* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2015 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 <vector>
-#include <iostream>
-#include <stdexcept>
-#include <limits>
-
#include "power.h"
#include "expairseq.h"
#include "add.h"
#include "lst.h"
#include "archive.h"
#include "utils.h"
+#include "relational.h"
+#include "compiler.h"
+
+#include <iostream>
+#include <limits>
+#include <stdexcept>
+#include <vector>
namespace GiNaC {
print_func<print_latex>(&power::do_print_latex).
print_func<print_csrc>(&power::do_print_csrc).
print_func<print_python>(&power::do_print_python).
- print_func<print_python_repr>(&power::do_print_python_repr))
-
-typedef std::vector<int> intvector;
+ print_func<print_python_repr>(&power::do_print_python_repr).
+ print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
//////////
// default constructor
//////////
-power::power() : inherited(TINFO_power) { }
+power::power() { }
//////////
// other constructors
// archiving
//////////
-power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
+void power::read_archive(const archive_node &n, lst &sym_lst)
{
+ inherited::read_archive(n, sym_lst);
n.find_ex("basis", basis, sym_lst);
n.find_ex("exponent", exponent, sym_lst);
}
n.add_ex("exponent", exponent);
}
-DEFAULT_UNARCHIVE(power)
-
//////////
// functions overriding virtual functions from base classes
//////////
static void print_sym_pow(const print_context & c, const symbol &x, int exp)
{
// Optimal output of integer powers of symbols to aid compiler CSE.
- // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
+ // C.f. ISO/IEC 14882:2011, section 1.9 [intro execution], paragraph 15
// to learn why such a parenthesation is really necessary.
if (exp == 1) {
x.print(c);
}
}
+void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
+{
+ if (exponent.is_equal(_ex_1)) {
+ c.s << "recip(";
+ basis.print(c);
+ c.s << ')';
+ return;
+ }
+ c.s << "expt(";
+ basis.print(c);
+ c.s << ", ";
+ exponent.print(c);
+ c.s << ')';
+}
+
void power::do_print_csrc(const print_csrc & c, unsigned level) const
{
// Integer powers of symbols are printed in a special, optimized way
- if (exponent.info(info_flags::integer)
- && (is_a<symbol>(basis) || is_a<constant>(basis))) {
+ if (exponent.info(info_flags::integer) &&
+ (is_a<symbol>(basis) || is_a<constant>(basis))) {
int exp = ex_to<numeric>(exponent).to_int();
if (exp > 0)
c.s << '(';
else {
exp = -exp;
- if (is_a<print_csrc_cl_N>(c))
- c.s << "recip(";
- else
- c.s << "1.0/(";
+ c.s << "1.0/(";
}
print_sym_pow(c, ex_to<symbol>(basis), exp);
c.s << ')';
// <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
} else if (exponent.is_equal(_ex_1)) {
- if (is_a<print_csrc_cl_N>(c))
- c.s << "recip(";
- else
- c.s << "1.0/(";
+ c.s << "1.0/(";
basis.print(c);
c.s << ')';
- // Otherwise, use the pow() or expt() (CLN) functions
+ // Otherwise, use the pow() function
} else {
- if (is_a<print_csrc_cl_N>(c))
- c.s << "expt(";
- else
- c.s << "pow(";
+ c.s << "pow(";
basis.print(c);
c.s << ',';
exponent.print(c);
case info_flags::cinteger_polynomial:
case info_flags::rational_polynomial:
case info_flags::crational_polynomial:
- return exponent.info(info_flags::nonnegint);
+ return exponent.info(info_flags::nonnegint) &&
+ basis.info(inf);
case info_flags::rational_function:
- return exponent.info(info_flags::integer);
+ return exponent.info(info_flags::integer) &&
+ basis.info(inf);
case info_flags::algebraic:
- return (!exponent.info(info_flags::integer) ||
- basis.info(inf));
+ return !exponent.info(info_flags::integer) ||
+ basis.info(inf);
+ case info_flags::expanded:
+ return (flags & status_flags::expanded);
+ case info_flags::positive:
+ return basis.info(info_flags::positive) && exponent.info(info_flags::real);
+ case info_flags::nonnegative:
+ return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
+ (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
+ case info_flags::has_indices: {
+ if (flags & status_flags::has_indices)
+ return true;
+ else if (flags & status_flags::has_no_indices)
+ return false;
+ else if (basis.info(info_flags::has_indices)) {
+ setflag(status_flags::has_indices);
+ clearflag(status_flags::has_no_indices);
+ return true;
+ } else {
+ clearflag(status_flags::has_indices);
+ setflag(status_flags::has_no_indices);
+ return false;
+ }
+ }
}
return inherited::info(inf);
}
return *this;
}
+bool power::is_polynomial(const ex & var) const
+{
+ if (basis.is_polynomial(var)) {
+ if (basis.has(var))
+ // basis is non-constant polynomial in var
+ return exponent.info(info_flags::nonnegint);
+ else
+ // basis is constant in var
+ return !exponent.has(var);
+ }
+ // basis is a non-polynomial function of var
+ return false;
+}
+
int power::degree(const ex & s) const
{
if (is_equal(ex_to<basic>(s)))
* - ^(0,c) -> 0 or exception (depending on the real part of c)
* - ^(1,x) -> 1
* - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
- * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
+ * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
+ * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0), case c1=1 should not happen, see below!)
* - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
* - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
* - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
const ex & ebasis = level==1 ? basis : basis.eval(level-1);
const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
- bool basis_is_numerical = false;
- bool exponent_is_numerical = false;
- const numeric *num_basis;
- const numeric *num_exponent;
+ const numeric *num_basis = nullptr;
+ const numeric *num_exponent = nullptr;
if (is_exactly_a<numeric>(ebasis)) {
- basis_is_numerical = true;
num_basis = &ex_to<numeric>(ebasis);
}
if (is_exactly_a<numeric>(eexponent)) {
- exponent_is_numerical = true;
num_exponent = &ex_to<numeric>(eexponent);
}
return ebasis;
// ^(0,c1) -> 0 or exception (depending on real value of c1)
- if (ebasis.is_zero() && exponent_is_numerical) {
+ if ( ebasis.is_zero() && num_exponent ) {
if ((num_exponent->real()).is_zero())
throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
else if ((num_exponent->real()).is_negative())
if (ebasis.is_equal(_ex1))
return _ex1;
- if (exponent_is_numerical) {
+ // power of a function calculated by separate rules defined for this function
+ if (is_exactly_a<function>(ebasis))
+ return ex_to<function>(ebasis).power(eexponent);
+
+ // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
+ if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
+ return power(ebasis.op(0), ebasis.op(1) * eexponent);
+
+ if ( num_exponent ) {
// ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
// except if c1,c2 are rational, but c1^c2 is not)
- if (basis_is_numerical) {
+ if ( num_basis ) {
const bool basis_is_crational = num_basis->is_crational();
const bool exponent_is_crational = num_exponent->is_crational();
if (!basis_is_crational || !exponent_is_crational) {
}
// ^(^(x,c1),c2) -> ^(x,c1*c2)
- // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
+ // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
// case c1==1 should not happen, see below!)
