* Implementation of GiNaC's products of expressions. */
/*
- * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <iostream>
-#include <vector>
-#include <stdexcept>
-#include <limits>
-
#include "mul.h"
#include "add.h"
#include "power.h"
#include "operators.h"
#include "matrix.h"
+#include "indexed.h"
#include "lst.h"
#include "archive.h"
#include "utils.h"
+#include "symbol.h"
+#include "compiler.h"
+
+#include <iostream>
+#include <limits>
+#include <stdexcept>
+#include <vector>
namespace GiNaC {
print_func<print_context>(&mul::do_print).
print_func<print_latex>(&mul::do_print_latex).
print_func<print_csrc>(&mul::do_print_csrc).
- print_func<print_tree>(&inherited::do_print_tree).
+ print_func<print_tree>(&mul::do_print_tree).
print_func<print_python_repr>(&mul::do_print_python_repr))
mul::mul()
{
- tinfo_key = TINFO_mul;
}
//////////
mul::mul(const ex & lh, const ex & rh)
{
- tinfo_key = TINFO_mul;
overall_coeff = _ex1;
construct_from_2_ex(lh,rh);
GINAC_ASSERT(is_canonical());
mul::mul(const exvector & v)
{
- tinfo_key = TINFO_mul;
overall_coeff = _ex1;
construct_from_exvector(v);
GINAC_ASSERT(is_canonical());
mul::mul(const epvector & v)
{
- tinfo_key = TINFO_mul;
overall_coeff = _ex1;
construct_from_epvector(v);
GINAC_ASSERT(is_canonical());
}
-mul::mul(const epvector & v, const ex & oc)
+mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
{
- tinfo_key = TINFO_mul;
overall_coeff = oc;
- construct_from_epvector(v);
+ construct_from_epvector(v, do_index_renaming);
+ GINAC_ASSERT(is_canonical());
+}
+
+mul::mul(epvector && vp)
+{
+ overall_coeff = _ex1;
+ construct_from_epvector(std::move(vp));
GINAC_ASSERT(is_canonical());
}
-mul::mul(std::auto_ptr<epvector> vp, const ex & oc)
+mul::mul(epvector && vp, const ex & oc, bool do_index_renaming)
{
- tinfo_key = TINFO_mul;
- GINAC_ASSERT(vp!=0);
overall_coeff = oc;
- construct_from_epvector(*vp);
+ construct_from_epvector(std::move(vp), do_index_renaming);
GINAC_ASSERT(is_canonical());
}
mul::mul(const ex & lh, const ex & mh, const ex & rh)
{
- tinfo_key = TINFO_mul;
exvector factors;
factors.reserve(3);
factors.push_back(lh);
// archiving
//////////
-DEFAULT_ARCHIVING(mul)
-
//////////
// functions overriding virtual functions from base classes
//////////
const numeric &coeff = ex_to<numeric>(overall_coeff);
if (coeff.csgn() == -1)
c.s << '-';
- if (!coeff.is_equal(_num1) &&
- !coeff.is_equal(_num_1)) {
+ if (!coeff.is_equal(*_num1_p) &&
+ !coeff.is_equal(*_num_1_p)) {
if (coeff.is_rational()) {
if (coeff.is_negative())
(-coeff).print(c);
print_overall_coeff(c, "*");
- epvector::const_iterator it = seq.begin(), itend = seq.end();
bool first = true;
- while (it != itend) {
+ for (auto & it : seq) {
if (!first)
c.s << '*';
else
first = false;
- recombine_pair_to_ex(*it).print(c, precedence());
- ++it;
+ recombine_pair_to_ex(it).print(c, precedence());
}
if (precedence() <= level)
// Separate factors into those with negative numeric exponent
// and all others
- epvector::const_iterator it = seq.begin(), itend = seq.end();
exvector neg_powers, others;
- while (it != itend) {
- GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
- if (ex_to<numeric>(it->coeff).is_negative())
- neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
+ for (auto & it : seq) {
+ GINAC_ASSERT(is_exactly_a<numeric>(it.coeff));
+ if (ex_to<numeric>(it.coeff).is_negative())
+ neg_powers.push_back(recombine_pair_to_ex(expair(it.rest, -it.coeff)));
else
- others.push_back(recombine_pair_to_ex(*it));
- ++it;
+ others.push_back(recombine_pair_to_ex(it));
}
if (!neg_powers.empty()) {
} else {
// All other factors are printed in the ordinary way
- exvector::const_iterator vit = others.begin(), vitend = others.end();
- while (vit != vitend) {
+ for (auto & vit : others) {
c.s << ' ';
- vit->print(c, precedence());
- ++vit;
+ vit.print(c, precedence());
}
}
c.s << "(";
if (!overall_coeff.is_equal(_ex1)) {
- overall_coeff.print(c, precedence());
- c.s << "*";
+ if (overall_coeff.is_equal(_ex_1))
+ c.s << "-";
+ else {
+ overall_coeff.print(c, precedence());
+ c.s << "*";
+ }
}
// Print arguments, separated by "*" or "/"
- epvector::const_iterator it = seq.begin(), itend = seq.end();
+ auto it = seq.begin(), itend = seq.end();
while (it != itend) {
// If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
it->rest.print(c, precedence());
else if (it->coeff.info(info_flags::negint))
- // Outer parens around ex needed for broken GCC parser:
- (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
+ ex(power(it->rest, -ex_to<numeric>(it->coeff))).print(c, level);
else
- // Outer parens around ex needed for broken GCC parser:
- (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
+ ex(power(it->rest, ex_to<numeric>(it->coeff))).print(c, level);
if (needclosingparenthesis)
c.s << ")";
case info_flags::integer_polynomial:
case info_flags::cinteger_polynomial:
case info_flags::rational_polynomial:
+ case info_flags::real:
+ case info_flags::rational:
+ case info_flags::integer:
+ case info_flags::crational:
+ case info_flags::cinteger:
+ case info_flags::even:
case info_flags::crational_polynomial:
case info_flags::rational_function: {
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- if (!(recombine_pair_to_ex(*i).info(inf)))
+ for (auto & it : seq) {
+ if (!recombine_pair_to_ex(it).info(inf))
return false;
- ++i;
}
+ if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
+ return true;
return overall_coeff.info(inf);
}
- case info_flags::algebraic: {
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- if ((recombine_pair_to_ex(*i).info(inf)))
- return true;
- ++i;
+ case info_flags::positive:
+ case info_flags::negative: {
+ if ((inf==info_flags::positive) && (flags & status_flags::is_positive))
+ return true;
+ else if ((inf==info_flags::negative) && (flags & status_flags::is_negative))
+ return true;
+ if (flags & status_flags::purely_indefinite)
+ return false;
+
+ bool pos = true;
+ for (auto & it : seq) {
+ const ex& factor = recombine_pair_to_ex(it);
+ if (factor.info(info_flags::positive))
+ continue;
+ else if (factor.info(info_flags::negative))
+ pos = !pos;
+ else
+ return false;
}
- return false;
+ if (overall_coeff.info(info_flags::negative))
+ pos = !pos;
+ setflag(pos ? status_flags::is_positive : status_flags::is_negative);
+ return (inf == info_flags::positive? pos : !pos);
+ }
+ case info_flags::nonnegative: {
+ if (flags & status_flags::is_positive)
+ return true;
+ bool pos = true;
+ for (auto & it : seq) {
+ const ex& factor = recombine_pair_to_ex(it);
+ if (factor.info(info_flags::nonnegative) || factor.info(info_flags::positive))
+ continue;
+ else if (factor.info(info_flags::negative))
+ pos = !pos;
+ else
+ return false;
+ }
+ return (overall_coeff.info(info_flags::negative)? !pos : pos);
+ }
+ case info_flags::posint:
+ case info_flags::negint: {
+ bool pos = true;
+ for (auto & it : seq) {
+ const ex& factor = recombine_pair_to_ex(it);
+ if (factor.info(info_flags::posint))
+ continue;
+ else if (factor.info(info_flags::negint))
+ pos = !pos;
+ else
+ return false;
+ }
+ if (overall_coeff.info(info_flags::negint))
+ pos = !pos;
+ else if (!overall_coeff.info(info_flags::posint))
+ return false;
+ return (inf ==info_flags::posint? pos : !pos);
+ }
+ case info_flags::nonnegint: {
+ bool pos = true;
+ for (auto & it : seq) {
+ const ex& factor = recombine_pair_to_ex(it);
+ if (factor.info(info_flags::nonnegint) || factor.info(info_flags::posint))
+ continue;
+ else if (factor.info(info_flags::negint))
+ pos = !pos;
+ else
+ return false;
+ }
+ if (overall_coeff.info(info_flags::negint))
+ pos = !pos;
+ else if (!overall_coeff.info(info_flags::posint))
+ return false;
+ return pos;
+ }
+ case info_flags::indefinite: {
+ if (flags & status_flags::purely_indefinite)
+ return true;
+ if (flags & (status_flags::is_positive | status_flags::is_negative))
+ return false;
+ for (auto & it : seq) {
+ const ex& term = recombine_pair_to_ex(it);
+ if (term.info(info_flags::positive) || term.info(info_flags::negative))
+ return false;
+ }
+ setflag(status_flags::purely_indefinite);
+ return true;
}
}
return inherited::info(inf);
}
+bool mul::is_polynomial(const ex & var) const
+{
+ for (auto & it : seq) {
+ if (!it.rest.is_polynomial(var) ||
+ (it.rest.has(var) && !it.coeff.info(info_flags::nonnegint))) {
+ return false;
+ }
+ }
+ return true;
+}
+
int mul::degree(const ex & s) const
{
// Sum up degrees of factors
int deg_sum = 0;
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- if (ex_to<numeric>(i->coeff).is_integer())
- deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
- ++i;
+ for (auto & it : seq) {
+ if (ex_to<numeric>(it.coeff).is_integer())
+ deg_sum += recombine_pair_to_ex(it).degree(s);
+ else {
+ if (it.rest.has(s))
+ throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
+ }
}
return deg_sum;
}
{
// Sum up degrees of factors
int deg_sum = 0;
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- if (ex_to<numeric>(i->coeff).is_integer())
- deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
- ++i;
+ for (auto & it : seq) {
+ if (ex_to<numeric>(it.coeff).is_integer())
+ deg_sum += recombine_pair_to_ex(it).ldegree(s);
+ else {
+ if (it.rest.has(s))
+ throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
+ }
}
return deg_sum;
}
if (n==0) {
// product of individual coeffs
// if a non-zero power of s is found, the resulting product will be 0
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
- ++i;
- }
+ for (auto & it : seq)
+ coeffseq.push_back(recombine_pair_to_ex(it).coeff(s,n));
coeffseq.push_back(overall_coeff);
- return (new mul(coeffseq))->setflag(status_flags::dynallocated);
+ return dynallocate<mul>(coeffseq);
}
