X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?a=blobdiff_plain;f=ginac%2Fpower.cpp;h=62fc3a8ddf070bd5c02a3fe0e2b75ad0006330e6;hb=e432b3743d3d32f6060600af580e49d7033dcae9;hp=72806768dea4564b754b912f3734651331559e22;hpb=acae7ab5a4dc94d1f54ba794f32f5764cdb4d704;p=ginac.git diff --git a/ginac/power.cpp b/ginac/power.cpp index 72806768..62fc3a8d 100644 --- a/ginac/power.cpp +++ b/ginac/power.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */ /* - * GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2018 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 @@ -42,6 +42,7 @@ #include #include #include +#include namespace GiNaC { @@ -139,7 +140,7 @@ void power::do_print_latex(const print_latex & c, unsigned level) const 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); @@ -228,21 +229,18 @@ 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) && - basis.info(inf); + return basis.info(inf) && exponent.info(info_flags::nonnegint); case info_flags::rational_function: - return exponent.info(info_flags::integer) && - basis.info(inf); - case info_flags::algebraic: - return !exponent.info(info_flags::integer) || - basis.info(inf); + return basis.info(inf) && exponent.info(info_flags::integer); + case info_flags::real: + return basis.info(inf) && exponent.info(info_flags::integer); 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)); + (basis.info(info_flags::real) && exponent.info(info_flags::even)); case info_flags::has_indices: { if (flags & status_flags::has_indices) return true; @@ -281,7 +279,7 @@ ex power::map(map_function & f) const 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); + return dynallocate(mapped_basis, mapped_exponent); else return *this; } @@ -372,42 +370,36 @@ ex power::coeff(const ex & s, int n) const * - ^(*(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) - * - * @param level cut-off in recursive evaluation */ -ex power::eval(int level) const + */ +ex power::eval() const { - if ((level==1) && (flags & status_flags::evaluated)) + if (flags & status_flags::evaluated) return *this; - else if (level == -max_recursion_level) - throw(std::runtime_error("max recursion level reached")); - - const ex & ebasis = level==1 ? basis : basis.eval(level-1); - const ex & eexponent = level==1 ? exponent : exponent.eval(level-1); - + const numeric *num_basis = nullptr; const numeric *num_exponent = nullptr; - - if (is_exactly_a(ebasis)) { - num_basis = &ex_to(ebasis); + + if (is_exactly_a(basis)) { + num_basis = &ex_to(basis); } - if (is_exactly_a(eexponent)) { - num_exponent = &ex_to(eexponent); + if (is_exactly_a(exponent)) { + num_exponent = &ex_to(exponent); } // ^(x,0) -> 1 (0^0 also handled here) - if (eexponent.is_zero()) { - if (ebasis.is_zero()) + if (exponent.is_zero()) { + if (basis.is_zero()) throw (std::domain_error("power::eval(): pow(0,0) is undefined")); else return _ex1; } // ^(x,1) -> x - if (eexponent.is_equal(_ex1)) - return ebasis; + if (exponent.is_equal(_ex1)) + return basis; // ^(0,c1) -> 0 or exception (depending on real value of c1) - if ( ebasis.is_zero() && num_exponent ) { + if (basis.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()) @@ -417,16 +409,16 @@ ex power::eval(int level) const } // ^(1,x) -> 1 - if (ebasis.is_equal(_ex1)) + if (basis.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); + if (is_exactly_a(basis)) + return ex_to(basis).power(exponent); // 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 (is_exactly_a(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real)) + return dynallocate(basis.op(0), basis.op(1) * exponent); if ( num_exponent ) { @@ -437,9 +429,7 @@ ex power::eval(int level) const const bool exponent_is_crational = num_exponent->is_crational(); if (!basis_is_crational || !exponent_is_crational) { // return a plain float - return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated | - status_flags::evaluated | - status_flags::expanded); + return dynallocate(num_basis->power(*num_exponent)); } const numeric res = num_basis->power(*num_exponent); @@ -469,9 +459,9 @@ ex power::eval(int level) const const numeric res_bnum = bnum.power(*num_exponent); const numeric res_bden = bden.power(*num_exponent); if (res_bnum.is_integer()) - return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated); + return dynallocate(dynallocate(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated); if (res_bden.is_integer()) - return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated); + return dynallocate(dynallocate(bnum,*num_exponent),res_bden.inverse()).setflag(status_flags::evaluated); } return this->hold(); } else { @@ -479,8 +469,7 @@ ex power::eval(int level) const // because otherwise we'll end up with something like // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3)) // instead of 7/16*7^(1/3). - ex prod = power(*num_basis,r.div(m)); - return prod*power(*num_basis,q); + return pow(basis, r.div(m)) * pow(basis, q); } } } @@ -488,8 +477,8 @@ ex power::eval(int level) const // ^(^(x,c1),c2) -> ^(x,c1*c2) // (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(ebasis)) { - const power & sub_power = ex_to(ebasis); + if (is_exactly_a(basis)) { + const power & sub_power = ex_to(basis); const ex & sub_basis = sub_power.basis; const ex & sub_exponent = sub_power.exponent; if (is_exactly_a(sub_exponent)) { @@ -497,21 +486,21 @@ ex power::eval(int level) const GINAC_ASSERT(num_sub_exponent!=numeric(1)); 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)); + return dynallocate(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, 0); + if (num_exponent->is_integer() && is_exactly_a(basis)) { + return expand_mul(ex_to(basis), *num_exponent, false); } // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4) - if (num_exponent->is_integer() && is_exactly_a(ebasis)) { - numeric icont = ebasis.integer_content(); + if (num_exponent->is_integer() && is_exactly_a(basis)) { + numeric icont = basis.integer_content(); const numeric lead_coeff = - ex_to(ex_to(ebasis).seq.begin()->coeff).div(icont); + ex_to(ex_to(basis).seq.begin()->coeff).div(icont); const bool canonicalizable = lead_coeff.is_integer(); const bool unit_normal = lead_coeff.is_pos_integer(); @@ -519,48 +508,43 @@ ex power::eval(int level) const icont = icont.mul(*_num_1_p); if (canonicalizable && (icont != *_num1_p)) { - const add& addref = ex_to(ebasis); - add* addp = new add(addref); - addp->setflag(status_flags::dynallocated); - addp->clearflag(status_flags::hash_calculated); - addp->overall_coeff = ex_to(addp->overall_coeff).div_dyn(icont); - for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i) - i->coeff = ex_to(i->coeff).div_dyn(icont); + const add& addref = ex_to(basis); + add & addp = dynallocate(addref); + addp.clearflag(status_flags::hash_calculated); + addp.overall_coeff = ex_to(addp.overall_coeff).div_dyn(icont); + for (auto & i : addp.seq) + i.coeff = ex_to(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); + return dynallocate(dynallocate(addp, *num_exponent), c); else - return power(*addp, *num_exponent); + return dynallocate(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(ebasis)) { + if (is_exactly_a(basis)) { GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above - const mul & mulref = ex_to(ebasis); + const mul & mulref = ex_to(basis); if (!mulref.overall_coeff.is_equal(_ex1)) { const numeric & num_coeff = ex_to(mulref.overall_coeff); if (num_coeff.is_real()) { 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); + mul & mulp = dynallocate(mulref); + mulp.overall_coeff = _ex1; + mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated); + return dynallocate(dynallocate(mulp, exponent), + dynallocate(num_coeff, *num_exponent)); } else { 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), - power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated); + mul & mulp = dynallocate(mulref); + mulp.overall_coeff = _ex_1; + mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated); + return dynallocate(dynallocate(mulp, exponent), + dynallocate(abs(num_coeff), *num_exponent)); } } } @@ -569,39 +553,26 @@ ex power::eval(int level) const // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix) if (num_exponent->is_pos_integer() && - ebasis.return_type() != return_types::commutative && - !is_a(ebasis)) { - return ncmul(exvector(num_exponent->to_int(), ebasis), true); + basis.return_type() != return_types::commutative && + !is_a(basis)) { + return ncmul(exvector(num_exponent->to_int(), basis)); } } - - if (are_ex_trivially_equal(ebasis,basis) && - are_ex_trivially_equal(eexponent,exponent)) { - return this->hold(); - } - return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated | - status_flags::evaluated); + + return this->hold(); } -ex power::evalf(int level) const +ex power::evalf() const { - ex