if (is_exactly_a<power>(ebasis)) {
const power & sub_power = ex_to<power>(ebasis);
if (is_exactly_a<numeric>(sub_exponent)) {
const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
GINAC_ASSERT(num_sub_exponent!=numeric(1));
- if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
+ if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
+ (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
return power(sub_basis,num_sub_exponent.mul(*num_exponent));
+ }
}
}
if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
}
-
+
+ // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
+ if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
+ numeric icont = ebasis.integer_content();
+ const numeric lead_coeff =
+ ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
+
+ const bool canonicalizable = lead_coeff.is_integer();
+ const bool unit_normal = lead_coeff.is_pos_integer();
+ if (canonicalizable && (! unit_normal))
+ icont = icont.mul(*_num_1_p);
+
+ if (canonicalizable && (icont != *_num1_p)) {
+ const add& addref = ex_to<add>(ebasis);
+ add* addp = new add(addref);
+ addp->setflag(status_flags::dynallocated);
+ addp->clearflag(status_flags::hash_calculated);
+ addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
+ for (auto & i : addp->seq)
+ i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
+
+ const numeric c = icont.power(*num_exponent);
+ if (likely(c != *_num1_p))
+ return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
+ else
+ return power(*addp, *num_exponent);
+ }
+ }
+
// ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
// ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
if (is_exactly_a<mul>(ebasis)) {
if (num_coeff.is_positive()) {
mul *mulp = new mul(mulref);
mulp->overall_coeff = _ex1;
+ mulp->setflag(status_flags::dynallocated);
mulp->clearflag(status_flags::evaluated);
mulp->clearflag(status_flags::hash_calculated);
return (new mul(power(*mulp,exponent),
power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
} else {
- GINAC_ASSERT(num_coeff.compare(_num0)<0);
- if (!num_coeff.is_equal(_num_1)) {
+ GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
+ if (!num_coeff.is_equal(*_num_1_p)) {
mul *mulp = new mul(mulref);
mulp->overall_coeff = _ex_1;
+ mulp->setflag(status_flags::dynallocated);
mulp->clearflag(status_flags::evaluated);
mulp->clearflag(status_flags::hash_calculated);
return (new mul(power(*mulp,exponent),
if (num_exponent->is_pos_integer() &&
ebasis.return_type() != return_types::commutative &&
!is_a<matrix>(ebasis)) {
- return ncmul(exvector(num_exponent->to_int(), ebasis), true);
+ return ncmul(exvector(num_exponent->to_int(), ebasis));
}
}
return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
}
+bool power::has(const ex & other, unsigned options) const
+{
+ if (!(options & has_options::algebraic))
+ return basic::has(other, options);
+ if (!is_a<power>(other))
+ return basic::has(other, options);
+ if (!exponent.info(info_flags::integer) ||
+ !other.op(1).info(info_flags::integer))
+ return basic::has(other, options);
+ if (exponent.info(info_flags::posint) &&
+ other.op(1).info(info_flags::posint) &&
+ ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
+ basis.match(other.op(0)))
+ return true;
+ if (exponent.info(info_flags::negint) &&
+ other.op(1).info(info_flags::negint) &&
+ ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
+ basis.match(other.op(0)))
+ return true;
+ return basic::has(other, options);
+}
+
// from mul.cpp
-extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
+extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
ex power::subs(const exmap & m, unsigned options) const
{
if (!(options & subs_options::algebraic))
return subs_one_level(m, options);
- for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
+ for (auto & it : m) {
int nummatches = std::numeric_limits<int>::max();
- lst repls;
- if (tryfactsubs(*this, it->first, nummatches, repls))
- return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
+ exmap repls;
+ if (tryfactsubs(*this, it.first, nummatches, repls)) {
+ ex anum = it.second.subs(repls, subs_options::no_pattern);
+ ex aden = it.first.subs(repls, subs_options::no_pattern);
+ ex result = (*this)*power(anum/aden, nummatches);
+ return (ex_to<basic>(result)).subs_one_level(m, options);
+ }
}
return subs_one_level(m, options);
ex power::conjugate() const
{
- ex newbasis = basis.conjugate();
- ex newexponent = exponent.conjugate();
- if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
+ // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
+ // branch cut which runs along the negative real axis.
+ if (basis.info(info_flags::positive)) {
+ ex newexponent = exponent.conjugate();
+ if (are_ex_trivially_equal(exponent, newexponent)) {
+ return *this;
+ }
+ return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
+ }
+ if (exponent.info(info_flags::integer)) {
+ ex newbasis = basis.conjugate();
+ if (are_ex_trivially_equal(basis, newbasis)) {
+ return *this;
+ }
+ return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
+ }
+ return conjugate_function(*this).hold();
+}
+
+ex power::real_part() const
+{
+ // basis == a+I*b, exponent == c+I*d
+ const ex a = basis.real_part();
+ const ex c = exponent.real_part();
+ if (basis.is_equal(a) && exponent.is_equal(c)) {
+ // Re(a^c)
return *this;
}
- return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
+
+ const ex b = basis.imag_part();
+ if (exponent.info(info_flags::integer)) {
+ // Re((a+I*b)^c) w/ c ∈ ℤ
+ long N = ex_to<numeric>(c).to_long();
+ // Use real terms in Binomial expansion to construct
+ // Re(expand(power(a+I*b, N))).