- epvector::const_iterator i = seq.begin(), end = seq.end();
bool coeff_found = false;
- while (i != end) {
- ex t = recombine_pair_to_ex(*i);
+ for (auto & it : seq) {
+ ex t = recombine_pair_to_ex(it);
ex c = t.coeff(s, n);
if (!c.is_zero()) {
coeffseq.push_back(c);
} else {
coeffseq.push_back(t);
}
- ++i;
}
if (coeff_found) {
coeffseq.push_back(overall_coeff);
- return (new mul(coeffseq))->setflag(status_flags::dynallocated);
+ return dynallocate<mul>(coeffseq);
}
return _ex0;
* - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
* - *(x;1) -> x
* - *(;c) -> c
- *
- * @param level cut-off in recursive evaluation */
-ex mul::eval(int level) const
+ */
+ex mul::eval() const
{
- std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
- if (evaled_seqp.get()) {
- // do more evaluation later
- return (new mul(evaled_seqp, overall_coeff))->
- setflag(status_flags::dynallocated);
- }
-
-#ifdef DO_GINAC_ASSERT
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
- (!(ex_to<numeric>(i->coeff).is_integer())));
- GINAC_ASSERT(!(i->is_canonical_numeric()));
- if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
- print(print_tree(std::cerr));
- GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
- /* for paranoia */
- expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
- GINAC_ASSERT(p.rest.is_equal(i->rest));
- GINAC_ASSERT(p.coeff.is_equal(i->coeff));
- /* end paranoia */
- ++i;
- }
-#endif // def DO_GINAC_ASSERT
-
if (flags & status_flags::evaluated) {
GINAC_ASSERT(seq.size()>0);
GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
return *this;
}
-
- int seq_size = seq.size();
+
+ const epvector evaled = evalchildren();
+ if (unlikely(!evaled.empty())) {
+ // start over evaluating a new object
+ return dynallocate<mul>(std::move(evaled), overall_coeff);
+ }
+
+ size_t seq_size = seq.size();
if (overall_coeff.is_zero()) {
// *(...,x;0) -> 0
return _ex0;
return recombine_pair_to_ex(*(seq.begin()));
} else if ((seq_size==1) &&
is_exactly_a<add>((*seq.begin()).rest) &&
- ex_to<numeric>((*seq.begin()).coeff).is_equal(_num1)) {
+ ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
// *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
const add & addref = ex_to<add>((*seq.begin()).rest);
- std::auto_ptr<epvector> distrseq(new epvector);
- distrseq->reserve(addref.seq.size());
- epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
- while (i != end) {
- distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
+ epvector distrseq;
+ distrseq.reserve(addref.seq.size());
+ for (auto & it : addref.seq) {
+ distrseq.push_back(addref.combine_pair_with_coeff_to_pair(it, overall_coeff));
+ }
+ return dynallocate<add>(std::move(distrseq),
+ ex_to<numeric>(addref.overall_coeff).mul_dyn(ex_to<numeric>(overall_coeff)))
+ .setflag(status_flags::evaluated);
+ } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
+ // Strip the content and the unit part from each term. Thus
+ // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
+
+ auto i = seq.begin(), last = seq.end();
+ auto j = seq.begin();
+ epvector s;
+ numeric oc = *_num1_p;
+ bool something_changed = false;
+ while (i!=last) {
+ if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
+ // power::eval has such a rule, no need to handle powers here
+ ++i;
+ continue;
+ }
+
+ // XXX: What is the best way to check if the polynomial is a primitive?
+ numeric c = i->rest.integer_content();
+ const numeric lead_coeff =
+ ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
+ const bool canonicalizable = lead_coeff.is_integer();
+
+ // XXX: The main variable is chosen in a random way, so this code
+ // does NOT transform the term into the canonical form (thus, in some
+ // very unlucky event it can even loop forever). Hopefully the main
+ // variable will be the same for all terms in *this
+ const bool unit_normal = lead_coeff.is_pos_integer();
+ if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
+ ++i;
+ continue;
+ }
+
+ if (! something_changed) {
+ s.reserve(seq_size);
+ something_changed = true;
+ }
+
+ while ((j!=i) && (j!=last)) {
+ s.push_back(*j);
+ ++j;
+ }
+
+ if (! unit_normal)
+ c = c.mul(*_num_1_p);
+
+ oc = oc.mul(c);
+
+ // divide add by the number in place to save at least 2 .eval() calls
+ const add& addref = ex_to<add>(i->rest);
+ add & primitive = dynallocate<add>(addref);
+ primitive.clearflag(status_flags::hash_calculated);
+ primitive.overall_coeff = ex_to<numeric>(primitive.overall_coeff).div_dyn(c);
+ for (auto & ai : primitive.seq)
+ ai.coeff = ex_to<numeric>(ai.coeff).div_dyn(c);
+
+ s.push_back(expair(primitive, _ex1));
+
++i;
+ ++j;
+ }
+ if (something_changed) {
+ while (j!=last) {
+ s.push_back(*j);
+ ++j;
+ }
+ return dynallocate<mul>(std::move(s), ex_to<numeric>(overall_coeff).mul_dyn(oc));
}
- return (new add(distrseq,
- ex_to<numeric>(addref.overall_coeff).