ebasis; + ex ebasis = basis.evalf(); ex eexponent; - if (level==1) { - ebasis = basis; + if (!is_exactly_a(exponent)) + eexponent = exponent.evalf(); + else eexponent = exponent; - } else if (level == -max_recursion_level) { - throw(std::runtime_error("max recursion level reached")); - } else { - ebasis = basis.evalf(level-1); - if (!is_exactly_a(exponent)) - eexponent = exponent.evalf(level-1); - else - eexponent = exponent; - } - return power(ebasis,eexponent); + return dynallocate(ebasis, eexponent); } ex power::evalm() const @@ -610,10 +581,10 @@ ex power::evalm() const const ex eexponent = exponent.evalm(); if (is_a(ebasis)) { if (is_exactly_a(eexponent)) { - return (new matrix(ex_to(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated); + return dynallocate(ex_to(ebasis).pow(eexponent)); } } - return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated); + return dynallocate(ebasis, eexponent); } bool power::has(const ex & other, unsigned options) const @@ -653,13 +624,13 @@ 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::max(); 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); + 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) * pow(anum/aden, nummatches); return (ex_to(result)).subs_one_level(m, options); } } @@ -681,66 +652,89 @@ ex power::conjugate() const if (are_ex_trivially_equal(exponent, newexponent)) { return *this; } - return (new power(basis, newexponent))->setflag(status_flags::dynallocated); + return dynallocate(basis, newexponent); } 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 dynallocate(newbasis, exponent); } 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) && + (a.info(info_flags::nonnegative) || c.info(info_flags::integer))) { + // Re(a^c) + return *this; + } + + const ex b = basis.imag_part(); 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() )); + // Re((a+I*b)^c) w/ c ∈ ℤ + long N = ex_to(c).to_long(); + // Use real terms in Binomial expansion to construct + // Re(expand(pow(a+I*b, N))). + long NN = N > 0 ? N : -N; + ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN); + ex result = 0; + for (long n = 0; n <= NN; n += 2) { + ex term = binomial(NN, n) * pow(a, NN-n) * pow(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; } - - 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))); + + // Re((a+I*b)^(c+I*d)) + const ex d = exponent.imag_part(); + return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis))); } ex power::imag_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) && + (a.info(info_flags::nonnegative) || c.info(info_flags::integer))) { + // Im(a^c) + return 0; + } + + const ex b = basis.imag_part(); 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() )); + // Im((a+I*b)^c) w/ c ∈ ℤ + long N = ex_to(c).to_long(); + // Use imaginary terms in Binomial expansion to construct + // Im(expand(pow(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 : pow(pow(a,2) + pow(b,2), NN); + ex result = 0; + for (long n = 1; n <= NN; n += 2) { + ex term = binomial(NN, n) * pow(a, NN-n) * pow(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; } - - 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))); + + // Im((a+I*b)^(c+I*d)) + const ex d = exponent.imag_part(); + return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis))); } // protected @@ -751,16 +745,11 @@ ex power::derivative(const symbol & s) const { if (is_a(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); + const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)}; + return dynallocate(std::move(newseq), exponent); } else { // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b) - return mul(*this, - add(mul(exponent.diff(s), log(basis)), - mul(mul(exponent, basis.diff(s)), power(basis, _ex_1)))); + return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1)); } } @@ -802,29 +791,26 @@ ex power::expand(unsigned options) const epvector powseq; prodseq.reserve(m.seq.size() + 1); powseq.reserve(m.seq.size() + 1); - epvector::const_iterator last = m.seq.end(); - epvector::const_iterator cit = m.seq.begin(); bool possign = true; // search for positive/negative factors - while (cit!=last) { - ex e=m.recombine_pair_to_ex(*cit); + 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); - ++cit; + 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)); + prodseq.push_back(pow(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)); + prodseq.push_back(pow(-m.overall_coeff, exponent)); else coeff *= m.overall_coeff; @@ -832,9 +818,9 @@ ex power::expand(unsigned options) const // 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(powseq); + ex newbasis = dynallocate(std::move(powseq), coeff); ex_to(newbasis).setflag(status_flags::purely_indefinite); - return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options); + return dynallocate(std::move(prodseq)) * pow(newbasis, exponent); } else ex_to(basis).setflag(status_flags::purely_indefinite); } @@ -847,26 +833,23 @@ ex power::expand(unsigned options) const const add &a = ex_to(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(pow(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(a.overall_coeff).is_integer()) { const numeric &num_exponent = ex_to(a.overall_coeff); - int int_exponent = num_exponent.to_int(); + long 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, options)); else - distrseq.push_back(power(expanded_basis, a.overall_coeff)); + distrseq.push_back(pow(expanded_basis, a.overall_coeff)); } else - distrseq.push_back(power(expanded_basis, a.overall_coeff)); + distrseq.push_back(pow(expanded_basis, a.overall_coeff)); // 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); + ex r = dynallocate(distrseq); return r.expand(options); } @@ -875,13 +858,13 @@ ex power::expand(unsigned options) const if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) { return this->hold(); } else { - return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)); + return dynallocate(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0); } } // integer numeric exponent const numeric & num_exponent = ex_to(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(expanded_basis)) @@ -895,7 +878,7 @@ ex power::expand(unsigned options) const if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) return this->hold(); else - return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)); + return dynallocate(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0); } ////////// @@ -908,192 +891,9 @@ ex power::expand(unsigned options) const // 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 x; - int n; // n>0 - int m; // 0 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& 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." ) - 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& partition) - { - head = 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 composition; // current compositions -public: - explicit composition_generator(const std::vector& partition) - : cmgen(partition), atend(false), trivial(true), composition(partition.size()) - { - for (unsigned i=1; i& 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 p) -{ - numeric n = 0, d = 1; - std::vector::const_iterator it = p.begin(), itend = p.end(); - while (it != itend) { - n += numeric(*it); - d *= factorial(numeric(*it)); - ++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) { // The special case power(+(x,...y;x),2) can be optimized better. if (n==2) @@ -1155,7 +955,7 @@ ex power::expand_add(const add & a, int n, unsigned options) const // 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: binomial(n+m-1,m-1). - size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int(); + 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; @@ -1181,21 +981,22 @@ ex power::expand_add(const add & a, int n, unsigned options) const // 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()); + partition_with_zero_parts_generator partitions(k, a.seq.size()); do { - const std::vector& partition = partitions.current(); + const std::vector& partition = partitions.get(); + // All monomials of this partition have the same number of terms and the same coefficient. + const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; }); const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient; // Iterate over all compositions of the current partition. composition_generator compositions(partition); do { - const std::vector& exponent = compositions.current(); - exvector term; - term.reserve(n); + const std::vector& exponent = compositions.get(); + epvector monomial; + monomial.reserve(msize); numeric factor = coeff; for (unsigned i = 0; i < exponent.size(); ++i) { const ex & r = a.seq[i].rest; - const ex & c = a.seq[i].coeff; GINAC_ASSERT(!is_exactly_a(r)); GINAC_ASSERT(!is_exactly_a(r) || !is_exactly_a(ex_to(r).exponent) || @@ -1203,46 +1004,52 @@ ex power::expand_add(const add & a, int n, unsigned options) const !is_exactly_a(ex_to(r).basis) || !is_exactly_a(ex_to(r).basis) || !is_exactly_a(ex_to(r).basis)); + GINAC_ASSERT(is_exactly_a(a.seq[i].coeff)); + const numeric & c = ex_to(a.seq[i].coeff); if (exponent[i] == 0) { // optimize away } else if (exponent[i] == 1) { // optimized - term.push_back(r); - factor = factor.mul(ex_to(c)); + monomial.push_back(expair(r, _ex1)); + 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)); - factor = factor.mul(ex_to(c).power(exponent[i])); + monomial.push_back(expair(r, exponent[i])); + 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)); + result.push_back(expair(mul(std::move(monomial)).expand(options), factor)); } while (compositions.next()); } while (partitions.next()); } GINAC_ASSERT(result.size() == result_size); - if (a.overall_coeff.is_zero()) { - return (new add(result))->setflag(status_flags::dynallocated | - status_flags::expanded); + return dynallocate(std::move(result)).setflag(status_flags::expanded); } else { - return (new add(result, ex_to(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated | - status_flags::expanded); + return dynallocate(std::move(result), ex_to(a.overall_coeff).power(n)).setflag(status_flags::expanded); } } /** 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, unsigned options) const +ex power::expand_add_2(const add & a, unsigned options) { - epvector sum; - size_t a_nops = a.nops(); - sum.reserve((a_nops*(a_nops+1))/2); - epvector::const_iterator last = a.seq.end(); + 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); + + auto last = a.seq.end(); // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c // first part: ignore overall_coeff and expand other terms - for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) { + for (auto cit0=a.seq.begin(); cit0!=last; ++cit0) { const ex & r = cit0->rest; const ex & c = cit0->coeff; @@ -1256,50 +1063,48 @@ ex power::expand_add_2(const add & a, unsigned options) const if (c.is_equal(_ex1)) { if (is_exactly_a(r)) { - sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to(r), *_num2_p, options, true), - _ex1)); + result.push_back(expair(expand_mul(ex_to(r), *_num2_p, options, true), + _ex1)); } else { - sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated), - _ex1)); + result.push_back(expair(dynallocate(r, _ex2), + _ex1)); } } else { if (is_exactly_a(r)) { - 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))); + result.push_back(expair(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_p))); + result.push_back(expair(dynallocate(r, _ex2), + ex_to(c).power_dyn(*_num2_p))); } } - for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) { + for (auto 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(mul(r,r1).expand(options), - _num2_p->mul(ex_to(c)).mul_dyn(ex_to(c1)))); + result.push_back(expair(mul(r,r1).expand(options), + _num2_p->mul(ex_to(c)).mul_dyn(ex_to(c1)))); } } - GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2); - // 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(a.overall_coeff).mul_dyn(*_num2_p))); - ++i; - } - sum.push_back(expair(ex_to(a.overall_coeff).power_dyn(*_num2_p),_ex1)); + for (auto & i : a.seq) + result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to(a.overall_coeff).mul_dyn(*_num2_p))); + } + + GINAC_ASSERT(result.size() == result_size); + + if (a.overall_coeff.is_zero()) { + return dynallocate(std::move(result)).setflag(status_flags::expanded); + } else { + return dynallocate(std::move(result), ex_to(a.overall_coeff).power(2)).setflag(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 an integer. * @see power::expand */ -ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const +ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) { GINAC_ASSERT(n.is_integer()); @@ -1327,20 +1132,17 @@ ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool fr 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) { - 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()) { + for (auto & cit : m.seq) { + 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; } - const mul & result = static_cast((new mul(distrseq, ex_to(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated)); + const mul & result = dynallocate(std::move(distrseq), ex_to(m.overall_coeff).power_dyn(n)); if (need_reexpand) return ex(result).expand(options); if (from_expand)