X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpower.cpp;h=fb5e351e7390204dbe6b024f5cc2f8a80c67625b;hp=1aade2e6363d74e62b5178fd116d6fce5d1f67c7;hb=09403bb1127f7ef58dce7b669e681c74dd77eaed;hpb=b4be7b0f30fbb6178cf4ee83e1b3952e084bd8ca diff --git a/ginac/power.cpp b/ginac/power.cpp index 1aade2e6..fb5e351e 100644 --- a/ginac/power.cpp +++ b/ginac/power.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */ /* - * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2007 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 @@ -17,7 +17,7 @@ * * 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 @@ -40,6 +40,7 @@ #include "lst.h" #include "archive.h" #include "utils.h" +#include "relational.h" namespace GiNaC { @@ -56,7 +57,7 @@ typedef std::vector intvector; // default constructor ////////// -power::power() : inherited(TINFO_power) { } +power::power() : inherited(&power::tinfo_static) { } ////////// // other constructors @@ -100,7 +101,7 @@ void power::print_power(const print_context & c, const char *powersymbol, const exponent.print(c, precedence()); c.s << closebrace; if (precedence() <= level) - c.s << closebrace << ')'; + c.s << ')' << closebrace; } void power::do_print_dflt(const print_dflt & c, unsigned level) const @@ -222,12 +223,14 @@ bool power::info(unsigned inf) const 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); } return inherited::info(inf); } @@ -246,7 +249,23 @@ ex power::op(size_t i) const ex power::map(map_function & f) const { - return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated); + const ex &mapped_basis = f(basis); + const ex &mapped_exponent = f(exponent); + + if (!are_ex_trivially_equal(basis, mapped_basis) + || !are_ex_trivially_equal(exponent, mapped_exponent)) + return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated); + else + return *this; +} + +bool power::is_polynomial(const ex & var) const +{ + if (exponent.has(var)) + return false; + if (!exponent.info(info_flags::nonnegint)) + return false; + return basis.is_polynomial(var); } int power::degree(const ex & s) const @@ -316,6 +335,7 @@ ex power::coeff(const ex & s, int n) const * - ^(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) if x is positive and c1 is real. * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, 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) @@ -372,6 +392,14 @@ ex power::eval(int level) const if (ebasis.is_equal(_ex1)) return _ex1; + // power of a function calculated by separate rules defined for this function + if (is_exactly_a(ebasis)) + return ex_to(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(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 (exponent_is_numerical) { // ^(c1,c2) -> c1^c2 (c1, c2 numeric(), @@ -439,14 +467,14 @@ ex power::eval(int level) const if (is_exactly_a(sub_exponent)) { const numeric & num_sub_exponent = ex_to(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()) return power(sub_basis,num_sub_exponent.mul(*num_exponent)); } } // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer) if (num_exponent->is_integer() && is_exactly_a(ebasis)) { - return expand_mul(ex_to(ebasis), *num_exponent); + return expand_mul(ex_to(ebasis), *num_exponent, 0); } // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0) @@ -465,8 +493,8 @@ ex power::eval(int level) const 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->clearflag(status_flags::evaluated); @@ -528,6 +556,30 @@ ex power::evalm() const 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(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(exponent).to_int() + > ex_to(other.op(1)).to_int() + && basis.match(other.op(0))) + return true; + if (exponent.info(info_flags::negint) + && other.op(1).info(info_flags::negint) + && ex_to(exponent).to_int() + < ex_to(other.op(1)).to_int() + && basis.match(other.op(0))) + return true; + return basic::has(other, options); +} + // from mul.cpp extern bool tryfactsubs(const ex &, const ex &, int &, lst &); @@ -558,13 +610,76 @@ ex power::eval_ncmul(const exvector & v) const return inherited::eval_ncmul(v); } +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)) { + return *this; + } + return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated); +} + +ex power::real_part() const +{ + if (exponent.info(info_flags::integer)) { + ex basis_real = basis.real_part(); + if (basis_real == basis) + return *this; + realsymbol a("a"),b("b"); + ex result; + if (exponent.info(info_flags::posint)) + result = power(a+I*b,exponent); + else + result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent); + result = result.expand(); + result = result.real_part(); + result = result.subs(lst( a==basis_real, b==basis.imag_part() )); + return result; + } + + ex a = basis.real_part(); + ex b = basis.imag_part(); + ex c = exponent.real_part(); + 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 +{ + if (exponent.info(info_flags::integer)) { + ex basis_real = basis.real_part(); + if (basis_real == basis) + return 0; + realsymbol a("a"),b("b"); + ex result; + if (exponent.info(info_flags::posint)) + result = power(a+I*b,exponent); + else + result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent); + result = result.expand(); + result = result.imag_part(); + result = result.subs(lst( a==basis_real, b==basis.imag_part() )); + return result; + } + + ex a=basis.real_part(); + ex b=basis.imag_part(); + ex c=exponent.real_part(); + 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 + // protected /** Implementation of ex::diff() for a power. * @see ex::diff */ ex power::derivative(const symbol & s) const { - if (exponent.info(info_flags::real)) { + if (is_a(exponent)) { // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below) epvector newseq; newseq.reserve(2); @@ -595,8 +710,8 @@ unsigned power::return_type() const { return basis.return_type(); } - -unsigned power::return_type_tinfo() const + +tinfo_t power::return_type_tinfo() const { return basis.return_type_tinfo(); } @@ -626,7 +741,7 @@ ex power::expand(unsigned options) const const numeric &num_exponent = ex_to(a.overall_coeff); int int_exponent = num_exponent.to_int(); if (int_exponent > 0 && is_exactly_a(expanded_basis)) - distrseq.push_back(expand_add(ex_to(expanded_basis), int_exponent)); + distrseq.push_back(expand_add(ex_to(expanded_basis), int_exponent, options)); else distrseq.push_back(power(expanded_basis, a.overall_coeff)); } else @@ -634,7 +749,7 @@ ex power::expand(unsigned options) const // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a ex r = (new mul(distrseq))->setflag(status_flags::dynallocated); - return r.expand(); + return r.expand(options); } if (!is_exactly_a(expanded_exponent) || @@ -652,11 +767,11 @@ ex power::expand(unsigned options) const // (x+y)^n, n>0 if (int_exponent > 0 && is_exactly_a(expanded_basis)) - return expand_add(ex_to(expanded_basis), int_exponent); + return expand_add(ex_to(expanded_basis), int_exponent, options); // (x*y)^n -> x^n * y^n if (is_exactly_a(expanded_basis)) - return expand_mul(ex_to(expanded_basis), num_exponent); + return expand_mul(ex_to(expanded_basis), num_exponent, options, true); // cannot expand further if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) @@ -677,15 +792,15 @@ ex power::expand(unsigned options) const /** 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) const +ex power::expand_add(const add & a, int n, unsigned options) const { if (n==2) - return expand_add_2(a); + return expand_add_2(a, options); const size_t m = a.nops(); exvector 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()); @@ -713,7 +828,7 @@ ex power::expand_add(const add & a, int n) const !is_exactly_a(ex_to(b).basis) || !is_exactly_a(ex_to(b).basis)); if (is_exactly_a(b)) - term.push_back(expand_mul(ex_to(b),numeric(k[l]))); + term.push_back(expand_mul(ex_to(b), numeric(k[l]), options, true)); else term.push_back(power(b,k[l])); } @@ -727,7 +842,7 @@ ex power::expand_add(const add & a, int n) const !is_exactly_a(ex_to(b).basis) || !is_exactly_a(ex_to(b).basis)); if (is_exactly_a(b)) - term.push_back(expand_mul(ex_to(b),numeric(n-k_cum[m-2]))); + term.push_back(expand_mul(ex_to(b), numeric(n-k_cum[m-2]), options, true)); else term.push_back(power(b,n-k_cum[m-2])); @@ -737,7 +852,7 @@ ex power::expand_add(const add & a, int n) const term.push_back(f); - result.push_back((new mul(term))->setflag(status_flags::dynallocated)); + result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options)); // increment k[] l = m-2; @@ -764,7 +879,7 @@ ex power::expand_add(const add & a, int n) const /** Special case of power::expand_add. Expands a^2 where a is an add. * @see power::expand_add */ -ex power::expand_add_2(const add & a) const +ex power::expand_add_2(const add & a, unsigned options) const { epvector sum; size_t a_nops = a.nops(); @@ -787,7 +902,7 @@ ex power::expand_add_2(const add & a) const if (c.is_equal(_ex1)) { if (is_exactly_a(r)) { - sum.push_back(expair(expand_mul(ex_to(r),_num2), + sum.push_back(expair(expand_mul(ex_to(r), *_num2_p, options, true), _ex1)); } else { sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated), @@ -795,11 +910,11 @@ ex power::expand_add_2(const add & a) const } } else { if (is_exactly_a(r)) { - sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to(r),_num2), - ex_to(c).power_dyn(_num2))); + sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to(r), *_num2_p, options, true), + ex_to(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(c).power_dyn(_num2))); + ex_to(c).power_dyn(*_num2_p))); } } @@ -807,7 +922,7 @@ ex power::expand_add_2(const add & a) const 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(c)).mul_dyn(ex_to(c1)))); + _num2_p->mul(ex_to(c)).mul_dyn(ex_to(c1)))); } } @@ -817,10 +932,10 @@ ex power::expand_add_2(const add & a) const 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(a.overall_coeff).mul_dyn(_num2))); + sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to(a.overall_coeff).mul_dyn(*_num2_p))); ++i; } - sum.push_back(expair(ex_to(a.overall_coeff).power_dyn(_num2),_ex1)); + sum.push_back(expair(ex_to(a.overall_coeff).power_dyn(*_num2_p),_ex1)); } GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2); @@ -828,30 +943,50 @@ ex power::expand_add_2(const add & a) const 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) const +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; + } + + // Leave it to multiplication since dummy indices have to be renamed + if (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(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 - distrseq.push_back(expair(cit->rest, ex_to(cit->coeff).mul(n))); + expair p = m.combine_pair_with_coeff_to_pair(*cit, n); + if (from_expand && is_exactly_a(cit->rest) && ex_to(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; } + distrseq.push_back(p); ++cit; } - return (new mul(distrseq, ex_to(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated); + + const mul & result = static_cast((new mul(distrseq, ex_to(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated)); + if (need_reexpand) + return ex(result).expand(options); + if (from_expand) + return result.setflag(status_flags::expanded); + return result; } } // namespace GiNaC