- mul_dyn(ex_to<numeric>(overall_coeff))))
- ->setflag(status_flags::dynallocated | status_flags::evaluated);
}
+
return this->hold();
}
-ex mul::evalf(int level) const
+ex mul::evalf() const
{
- if (level==1)
- return mul(seq,overall_coeff);
-
- if (level==-max_recursion_level)
- throw(std::runtime_error("max recursion level reached"));
-
- std::auto_ptr<epvector> s(new epvector);
- s->reserve(seq.size());
+ epvector s;
+ s.reserve(seq.size());
- --level;
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
- i->coeff));
- ++i;
+ for (auto & it : seq)
+ s.push_back(expair(it.rest.evalf(), it.coeff));
+ return dynallocate<mul>(std::move(s), overall_coeff.evalf());
+}
+
+void mul::find_real_imag(ex & rp, ex & ip) const
+{
+ rp = overall_coeff.real_part();
+ ip = overall_coeff.imag_part();
+ for (auto & it : seq) {
+ ex factor = recombine_pair_to_ex(it);
+ ex new_rp = factor.real_part();
+ ex new_ip = factor.imag_part();
+ if (new_ip.is_zero()) {
+ rp *= new_rp;
+ ip *= new_rp;
+ } else {
+ ex temp = rp*new_rp - ip*new_ip;
+ ip = ip*new_rp + rp*new_ip;
+ rp = temp;
+ }
}
- return mul(s, overall_coeff.evalf(level));
+ rp = rp.expand();
+ ip = ip.expand();
+}
+
+ex mul::real_part() const
+{
+ ex rp, ip;
+ find_real_imag(rp, ip);
+ return rp;
+}
+
+ex mul::imag_part() const
+{
+ ex rp, ip;
+ find_real_imag(rp, ip);
+ return ip;
}
ex mul::evalm() const
// Evaluate children first, look whether there are any matrices at all
// (there can be either no matrices or one matrix; if there were more
// than one matrix, it would be a non-commutative product)
- std::auto_ptr<epvector> s(new epvector);
- s->reserve(seq.size());
+ epvector s;
+ s.reserve(seq.size());
bool have_matrix = false;
epvector::iterator the_matrix;
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- const ex &m = recombine_pair_to_ex(*i).evalm();
- s->push_back(split_ex_to_pair(m));
+ for (auto & it : seq) {
+ const ex &m = recombine_pair_to_ex(it).evalm();
+ s.push_back(split_ex_to_pair(m));
if (is_a<matrix>(m)) {
have_matrix = true;
- the_matrix = s->end() - 1;
+ the_matrix = s.end() - 1;
}
- ++i;
}
if (have_matrix) {
// The product contained a matrix. We will multiply all other factors
// into that matrix.
matrix m = ex_to<matrix>(the_matrix->rest);
- s->erase(the_matrix);
- ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
+ s.erase(the_matrix);
+ ex scalar = dynallocate<mul>(std::move(s), overall_coeff);
return m.mul_scalar(scalar);
} else
- return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
+ return dynallocate<mul>(std::move(s), overall_coeff);
}
ex mul::eval_ncmul(const exvector & v) const
return inherited::eval_ncmul(v);
// Find first noncommutative element and call its eval_ncmul()
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- if (i->rest.return_type() == return_types::noncommutative)
- return i->rest.eval_ncmul(v);
- ++i;
- }
+ for (auto & it : seq)
+ if (it.rest.return_type() == return_types::noncommutative)
+ return it.rest.eval_ncmul(v);
return inherited::eval_ncmul(v);
}
-bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
+bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
{
ex origbase;
int origexponent;
patternexpsign = 1;
}
- lst saverepls = repls;
+ exmap saverepls = repls;
if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
return false;
repls = saverepls;
return true;
}
+/** Checks whether e matches to the pattern pat and the (possibly to be updated)
+ * list of replacements repls. This matching is in the sense of algebraic
+ * substitutions. Matching starts with pat.op(factor) of the pattern because
+ * the factors before this one have already been matched. The (possibly
+ * updated) number of matches is in nummatches. subsed[i] is true for factors
+ * that already have been replaced by previous substitutions and matched[i]
+ * is true for factors that have been matched by the current match.
+ */
+bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
+ int factor, int &nummatches, const std::vector<bool> &subsed,
+ std::vector<bool> &matched)
+{
+ GINAC_ASSERT(subsed.size() == e.nops());
+ GINAC_ASSERT(matched.size() == e.nops());
+
+ if (factor == (int)pat.nops())
+ return true;
+
+ for (size_t i=0; i<e.nops(); ++i) {
+ if(subsed[i] || matched[i])
+ continue;
+ exmap newrepls = repls;
+ int newnummatches = nummatches;
+ if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
+ matched[i] = true;
+ if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
+ newnummatches, subsed, matched)) {
+ repls = newrepls;
+ nummatches = newnummatches;
+ return true;
+ }
+ else
+ matched[i] = false;
+ }
+ }
+
+ return false;
+}
+
+bool mul::has(const ex & pattern, unsigned options) const
+{
+ if(!(options & has_options::algebraic))
+ return basic::has(pattern,options);
+ if(is_a<mul>(pattern)) {
+ exmap repls;
+ int nummatches = std::numeric_limits<int>::max();
+ std::vector<bool> subsed(nops(), false);
+ std::vector<bool> matched(nops(), false);
+ if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
+ subsed, matched))
+ return true;
+ }
+ return basic::has(pattern, options);
+}
+
ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
{
- std::vector<bool> subsed(seq.size(), false);
- exvector subsresult(seq.size());
-
- for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
+ std::vector<bool> subsed(nops(), false);
+ ex divide_by = 1;
+ ex multiply_by = 1;
- if (is_exactly_a<mul>(it->first)) {
+ for (auto & it : m) {
+ if (is_exactly_a<mul>(it.first)) {
+retry1:
int nummatches = std::numeric_limits<int>::max();
- std::vector<bool> currsubsed(seq.size(), false);
- bool succeed = true;
- lst repls;
-
- for (size_t j=0; j<it->first.nops(); j++) {
- bool found=false;
- for (size_t k=0; k<nops(); k++) {
- if (currsubsed[k] || subsed[k])
- continue;
- if (tryfactsubs(op(k), it->first.op(j), nummatches, repls)) {
- currsubsed[k] = true;
- found = true;
- break;
- }
- }
- if (!found) {
- succeed = false;
- break;
- }
- }
- if (!succeed)
+ std::vector<bool> currsubsed(nops(), false);
+ exmap repls;
+
+ if (!algebraic_match_mul_with_mul(*this, it.first, repls, 0, nummatches, subsed, currsubsed))
continue;
- bool foundfirstsubsedfactor = false;
- for (size_t j=0; j<subsed.size(); j++) {
- if (currsubsed[j]) {
- if (foundfirstsubsedfactor)
- subsresult[j] = op(j);
- else {
- foundfirstsubsedfactor = true;
- subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
- }
+ for (size_t j=0; j<subsed.size(); j++)
+ if (currsubsed[j])
subsed[j] = true;
- }
- }
+ ex subsed_pattern
+ = it.first.subs(repls, subs_options::no_pattern);
+ divide_by *= pow(subsed_pattern, nummatches);
+ ex subsed_result
+ = it.second.subs(repls, subs_options::no_pattern);
+ multiply_by *= pow(subsed_result, nummatches);
+ goto retry1;
} else {
- int nummatches = std::numeric_limits<int>::max();
- lst repls;
-
for (size_t j=0; j<this->nops(); j++) {
- if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)) {
+ int nummatches = std::numeric_limits<int>::max();
+ exmap repls;
+ if (!subsed[j] && tryfactsubs(op(j), it.first, nummatches, repls)){
subsed[j] = true;
- subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
+ ex subsed_pattern
+ = it.first.subs(repls, subs_options::no_pattern);
+ divide_by *= pow(subsed_pattern, nummatches);
+ ex subsed_result
+ = it.second.subs(repls, subs_options::no_pattern);
+ multiply_by *= pow(subsed_result, nummatches);
}
}
}
if (!subsfound)
return subs_one_level(m, options | subs_options::algebraic);
- exvector ev; ev.reserve(nops());
- for (size_t i=0; i<nops(); i++) {
- if (subsed[i])
- ev.push_back(subsresult[i]);
- else
- ev.push_back(op(i));
- }
+ return ((*this)/divide_by)*multiply_by;
+}
- return (new mul(ev))->setflag(status_flags::dynallocated);
+ex mul::conjugate() const
+{
+ // The base class' method is wrong here because we have to be careful at
+ // branch cuts. power::conjugate takes care of that already, so use it.
+ std::unique_ptr<epvector> newepv(nullptr);
+ for (auto i=seq.begin(); i!=seq.end(); ++i) {
+ if (newepv) {
+ newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
+ continue;
+ }
+ ex x = recombine_pair_to_ex(*i);
+ ex c = x.conjugate();
+ if (c.is_equal(x)) {
+ continue;
+ }
+ newepv.reset(new epvector);
+ newepv->reserve(seq.size());
+ for (auto j=seq.begin(); j!=i; ++j) {
+ newepv->push_back(*j);
+ }
+ newepv->push_back(split_ex_to_pair(c));
+ }
+ ex x = overall_coeff.conjugate();
+ if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
+ return *this;
+ }
+ return thisexpairseq(newepv ? std::move(*newepv) : seq, x);
}
+
// protected
/** Implementation of ex::diff() for a product. It applies the product rule.