+ long NN = N > 0 ? N : -N;
+ ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
+ ex result = 0;
+ for (long n = 0; n <= NN; n += 2) {
+ ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
+ if (n % 4 == 0) {
+ result += term; // sign: I^n w/ n == 4*m
+ } else {
+ result -= term; // sign: I^n w/ n == 4*m+2
+ }
+ }
+ return result;
+ }
+
+ // Re((a+I*b)^(c+I*d))
+ const ex d = exponent.imag_part();
+ return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
+}
+
+ex power::imag_part() const
+{
+ const ex a = basis.real_part();
+ const ex c = exponent.real_part();
+ if (basis.is_equal(a) && exponent.is_equal(c)) {
+ // Im(a^c)
+ return 0;
+ }
+
+ const ex b = basis.imag_part();
+ if (exponent.info(info_flags::integer)) {
+ // Im((a+I*b)^c) w/ c ∈ ℤ
+ long N = ex_to<numeric>(c).to_long();
+ // Use imaginary terms in Binomial expansion to construct
+ // Im(expand(power(a+I*b, N))).
+ long p = N > 0 ? 1 : 3; // modulus for positive sign
+ long NN = N > 0 ? N : -N;
+ ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
+ ex result = 0;
+ for (long n = 1; n <= NN; n += 2) {
+ ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
+ if (n % 4 == p) {
+ result += term; // sign: I^n w/ n == 4*m+p
+ } else {
+ result -= term; // sign: I^n w/ n == 4*m+2+p
+ }
+ }
+ return result;
+ }
+
+ // Im((a+I*b)^(c+I*d))
+ const ex d = exponent.imag_part();
+ return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
}
// protected
* @see ex::diff */
ex power::derivative(const symbol & s) const
{
- if (exponent.info(info_flags::real)) {
+ if (is_a<numeric>(exponent)) {
// D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
epvector newseq;
newseq.reserve(2);
newseq.push_back(expair(basis, exponent - _ex1));
newseq.push_back(expair(basis.diff(s), _ex1));
- return mul(newseq, exponent);
+ return mul(std::move(newseq), exponent);
} else {
// D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
return mul(*this,
{
return basis.return_type();
}
-
-unsigned power::return_type_tinfo() const
+
+return_type_t power::return_type_tinfo() const
{
return basis.return_type_tinfo();
}
ex power::expand(unsigned options) const
{
- if (options == 0 && (flags & status_flags::expanded))
+ if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
+ // A special case worth optimizing.
+ setflag(status_flags::expanded);
return *this;
-
+ }
+
+ // (x*p)^c -> x^c * p^c, if p>0
+ // makes sense before expanding the basis
+ if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
+ const mul &m = ex_to<mul>(basis);
+ exvector prodseq;
+ epvector powseq;
+ prodseq.reserve(m.seq.size() + 1);
+ powseq.reserve(m.seq.size() + 1);
+ bool possign = true;
+
+ // search for positive/negative factors
+ for (auto & cit : m.seq) {
+ ex e=m.recombine_pair_to_ex(cit);
+ if (e.info(info_flags::positive))
+ prodseq.push_back(pow(e, exponent).expand(options));
+ else if (e.info(info_flags::negative)) {
+ prodseq.push_back(pow(-e, exponent).expand(options));
+ possign = !possign;
+ } else
+ powseq.push_back(cit);
+ }
+
+ // take care on the numeric coefficient
+ ex coeff=(possign? _ex1 : _ex_1);
+ if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
+ prodseq.push_back(power(m.overall_coeff, exponent));
+ else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
+ prodseq.push_back(power(-m.overall_coeff, exponent));
+ else
+ coeff *= m.overall_coeff;
+
+ // If positive/negative factors are found, then extract them.
+ // In either case we set a flag to avoid the second run on a part
+ // which does not have positive/negative terms.
+ if (prodseq.size() > 0) {
+ ex newbasis = coeff*mul(std::move(powseq));
+ ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
+ return ((new mul(std::move(prodseq)))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
+ } else
+ ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
+ }
+
const ex expanded_basis = basis.expand(options);
const ex expanded_exponent = exponent.expand(options);
const add &a = ex_to<add>(expanded_exponent);
exvector distrseq;
distrseq.reserve(a.seq.size() + 1);
- epvector::const_iterator last = a.seq.end();
- epvector::const_iterator cit = a.seq.begin();
- while (cit!=last) {
- distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
- ++cit;
+ for (auto & cit : a.seq) {
+ distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(cit)));
}
// Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
if (ex_to<numeric>(a.overall_coeff).is_integer()) {
const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
- int int_exponent = num_exponent.to_int();
+ long int_exponent = num_exponent.to_int();
if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
else
// integer numeric exponent
const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
- int int_exponent = num_exponent.to_int();
+ long int_exponent = num_exponent.to_long();
// (x+y)^n, n>0
if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
// non-virtual functions in this class
//////////
+namespace { // anonymous namespace for power::expand_add() helpers
+
+/** Helper class to generate all bounded combinatorial partitions of an integer
+ * n with exactly m parts (including zero parts) in non-decreasing order.
+ */
+class partition_generator {
+private:
+ // Partitions n into m parts, not including zero parts.
+ // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
+ // FXT library)
+ struct mpartition2
+ {
+ // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
+ std::vector<int> x;
+ int n; // n>0
+ int m; // 0<m<=n
+ mpartition2(unsigned n_, unsigned m_)
+ : x(m_+1), n(n_), m(m_)
+ {
+ for (int k=1; k<m; ++k)
+ x[k] = 1;
+ x[m] = n - m + 1;
+ }
+ bool next_partition()
+ {
+ int u = x[m]; // last element
+ int k = m;
+ int s = u;
+ while (--k) {
+ s += x[k];
+ if (x[k] + 2 <= u)
+ break;
+ }
+ if (k==0)
+ return false; // current is last
+ int f = x[k] + 1;
+ while (k < m) {
+ x[k] = f;
+ s -= f;
+ ++k;
+ }
+ x[m] = s;
+ return true;
+ }
+ } mpgen;
+ int m; // number of parts 0<m<=n
+ mutable std::vector<int> partition; // current partition
+public:
+ partition_generator(unsigned n_, unsigned m_)
+ : mpgen(n_, 1), m(m_), partition(m_)
+ { }
+ // returns current partition in non-decreasing order, padded with zeros
+ const std::vector<int>& current() const
+ {
+ for (int i = 0; i < m - mpgen.m; ++i)
+ partition[i] = 0; // pad with zeros
+
+ for (int i = m - mpgen.m; i < m; ++i)
+ partition[i] = mpgen.x[i - m + mpgen.m + 1];
+
+ return partition;
+ }
+ bool next()
+ {
+ if (!mpgen.next_partition()) {
+ if (mpgen.m == m || mpgen.m == mpgen.n)
+ return false; // current is last
+ // increment number of parts
+ mpgen = mpartition2(mpgen.n, mpgen.m + 1);
+ }
+ return true;
+ }
+};
+
+/** Helper class to generate all compositions of a partition of an integer n,
+ * starting with the compositions which has non-decreasing order.
+ */
+class composition_generator {
+private:
+ // Generates all distinct permutations of a multiset.
+ // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
+ // Multiset Permutations using a Constant Number of Variables by Prefix
+ // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
+ struct coolmulti {
+ // element of singly linked list
+ struct element {
+ int value;
+ element* next;
+ element(int val, element* n)
+ : value(val), next(n) {}
+ ~element()
+ { // recurses down to the end of the singly linked list
+ delete next;
+ }
+ };
+ element *head, *i, *after_i;
+ // NB: Partition must be sorted in non-decreasing order.
+ explicit coolmulti(const std::vector<int>& partition)
+ : head(nullptr), i(nullptr), after_i(nullptr)
+ {
+ for (unsigned n = 0; n < partition.size(); ++n) {
+ head = new element(partition[n], head);
+ if (n <= 1)
+ i = head;
+ }
+ after_i = i->next;
+ }
+ ~coolmulti()
+ { // deletes singly linked list
+ delete head;
+ }
+ void next_permutation()
+ {
+ element *before_k;
+ if (after_i->next != nullptr && i->value >= after_i->next->value)
+ before_k = after_i;
+ else
+ before_k = i;
+ element *k = before_k->next;
+ before_k->next = k->next;
+ k->next = head;
+ if (k->value < head->value)
+ i = k;
+ after_i = i->next;
+ head = k;
+ }
+ bool finished() const
+ {
+ return after_i->next == nullptr && after_i->value >= head->value;
+ }
+ } cmgen;
+ bool atend; // needed for simplifying iteration over permutations
+ bool trivial; // likewise, true if all elements are equal
+ mutable std::vector<int> composition; // current compositions
+public:
+ explicit composition_generator(const std::vector<int>& partition)
+ : cmgen(partition), atend(false), trivial(true), composition(partition.size())
+ {
+ for (unsigned i=1; i<partition.size(); ++i)
+ trivial = trivial && (partition[0] == partition[i]);
+ }
+ const std::vector<int>& current() const
+ {
+ coolmulti::element* it = cmgen.head;
+ size_t i = 0;
+ while (it != nullptr) {
+ composition[i] = it->value;
+ it = it->next;
+ ++i;
+ }
+ return composition;
+ }
+ bool next()
+ {
+ // This ugly contortion is needed because the original coolmulti
+ // algorithm requires code duplication of the payload procedure,
+ // one before the loop and one inside it.
+ if (trivial || atend)
+ return false;
+ cmgen.next_permutation();
+ atend = cmgen.finished();
+ return true;
+ }
+};
+
+/** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
+ * where n = p1+p2+...+pk, i.e. p is a partition of n.
+ */
+const numeric
+multinomial_coefficient(const std::vector<int> & p)
+{
+ numeric n = 0, d = 1;
+ for (auto & it : p) {
+ n += numeric(it);
+ d *= factorial(numeric(it));
+ }
+ return factorial(numeric(n)) / d;
+}
+
+} // anonymous namespace
+
/** expand a^n where a is an add and n is a positive integer.
* @see power::expand */
-ex power::expand_add(const add & a, int n, unsigned options) const
+ex power::expand_add(const add & a, long n, unsigned options) const
{
+ // The special case power(+(x,...y;x),2) can be optimized better.
if (n==2)
return expand_add_2(a, options);
- const size_t m = a.nops();
- exvector result;
+ // method:
+ //
+ // Consider base as the sum of all symbolic terms and the overall numeric
+ // coefficient and apply the binomial theorem:
+ // S = power(+(x,...,z;c),n)
+ // = power(+(+(x,...,z;0);c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
+ // The multinomial theorem is computed by an outer loop over all
+ // partitions of the exponent and an inner loop over all compositions of
+ // that partition. This method makes the expansion a combinatorial
+ // problem and allows us to directly construct the expanded sum and also
+ // to re-use the multinomial coefficients (since they depend only on the
+ // partition, not on the composition).
+ //
+ // multinomial power(+(x,y,z;0),3) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
+ // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
+ // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
+ // => (x + y + z)^3 =
+ // x^3 + y^3 + z^3
+ // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
+ // + 6*x*y*z
+ //
+ // multinomial power(+(x,y,z;0),4) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
+ // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
+ // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
+ // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
+ // (no [1,1,1,1] partition since it has too many parts)
+ // => (x + y + z)^4 =
+ // x^4 + y^4 + z^4
+ // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
+ // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
+ // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
+ //
+ // Summary:
+ // r = 0
+ // for k from 0 to n:
+ // f = c^(n-k)*binomial(n,k)
+ // for p in all partitions of n with m parts (including zero parts):
+ // h = f * multinomial coefficient of p
+ // for c in all compositions of p:
+ // t = 1
+ // for e in all elements of c:
+ // t = t * a[e]^e
+ // r = r + h*t
+ // return r
+
+ epvector result;
// The number of terms will be the number of combinatorial compositions,
- // i.e. the number of unordered arrangement of m nonnegative integers
+ // i.e. the number of unordered arrangements of m nonnegative integers
// which sum up to n. It is frequently written as C_n(m) and directly
- // related with binomial coefficients:
- result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
- intvector k(m-1);
- intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
- intvector upper_limit(m-1);
- int l;
-
- for (size_t l=0; l<m-1; ++l) {
- k[l] = 0;
- k_cum[l] = 0;
- upper_limit[l] = n;
- }
-
- while (true) {
- exvector term;
- term.reserve(m+1);
- for (l=0; l<m-1; ++l) {
- const ex & b = a.op(l);
- GINAC_ASSERT(!is_exactly_a<add>(b));
- GINAC_ASSERT(!is_exactly_a<power>(b) ||
- !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
- !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_exactly_a<add>(ex_to<power>(b).basis) ||
- !is_exactly_a<mul>(ex_to<power>(b).basis) ||
- !is_exactly_a<power>(ex_to<power>(b).basis));
- if (is_exactly_a<mul>(b))
- term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
- else
- term.push_back(power(b,k[l]));
- }
-
- const ex & b = a.op(l);
- GINAC_ASSERT(!is_exactly_a<add>(b));
- GINAC_ASSERT(!is_exactly_a<power>(b) ||
- !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
- !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_exactly_a<add>(ex_to<power>(b).basis) ||
- !is_exactly_a<mul>(ex_to<power>(b).basis) ||
- !is_exactly_a<power>(ex_to<power>(b).basis));
- if (is_exactly_a<mul>(b))
- term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
- else
- term.push_back(power(b,n-k_cum[m-2]));
-
- numeric f = binomial(numeric(n),numeric(k[0]));
- for (l=1; l<m-1; ++l)
- f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
-
- term.push_back(f);
-
- result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
-
- // increment k[]
- l = m-2;
- while ((l>=0) && ((++k[l])>upper_limit[l])) {
- k[l] = 0;
- --l;
+ // related with binomial coefficients: binomial(n+m-1,m-1).
+ size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
+ }
+ result.reserve(result_size);
+
+ // Iterate over all terms in binomial expansion of
+ // S = power(+(x,...,z;c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ for (int k = 1; k <= n; ++k) {
+ numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
+ if (a.overall_coeff.is_zero()) {
+ // degenerate case with zero overall_coeff:
+ // apply multinomial theorem directly to power(+(x,...z;0),n)
+ binomial_coefficient = 1;
+ if (k < n) {
+ continue;
+ }
+ } else {
+ binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
}
- if (l<0) break;
- // recalc k_cum[] and upper_limit[]
- k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
+ // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
+ // Iterate over all partitions of k with exactly as many parts as
+ // there are symbolic terms in the basis (including zero parts).
+ partition_generator partitions(k, a.seq.size());
+ do {
+ const std::vector<int>& partition = partitions.current();
+ const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
+
+ // Iterate over all compositions of the current partition.
+ composition_generator compositions(partition);
+ do {
+ const std::vector<int>& exponent = compositions.current();
+ exvector term;
+ term.reserve(n);
+ numeric factor = coeff;
+ for (unsigned i = 0; i < exponent.size(); ++i) {
+ const ex & r = a.seq[i].rest;
+ GINAC_ASSERT(!is_exactly_a<add>(r));
+ GINAC_ASSERT(!is_exactly_a<power>(r) ||
+ !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
+ !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
+ !is_exactly_a<add>(ex_to<power>(r).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(r).basis) ||
+ !is_exactly_a<power>(ex_to<power>(r).basis));
+ GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
+ const numeric & c = ex_to<numeric>(a.seq[i].coeff);
+ if (exponent[i] == 0) {
+ // optimize away
+ } else if (exponent[i] == 1) {
+ // optimized
+ term.push_back(r);
+ if (c != *_num1_p)
+ factor = factor.mul(c);
+ } else { // general case exponent[i] > 1
+ term.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
+ if (c != *_num1_p)
+ factor = factor.mul(c.power(exponent[i]));
+ }
+ }
+ result.push_back(a.combine_ex_with_coeff_to_pair(mul(term).expand(options), factor));
+ } while (compositions.next());
+ } while (partitions.next());
+ }
- for (size_t i=l+1; i<m-1; ++i)
- k_cum[i] = k_cum[i-1]+k[i];
+ GINAC_ASSERT(result.size() == result_size);
- for (size_t i=l+1; i<m-1; ++i)
- upper_limit[i] = n-k_cum[i-1];
+ if (a.overall_coeff.is_zero()) {
+ return (new add(std::move(result)))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
+ } else {
+ return (new add(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
}
-
- return (new add(result))->setflag(status_flags::dynallocated |
- status_flags::expanded);
}
* @see power::expand_add */
ex power::expand_add_2(const add & a, unsigned options) const
{
- epvector sum;
- size_t a_nops = a.nops();
- sum.reserve((a_nops*(a_nops+1))/2);
+ epvector result;
+ size_t result_size = (a.nops() * (a.nops()+1)) / 2;
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
+ }
+ result.reserve(result_size);
+
epvector::const_iterator last = a.seq.end();
// power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
if (c.is_equal(_ex1)) {
if (is_exactly_a<mul>(r)) {
- sum.push_back(expair(expand_mul(ex_to<mul>(r), _num2, options, true),
- _ex1));
+ result.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ _ex1));
} else {
- sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
- _ex1));
+ result.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
+ _ex1));
}
} else {
if (is_exactly_a<mul>(r)) {
- sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), _num2, options, true),
- ex_to<numeric>(c).power_dyn(_num2)));
+ result.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ ex_to<numeric>(c).power_dyn(*_num2_p)));
} else {
- sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
- ex_to<numeric>(c).power_dyn(_num2)));
+ result.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
+ ex_to<numeric>(c).power_dyn(*_num2_p)));
}
}
for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
const ex & r1 = cit1->rest;
const ex & c1 = cit1->coeff;
- sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
- _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
+ result.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
+ _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
}
}
- GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
-
- // second part: add terms coming from overall_factor (if != 0)
+ // second part: add terms coming from overall_coeff (if != 0)
if (!a.overall_coeff.is_zero()) {
- epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
- while (i != end) {
- sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
- ++i;
- }
- sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
+ for (auto & i : a.seq)
+ result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
+ }
+
+ GINAC_ASSERT(result.size() == result_size);
+
+ if (a.overall_coeff.is_zero()) {
+ return (new add(std::move(result)))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
+ } else {
+ return (new add(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
}
-
- GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
-
- return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
}
-/** Expand factors of m in m^n where m is a mul and n is and integer.
+/** Expand factors of m in m^n where m is a mul and n is an integer.
* @see power::expand */
ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
{
GINAC_ASSERT(n.is_integer());
- if (n.is_zero())
+ if (n.is_zero()) {
return _ex1;
+ }
+
+ // do not bother to rename indices if there are no any.
+ if (!(options & expand_options::expand_rename_idx) &&
+ m.info(info_flags::has_indices))
+ options |= expand_options::expand_rename_idx;
+ // Leave it to multiplication since dummy indices have to be renamed
+ if ((options & expand_options::expand_rename_idx) &&
+ (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
+ ex result = m;
+ exvector va = get_all_dummy_indices(m);
+ sort(va.begin(), va.end(), ex_is_less());
+
+ for (int i=1; i < n.to_int(); i++)
+ result *= rename_dummy_indices_uniquely(va, m);
+ return result;
+ }
epvector distrseq;
distrseq.reserve(m.seq.size());
bool need_reexpand = false;
- epvector::const_iterator last = m.seq.end();
- epvector::const_iterator cit = m.seq.begin();
- while (cit!=last) {
- if (is_exactly_a<numeric>(cit->rest)) {
- distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
- } else {
- // it is safe not to call mul::combine_pair_with_coeff_to_pair()
- // since n is an integer
- numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
- if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
- // this happens when e.g. (a+b)^(1/2) gets squared and
- // the resulting product needs to be reexpanded
- need_reexpand = true;
- }
- distrseq.push_back(expair(cit->rest, new_coeff));
+ for (auto & cit : m.seq) {
+ expair p = m.combine_pair_with_coeff_to_pair(cit, n);
+ if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
+ // this happens when e.g. (a+b)^(1/2) gets squared and
+ // the resulting product needs to be reexpanded
+ need_reexpand = true;
}
- ++cit;
+ distrseq.push_back(p);
}
const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
return result;
}
+GINAC_BIND_UNARCHIVER(power);
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff