
int compare(const ex &other) const
Definition: ex.h:322
GiNaC::is_pos_integer
bool is_pos_integer(const numeric &x)
Definition: numeric.h:275
GiNaC::ex::subs
ex subs(const exmap &m, unsigned options=0) const
Definition: ex.h:826
GiNaC::print_context
Base class for print_contexts.
Definition: print.h:103
GiNaC::info_flags::even
@ even
Definition: flags.h:232
GiNaC::numeric::is_rational
bool is_rational() const
True if object is an exact rational number, may even be complex (denominator may be unity).
Definition: numeric.cpp:1201
add.h
Interface to GiNaC's sums of expressions.
k
vector< int > k
Definition: factor.cpp:1466
GiNaC::print_csrc
Base context for C source output.
Definition: print.h:158
GiNaC::ex::conjugate
ex conjugate() const
Definition: ex.h:146
power.h
Interface to GiNaC's symbolic exponentiation (basis^exponent).
GiNaC::exvector
std::vector< ex > exvector
Definition: basic.h:46
GiNaC::info_flags::rational_polynomial
@ rational_polynomial
Definition: flags.h:258
GiNaC::info_flags::cinteger_polynomial
@ cinteger_polynomial
Definition: flags.h:257
GiNaC::power
This class holds a two-component object, a basis and and exponent representing exponentiation.
Definition: power.h:39
GiNaC::expand_options::expand_rename_idx
@ expand_rename_idx
used internally by mul::expand()
Definition: flags.h:34
GiNaC::status_flags::evaluated
@ evaluated
.eval() has already done its job
Definition: flags.h:203
GiNaC::has_options::algebraic
@ algebraic
enable algebraic matching
Definition: flags.h:43
GiNaC::numeric::is_real
bool is_real() const
True if object is a real integer, rational or float (but not complex).
Definition: numeric.cpp:1208
GiNaC::info_flags::nonnegative
@ nonnegative
Definition: flags.h:228
GiNaC::ex::degree
int degree(const ex &s) const
Definition: ex.h:173
GiNaC::numeric::to_long
long to_long() const
Converts numeric types to machine's long.
Definition: numeric.cpp:1313
GiNaC::power::exponent
ex exponent
Definition: power.h:106
GiNaC::archive_node
This class stores all properties needed to record/retrieve the state of one object of class basic (or...
Definition: archive.h:49
GiNaC::power::coeff
ex coeff(const ex &s, int n=1) const override
Return coefficient of degree n in object s.
Definition: power.cpp:331
GiNaC::_ex1
const ex _ex1
Definition: utils.cpp:193
relational.h
Interface to relations between expressions.
GiNaC::info_flags::positive
@ positive
Definition: flags.h:226
GiNaC::power::evalm
ex evalm() const override
Evaluate sums, products and integer powers of matrices.
Definition: power.cpp:578
options
unsigned options
Definition: factor.cpp:2480
GiNaC::ex::is_equal
bool is_equal(const ex &other) const
Definition: ex.h:345
m
mvec m
Definition: factor.cpp:771
GiNaC::add
Sum of expressions.
Definition: add.h:32
GiNaC::GINAC_BIND_UNARCHIVER
GINAC_BIND_UNARCHIVER(add)
GiNaC::print_func< print_dflt >
print_func< print_dflt >(&diracone::do_print). print_func< print_latex >(&diracone
Definition: clifford.cpp:51
GiNaC::power::power
power(const ex &lh, const ex &rh)
Definition: power.h:48
GiNaC::status_flags::has_indices
@ has_indices
Definition: flags.h:207
GiNaC
Definition: add.cpp:38
GiNaC::power::do_print_csrc
void do_print_csrc(const print_csrc &c, unsigned level) const
Definition: power.cpp:179
GiNaC::power::precedence
unsigned precedence() const override
Return relative operator precedence (for parenthezing output).
Definition: power.h:53
GiNaC::power::do_print_latex
void do_print_latex(const print_latex &c, unsigned level) const
Definition: power.cpp:120
GiNaC::basic::is_equal
bool is_equal(const basic &other) const
Test for syntactic equality.
Definition: basic.cpp:863
GiNaC::power::eval
ex eval() const override
Perform automatic term rewriting rules in this class.
Definition: power.cpp:374
GiNaC::info_flags::nonnegint
@ nonnegint
Definition: flags.h:231
GiNaC::power::expand_add
static ex expand_add(const add &a, long n, unsigned options)
expand a^n where a is an add and n is a positive integer.
Definition: power.cpp:897
GiNaC::basic::clearflag
const basic & clearflag(unsigned f) const
Clear some status_flags.
Definition: basic.h:291
GiNaC::are_ex_trivially_equal
bool are_ex_trivially_equal(const ex &e1, const ex &e2)
Compare two objects of class quickly without doing a deep tree traversal.
Definition: ex.h:684
matrix.h
Interface to symbolic matrices.
x
ex x
Definition: factor.cpp:1641
GiNaC::ex::op
ex op(size_t i) const
Definition: ex.h:136
GiNaC::ex::info
bool info(unsigned inf) const
Definition: ex.h:132
utils.h
Interface to several small and furry utilities needed within GiNaC but not of any interest to the use...
GiNaC::ex::match
bool match(const ex &pattern) const
Check whether expression matches a specified pattern.
Definition: ex.cpp:95
GiNaC::power::archive
void archive(archive_node &n) const override
Save (a.k.a.
Definition: power.cpp:80
last
size_t last
Definition: factor.cpp:1465
GiNaC::info_flags::crational_polynomial
@ crational_polynomial
Definition: flags.h:259
GiNaC::power::derivative
ex derivative(const symbol &s) const override
Implementation of ex::diff() for a power.
Definition: power.cpp:744
GiNaC::sin
const numeric sin(const numeric &x)
Numeric sine (trigonometric function).
Definition: numeric.cpp:1461
GiNaC::power::return_type
unsigned return_type() const override
Definition: power.cpp:768
GiNaC::expair::coeff
ex coeff
second member of pair, must be numeric
Definition: expair.h:91
GiNaC::ex
Lightweight wrapper for GiNaC's symbolic objects.
Definition: ex.h:72
lst.h
Definition of GiNaC's lst.
GiNaC::numeric::is_zero
bool is_zero() const
True if object is zero.
Definition: numeric.cpp:1129
GiNaC::_ex2
const ex _ex2
Definition: utils.cpp:197
GiNaC::basic::compare_same_type
virtual int compare_same_type(const basic &other) const
Returns order relation between two objects of same type.
Definition: basic.cpp:719
GiNaC::print_python
Context for python pretty-print output.
Definition: print.h:131
GiNaC::partition_with_zero_parts_generator::get
const std::vector< unsigned > & get() const
Definition: utils.h:337
GiNaC::subs_options::algebraic
@ algebraic
enable algebraic substitutions
Definition: flags.h:53
GiNaC::ex::imag_part
ex imag_part() const
Definition: ex.h:148
GiNaC::power::do_print_dflt
void do_print_dflt(const print_dflt &c, unsigned level) const
Definition: power.cpp:107
GiNaC::numeric::is_pos_integer
bool is_pos_integer() const
True if object is an exact integer greater than zero.
Definition: numeric.cpp:1161
GiNaC::numer
ex numer(const ex &thisex)
Definition: ex.h:745
GiNaC::exp
const numeric exp(const numeric &x)
Exponential function.
Definition: numeric.cpp:1439
GiNaC::composition_generator
Generate all compositions of a partition of an integer n, starting with the compositions which has no...
Definition: utils.h:395
GiNaC::status_flags::hash_calculated
@ hash_calculated
.calchash() has already done its job
Definition: flags.h:205
GiNaC::ex::diff
ex diff(const symbol &s, unsigned nth=1) const
Compute partial derivative of an expression.
Definition: ex.cpp:86
GiNaC::cos
const numeric cos(const numeric &x)
Numeric cosine (trigonometric function).
Definition: numeric.cpp:1470
GiNaC::basic::has
virtual bool has(const ex &other, unsigned options=0) const
Test for occurrence of a pattern.
Definition: basic.cpp:280
GiNaC::status_flags::has_no_indices
@ has_no_indices
Definition: flags.h:208
GiNaC::ex::ldegree
int ldegree(const ex &s) const
Definition: ex.h:174
GiNaC::numeric::inverse
const numeric inverse() const
Inverse of a number.
Definition: numeric.cpp:1053
GiNaC::basic::hold
const basic & hold() const
Stop further evaluation.
Definition: basic.cpp:887
GiNaC::expairseq::seq
epvector seq
Definition: expairseq.h:127
GiNaC::ex::evalm
ex evalm() const
Definition: ex.h:122
GiNaC::print_latex
Context for latex-parsable output.
Definition: print.h:123
GiNaC::power::ldegree
int ldegree(const ex &s) const override
Return degree of lowest power in object s.
Definition: power.cpp:316
symbol.h
Interface to GiNaC's symbolic objects.
GiNaC::composition_generator::get
const std::vector< unsigned > & get() const
Definition: utils.h:460
GiNaC::basic::subs_one_level
ex subs_one_level(const exmap &m, unsigned options) const
Helper function for subs().
Definition: basic.cpp:585
compiler.h
Definition of optimizing macros.
GiNaC::ex::print
void print(const print_context &c, unsigned level=0) const
Print expression to stream.
Definition: ex.cpp:56
GiNaC::add::recombine_pair_to_ex
ex recombine_pair_to_ex(const expair &p) const override
Definition: add.cpp:564
GiNaC::ex::real_part
ex real_part() const
Definition: ex.h:147
GiNaC::multinomial_coefficient
const numeric multinomial_coefficient(const std::vector< unsigned > &p)
Compute the multinomial coefficient n!/(p1!*p2!*...*pk!) where n = p1+p2+...+pk, i....
Definition: utils.cpp:60
GiNaC::power::nops
size_t nops() const override
Number of operands/members.
Definition: power.cpp:263
GiNaC::ncmul
Non-commutative product of expressions.
Definition: ncmul.h:33
GiNaC::ex::return_type_tinfo
return_type_t return_type_tinfo() const
Definition: ex.h:231
GiNaC::return_type_t
To distinguish between different kinds of non-commutative objects.
Definition: registrar.h:44
GiNaC::power::real_part
ex real_part() const override
Definition: power.cpp:667
ncmul.h
Interface to GiNaC's non-commutative products of expressions.
GiNaC::power::info
bool info(unsigned inf) const override
Information about the object.
Definition: power.cpp:224
GiNaC::info_flags::polynomial
@ polynomial
Definition: flags.h:255
GiNaC::return_types::commutative
@ commutative
Definition: flags.h:280
GiNaC::basic::setflag
const basic & setflag(unsigned f) const
Set some status_flags.
Definition: basic.h:288
GiNaC::numeric::is_crational
bool is_crational() const
True if object is an exact rational number, may even be complex (denominator may be unity).
Definition: numeric.cpp:1243
GiNaC::_num_1_p
const numeric * _num_1_p
Definition: utils.cpp:159
GiNaC::container
Wrapper template for making GiNaC classes out of STL containers.
Definition: container.h:73
GiNaC::info_flags::expanded
@ expanded
Definition: flags.h:270
GiNaC::subs_options::no_pattern
@ no_pattern
disable pattern matching
Definition: flags.h:51
GiNaC::pole_error
Exception class thrown when a singularity is encountered.
Definition: numeric.h:70
GiNaC::tryfactsubs
bool tryfactsubs(const ex &origfactor, const ex &patternfactor, int &nummatches, exmap &repls)
Definition: mul.cpp:672
c
size_t c
Definition: factor.cpp:770
GiNaC::power::map
ex map(map_function &f) const override
Construct new expression by applying the specified function to all sub-expressions (one level only,...
Definition: power.cpp:275
GiNaC::exmap
std::map< ex, ex, ex_is_less > exmap
Definition: basic.h:50
GiNaC::factor
ex factor(const ex &poly, unsigned options)
Interface function to the outside world.
Definition: factor.cpp:2581
GiNaC::numeric::is_equal
bool is_equal(const numeric &other) const
Definition: numeric.cpp:1122
GiNaC::expairseq::overall_coeff
ex overall_coeff
Definition: expairseq.h:128
GiNaC::_ex1_2
const ex _ex1_2
Definition: utils.cpp:189
GiNaC::status_flags::purely_indefinite
@ purely_indefinite
Definition: flags.h:211
GiNaC::power::expand_mul
static ex expand_mul(const mul &m, const numeric &n, unsigned options, bool from_expand=false)
Expand factors of m in m^n where m is a mul and n is an integer.
Definition: power.cpp:1108
GiNaC::power::is_polynomial
bool is_polynomial(const ex &var) const override
Check whether this is a polynomial in the given variables.
Definition: power.cpp:287
GiNaC::power::eval_ncmul
ex eval_ncmul(const exvector &v) const override
Definition: power.cpp:641
GiNaC::numeric::is_positive
bool is_positive() const
True if object is not complex and greater than zero.
Definition: numeric.cpp:1136
archive.h
Archiving of GiNaC expressions.
GiNaC::symbol
Basic CAS symbol.
Definition: symbol.h:39
expairseq.h
Interface to sequences of expression pairs.
n
size_t n
Definition: factor.cpp:1463
GiNaC::info_flags::rational_function
@ rational_function
Definition: flags.h:260
GiNaC::pow
const numeric pow(const numeric &x, const numeric &y)
Definition: numeric.h:251
GiNaC::info_flags::has_indices
@ has_indices
Definition: flags.h:264
GiNaC::binomial
const numeric binomial(const numeric &n, const numeric &k)
The Binomial coefficients.
Definition: numeric.cpp:2143
GiNaC::ex::is_polynomial
bool is_polynomial(const ex &vars) const
Check whether expression is a polynomial.
Definition: ex.cpp:241
GiNaC::power::expand
ex expand(unsigned options=0) const override
Expand expression, i.e.
Definition: power.cpp:778
GiNaC::basic
This class is the ABC (abstract base class) of GiNaC's class hierarchy.
Definition: basic.h:105
GiNaC::power::op
ex op(size_t i) const override
Return operand/member at position i.
Definition: power.cpp:268
GiNaC::basic::ex
friend class ex
Definition: basic.h:108
GiNaC::power::evalf
ex evalf() const override
Evaluate object numerically.
Definition: power.cpp:565
GiNaC::is_negative
bool is_negative(const numeric &x)
Definition: numeric.h:269
GiNaC::expair
A pair of expressions.
Definition: expair.h:38
GiNaC::_num0_p
const numeric * _num0_p
Definition: utils.cpp:175
GiNaC::info_flags::negint
@ negint
Definition: flags.h:230
GiNaC::power::basis
ex basis
Definition: power.h:105
GiNaC::numeric::div
const numeric div(const numeric &other) const
Numerical division method.
Definition: numeric.cpp:890
GiNaC::ex::is_zero
bool is_zero() const
Definition: ex.h:213
GiNaC::power::do_print_csrc_cl_N
void do_print_csrc_cl_N(const print_csrc_cl_N &c, unsigned level) const
Definition: power.cpp:164
indexed.h
Interface to GiNaC's indexed expressions.
GiNaC::info_flags::posint
@ posint
Definition: flags.h:229
GiNaC::log
const numeric log(const numeric &x)
Natural logarithm.
Definition: numeric.cpp:1450
GiNaC::print_csrc_cl_N
Context for C source output using CLN numbers.
Definition: print.h:182
GiNaC::ex::return_type
unsigned return_type() const
Definition: ex.h:230
GiNaC::power::do_print_python
void do_print_python(const print_python &c, unsigned level) const
Definition: power.cpp:210
GiNaC::composition_generator::next
bool next()
Definition: utils.h:474
GiNaC::info_flags::integer_polynomial
@ integer_polynomial
Definition: flags.h:256
GiNaC::power::read_archive
void read_archive(const archive_node &n, lst &syms) override
Read (a.k.a.
Definition: power.cpp:73
GiNaC::abs
const numeric abs(const numeric &x)
Absolute value.
Definition: numeric.cpp:2315
GiNaC::numeric::real
const numeric real() const
Real part of a number.
Definition: numeric.cpp:1339
GiNaC::is_integer
bool is_integer(const numeric &x)
Definition: numeric.h:272
GiNaC::_num1_p
const numeric * _num1_p
Definition: utils.cpp:192
GiNaC::basic::flags
unsigned flags
of type status_flags
Definition: basic.h:302
GiNaC::numeric::denom
const numeric denom() const
Denominator.
Definition: numeric.cpp:1387
GiNaC::get_all_dummy_indices
exvector get_all_dummy_indices(const ex &e)
Returns all dummy indices from the exvector.
Definition: indexed.cpp:1435
GiNaC::numeric::numer
const numeric numer() const
Numerator.
Definition: numeric.cpp:1356
GiNaC::partition_with_zero_parts_generator
Generate all bounded combinatorial partitions of an integer n with exactly m parts (including zero pa...
Definition: utils.h:327
operators.h
Interface to GiNaC's overloaded operators.
GiNaC::ex::integer_content
numeric integer_content() const
Compute the integer content (= GCD of all numeric coefficients) of an expanded polynomial.
Definition: normal.cpp:318
GiNaC::_ex_1
const ex _ex_1
Definition: utils.cpp:160
GiNaC::numeric
This class is a wrapper around CLN-numbers within the GiNaC class hierarchy.
Definition: numeric.h:82
GINAC_ASSERT
#define GINAC_ASSERT(X)
Assertion macro for checking invariances.
Definition: assertion.h:33
GiNaC::numeric::mul_dyn
const numeric & mul_dyn(const numeric &other) const
Numerical multiplication method.
Definition: numeric.cpp:957
GiNaC::_num2_p
const numeric * _num2_p
Definition: utils.cpp:196
GiNaC::ex_is_less
Definition: ex.h:690
GiNaC::mul
Product of expressions.
Definition: mul.h:32
GiNaC::numeric::power
const numeric power(const numeric &other) const
Numerical exponentiation.
Definition: numeric.cpp:900
GiNaC::info_flags::indefinite
@ indefinite
Definition: flags.h:273
likely
#define likely(cond)
Definition: compiler.h:32
GiNaC::power::expand_add_2
static ex expand_add_2(const add &a, unsigned options)
Special case of power::expand_add.
Definition: power.cpp:1039
GiNaC::numeric::compare
int compare(const numeric &other) const
This method establishes a canonical order on all numbers.
Definition: numeric.cpp:1104

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