// D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
epvector mulseq = seq;
- epvector::const_iterator i = seq.begin(), end = seq.end();
- epvector::iterator i2 = mulseq.begin();
+ auto i = seq.begin(), end = seq.end();
+ auto i2 = mulseq.begin();
while (i != end) {
- expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
+ expair ep = split_ex_to_pair(pow(i->rest, i->coeff - _ex1) *
i->rest.diff(s));
ep.swap(*i2);
- addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
+ addseq.push_back(dynallocate<mul>(mulseq, overall_coeff * i->coeff));
ep.swap(*i2);
++i; ++i2;
}
- return (new add(addseq))->setflag(status_flags::dynallocated);
+ return dynallocate<add>(addseq);
}
int mul::compare_same_type(const basic & other) const
unsigned mul::return_type() const
{
if (seq.empty()) {
- // mul without factors: should not happen, but commutes
+ // mul without factors: should not happen, but commutates
return return_types::commutative;
}
if ((rt == return_types::noncommutative) && (!all_commutative)) {
// another nc element found, compare type_infos
if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
- // diffent types -> mul is ncc
- return return_types::noncommutative_composite;
+ // different types -> mul is ncc
+ return return_types::noncommutative_composite;
}
}
++i;
// all factors checked
return all_commutative ? return_types::commutative : return_types::noncommutative;
}
-
-unsigned mul::return_type_tinfo() const
+
+return_type_t mul::return_type_tinfo() const
{
if (seq.empty())
- return tinfo_key; // mul without factors: should not happen
+ return make_return_type_t<mul>(); // mul without factors: should not happen
// return type_info of first noncommutative element
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- if (i->rest.return_type() == return_types::noncommutative)
- return i->rest.return_type_tinfo();
- ++i;
- }
+ for (auto & it : seq)
+ if (it.rest.return_type() == return_types::noncommutative)
+ return it.rest.return_type_tinfo();
+
// no noncommutative element found, should not happen
- return tinfo_key;
+ return make_return_type_t<mul>();
}
-ex mul::thisexpairseq(const epvector & v, const ex & oc) const
+ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
{
- return (new mul(v, oc))->setflag(status_flags::dynallocated);
+ return dynallocate<mul>(v, oc, do_index_renaming);
}
-ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc) const
+ex mul::thisexpairseq(epvector && vp, const ex & oc, bool do_index_renaming) const
{
- return (new mul(vp, oc))->setflag(status_flags::dynallocated);
+ return dynallocate<mul>(std::move(vp), oc, do_index_renaming);
}
expair mul::split_ex_to_pair(const ex & e) const
}
return expair(e,_ex1);
}
-
+
expair mul::combine_ex_with_coeff_to_pair(const ex & e,
const ex & c) const
{
+ GINAC_ASSERT(is_exactly_a<numeric>(c));
+
+ // First, try a common shortcut:
+ if (is_exactly_a<symbol>(e))
+ return expair(e, c);
+
+ // trivial case: exponent 1
+ if (c.is_equal(_ex1))
+ return split_ex_to_pair(e);
+
// to avoid duplication of power simplification rules,
// we create a temporary power object
// otherwise it would be hard to correctly evaluate
// expression like (4^(1/3))^(3/2)
- if (c.is_equal(_ex1))
- return split_ex_to_pair(e);
-
- return split_ex_to_pair(power(e,c));
+ return split_ex_to_pair(pow(e,c));
}
-
+
expair mul::combine_pair_with_coeff_to_pair(const expair & p,
const ex & c) const
{
+ GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
+ GINAC_ASSERT(is_exactly_a<numeric>(c));
+
+ // First, try a common shortcut:
+ if (is_exactly_a<symbol>(p.rest))
+ return expair(p.rest, p.coeff * c);
+
+ // trivial case: exponent 1
+ if (c.is_equal(_ex1))
+ return p;
+ if (p.coeff.is_equal(_ex1))
+ return expair(p.rest, c);
+
// to avoid duplication of power simplification rules,
// we create a temporary power object
// otherwise it would be hard to correctly evaluate
// expression like (4^(1/3))^(3/2)
- if (c.is_equal(_ex1))
- return p;
-
- return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
+ return split_ex_to_pair(pow(recombine_pair_to_ex(p),c));
}
-
+
ex mul::recombine_pair_to_ex(const expair & p) const
{
- if (ex_to<numeric>(p.coeff).is_equal(_num1))
+ if (p.coeff.is_equal(_ex1))
return p.rest;
else
- return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
+ return dynallocate<power>(p.rest, p.coeff);
}
bool mul::expair_needs_further_processing(epp it)
{
if (is_exactly_a<mul>(it->rest) &&
- ex_to<numeric>(it->coeff).is_integer()) {
+ ex_to<numeric>(it->coeff).is_integer()) {
// combined pair is product with integer power -> expand it
*it = split_ex_to_pair(recombine_pair_to_ex(*it));
return true;
}
if (is_exactly_a<numeric>(it->rest)) {
+ if (it->coeff.is_equal(_ex1)) {
+ // pair has coeff 1 and must be moved to the end
+ return true;
+ }
expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
if (!ep.is_equal(*it)) {
// combined pair is a numeric power which can be simplified
*it = ep;
return true;
}
- if (it->coeff.is_equal(_ex1)) {
- // combined pair has coeff 1 and must be moved to the end
- return true;
- }
}
return false;
}
bool mul::can_make_flat(const expair & p) const
{
GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
- // this assertion will probably fail somewhere
- // it would require a more careful make_flat, obeying the power laws
- // probably should return true only if p.coeff is integer
- return ex_to<numeric>(p.coeff).is_equal(_num1);
+
+ // (x*y)^c == x^c*y^c if c ∈ ℤ
+ return p.coeff.info(info_flags::integer);
+}
+
+bool mul::can_be_further_expanded(const ex & e)
+{
+ if (is_exactly_a<mul>(e)) {
+ for (auto & it : ex_to<mul>(e).seq) {
+ if (is_exactly_a<add>(it.rest) && it.coeff.info(info_flags::posint))
+ return true;
+ }
+ } else if (is_exactly_a<power>(e)) {
+ if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
+ return true;
+ }
+ return false;
}
ex mul::expand(unsigned options) const
{
+ // Check for trivial case: expanding the monomial (~ 30% of all calls)
+ bool monomial_case = true;
+ for (const auto & i : seq) {
+ if (!is_a<symbol>(i.rest) || !i.coeff.info(info_flags::integer)) {
+ monomial_case = false;
+ break;
+ }
+ }
+ if (monomial_case) {
+ setflag(status_flags::expanded);
+ return *this;
+ }
+
+ // do not rename indices if the object has no indices at all
+ if ((!(options & expand_options::expand_rename_idx)) &&
+ this->info(info_flags::has_indices))
+ options |= expand_options::expand_rename_idx;
+
+ const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
+
// First, expand the children
- std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
- const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
+ epvector expanded = expandchildren(options);
+ const epvector & expanded_seq = (expanded.empty() ? seq : expanded);
// Now, look for all the factors that are sums and multiply each one out
// with the next one that is found while collecting the factors which are
// not sums
- int number_of_adds = 0;
ex last_expanded = _ex1;
+
epvector non_adds;
non_adds.reserve(expanded_seq.size());
- epvector::const_iterator cit = expanded_seq.begin(), last = expanded_seq.end();
- while (cit != last) {
- if (is_exactly_a<add>(cit->rest) &&
- (cit->coeff.is_equal(_ex1))) {
- ++number_of_adds;
+
+ for (const auto & cit : expanded_seq) {
+ if (is_exactly_a<add>(cit.rest) &&
+ (cit.coeff.is_equal(_ex1))) {
if (is_exactly_a<add>(last_expanded)) {
// Expand a product of two sums, aggressive version.
// Caring for the overall coefficients in separate loops can
// sometimes give a performance gain of up to 15%!
- const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
+ const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit.rest).seq.size();
// add2 is for the inner loop and should be the bigger of the two sums
// in the presence of asymptotically good sorting:
- const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
- const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
- const epvector::const_iterator add1begin = add1.seq.begin();
- const epvector::const_iterator add1end = add1.seq.end();
- const epvector::const_iterator add2begin = add2.seq.begin();
- const epvector::const_iterator add2end = add2.seq.end();
+ const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit.rest));
+ const add& add2 = (sizedifference<0 ? ex_to<add>(cit.rest) : ex_to<add>(last_expanded));
epvector distrseq;
distrseq.reserve(add1.seq.size()+add2.seq.size());
+
// Multiply add2 with the overall coefficient of add1 and append it to distrseq:
if (!add1.overall_coeff.is_zero()) {
if (add1.overall_coeff.is_equal(_ex1))
- distrseq.insert(distrseq.end(),add2begin,add2end);
+ distrseq.insert(distrseq.end(), add2.seq.begin(), add2.seq.end());
else
- for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
- distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
+ for (const auto & i : add2.seq)
+ distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
}
+
// Multiply add1 with the overall coefficient of add2 and append it to distrseq:
if (!add2.overall_coeff.is_zero()) {
if (add2.overall_coeff.is_equal(_ex1))
- distrseq.insert(distrseq.end(),add1begin,add1end);
+ distrseq.insert(distrseq.end(), add1.seq.begin(), add1.seq.end());
else
- for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
- distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
+ for (const auto & i : add1.seq)
+ distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
}
+
// Compute the new overall coefficient and put it together:
- ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
+ ex tmp_accu = dynallocate<add>(distrseq, add1.overall_coeff*add2.overall_coeff);
+
+ exvector add1_dummy_indices, add2_dummy_indices, add_indices;
+ lst dummy_subs;
+
+ if (!skip_idx_rename) {
+ for (const auto & i : add1.seq) {
+ add_indices = get_all_dummy_indices_safely(i.rest);
+ add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
+ }
+ for (const auto & i : add2.seq) {
+ add_indices = get_all_dummy_indices_safely(i.rest);
+ add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
+ }
+
+ sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
+ sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
+ dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
+ }
+
// Multiply explicitly all non-numeric terms of add1 and add2:
- for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
+ for (const auto & i2 : add2.seq) {
// We really have to combine terms here in order to compactify
// the result. Otherwise it would become waayy tooo bigg.
- numeric oc;
- distrseq.clear();
- for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
+ numeric oc(*_num0_p);
+ epvector distrseq2;
+ distrseq2.reserve(add1.seq.size());
+ const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
+ i2.rest :
+ i2.rest.subs(ex_to<lst>(dummy_subs.op(0)),
+ ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
+ for (const auto & i1 : add1.seq) {
// Don't push_back expairs which might have a rest that evaluates to a numeric,
// since that would violate an invariant of expairseq:
- const ex rest = (new mul(i1->rest, i2->rest))->setflag(status_flags::dynallocated);
- if (is_exactly_a<numeric>(rest))
- oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
- else
- distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
+ const ex rest = dynallocate<mul>(i1.rest, i2_new);
+ if (is_exactly_a<numeric>(rest)) {
+ oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1.coeff).mul(ex_to<numeric>(i2.coeff)));
+ } else {
+ distrseq2.push_back(expair(rest, ex_to<numeric>(i1.coeff).mul_dyn(ex_to<numeric>(i2.coeff))));
+ }
}
- tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
+ tmp_accu += dynallocate<add>(std::move(distrseq2), oc);
}
last_expanded = tmp_accu;
-
} else {
- non_adds.push_back(split_ex_to_pair(last_expanded));
- last_expanded = cit->rest;
+ if (!last_expanded.is_equal(_ex1))
+ non_adds.push_back(split_ex_to_pair(last_expanded));
+ last_expanded = cit.rest;
}
+
} else {
- non_adds.push_back(*cit);
+ non_adds.push_back(cit);
}
- ++cit;
}
-
+
// Now the only remaining thing to do is to multiply the factors which
// were not sums into the "last_expanded" sum
if (is_exactly_a<add>(last_expanded)) {
- const add & finaladd = ex_to<add>(last_expanded);
+ size_t n = last_expanded.nops();
exvector distrseq;
- size_t n = finaladd.nops();
distrseq.reserve(n);
+ exvector va;
+ if (! skip_idx_rename) {
+ va = get_all_dummy_indices_safely(mul(non_adds));
+ sort(va.begin(), va.end(), ex_is_less());
+ }
+
for (size_t i=0; i<n; ++i) {
epvector factors = non_adds;
- factors.push_back(split_ex_to_pair(finaladd.op(i)));
- distrseq.push_back((new mul(factors, overall_coeff))->
- setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
+ if (skip_idx_rename)
+ factors.push_back(split_ex_to_pair(last_expanded.op(i)));
+ else
+ factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
+ ex term = dynallocate<mul>(factors, overall_coeff);
+ if (can_be_further_expanded(term)) {
+ distrseq.push_back(term.expand());
+ } else {
+ if (options == 0)
+ ex_to<basic>(term).setflag(status_flags::expanded);
+ distrseq.push_back(term);
+ }
}
- return ((new add(distrseq))->
- setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
+
+ return dynallocate<add>(distrseq).setflag(options == 0 ? status_flags::expanded : 0);
}
+
non_adds.push_back(split_ex_to_pair(last_expanded));
- return (new mul(non_adds, overall_coeff))->
- setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
+ ex result = dynallocate<mul>(non_adds, overall_coeff);
+ if (can_be_further_expanded(result)) {
+ return result.expand();
+ } else {
+ if (options == 0)
+ ex_to<basic>(result).setflag(status_flags::expanded);
+ return result;
+ }
}
/** Member-wise expand the expairs representing this sequence. This must be
* overridden from expairseq::expandchildren() and done iteratively in order
- * to allow for early cancallations and thus safe memory.
+ * to allow for early cancellations and thus safe memory.
*
* @see mul::expand()
- * @return pointer to epvector containing expanded representation or zero
- * pointer, if sequence is unchanged. */
-std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
+ * @return epvector containing expanded pairs, empty if no members
+ * had to be changed. */
+epvector mul::expandchildren(unsigned options) const
{
- const epvector::const_iterator last = seq.end();
- epvector::const_iterator cit = seq.begin();
+ auto cit = seq.begin(), last = seq.end();
while (cit!=last) {
const ex & factor = recombine_pair_to_ex(*cit);
const ex & expanded_factor = factor.expand(options);
if (!are_ex_trivially_equal(factor,expanded_factor)) {
// something changed, copy seq, eval and return it
- std::auto_ptr<epvector> s(new epvector);
- s->reserve(seq.size());
+ epvector s;
+ s.reserve(seq.size());
// copy parts of seq which are known not to have changed
- epvector::const_iterator cit2 = seq.begin();
+ auto cit2 = seq.begin();
while (cit2!=cit) {
- s->push_back(*cit2);
+ s.push_back(*cit2);
++cit2;
}
// copy first changed element
- s->push_back(split_ex_to_pair(expanded_factor));
+ s.push_back(split_ex_to_pair(expanded_factor));
++cit2;
// copy rest
while (cit2!=last) {
- s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
+ s.push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
++cit2;
}
return s;
}
++cit;
}
-
- return std::auto_ptr<epvector>(0); // nothing has changed
+
+ return epvector(); // nothing has changed
}
+GINAC_BIND_UNARCHIVER(mul);
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff