X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpower.cpp;h=5cddca8995bd475c6b6f56f5a7c8d5670280fa6a;hp=0ebc8dc29a20db47796cde8ab228eb293c76d3da;hb=c12c8ec3c5cf0c75f061f6c52d04206277bbdcca;hpb=2893c8e04685b99317384ff02a0fefc899f499f9 diff --git a/ginac/power.cpp b/ginac/power.cpp index 0ebc8dc2..5cddca89 100644 --- a/ginac/power.cpp +++ b/ginac/power.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */ /* - * GiNaC Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2016 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 @@ -20,11 +20,6 @@ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include -#include -#include - #include "power.h" #include "expairseq.h" #include "add.h" @@ -40,6 +35,14 @@ #include "lst.h" #include "archive.h" #include "utils.h" +#include "relational.h" +#include "compiler.h" + +#include +#include +#include +#include +#include namespace GiNaC { @@ -48,15 +51,14 @@ GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic, print_func(&power::do_print_latex). print_func(&power::do_print_csrc). print_func(&power::do_print_python). - print_func(&power::do_print_python_repr)) - -typedef std::vector intvector; + print_func(&power::do_print_python_repr). + print_func(&power::do_print_csrc_cl_N)) ////////// // default constructor ////////// -power::power() : inherited(&power::tinfo_static) { } +power::power() { } ////////// // other constructors @@ -68,8 +70,9 @@ power::power() : inherited(&power::tinfo_static) { } // 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); } @@ -81,8 +84,6 @@ void power::archive(archive_node &n) const n.add_ex("exponent", exponent); } -DEFAULT_UNARCHIVE(power) - ////////// // functions overriding virtual functions from base classes ////////// @@ -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); @@ -160,39 +161,45 @@ static void print_sym_pow(const print_context & c, const symbol &x, int exp) } } +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(basis) || is_a(basis))) { + if (exponent.info(info_flags::integer) && + (is_a(basis) || is_a(basis))) { int exp = ex_to(exponent).to_int(); if (exp > 0) c.s << '('; else { exp = -exp; - if (is_a(c)) - c.s << "recip("; - else - c.s << "1.0/("; + c.s << "1.0/("; } print_sym_pow(c, ex_to(basis), exp); c.s << ')'; // ^-1 is printed as "1.0/" or with the recip() function of CLN } else if (exponent.is_equal(_ex_1)) { - if (is_a(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(c)) - c.s << "expt("; - else - c.s << "pow("; + c.s << "pow("; basis.print(c); c.s << ','; exponent.print(c); @@ -222,14 +229,33 @@ 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::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); } @@ -253,18 +279,23 @@ 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; } 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); + 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 @@ -334,50 +365,41 @@ 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) (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) - * - * @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); - - bool basis_is_numerical = false; - bool exponent_is_numerical = false; - const numeric *num_basis; - const numeric *num_exponent; - - if (is_exactly_a(ebasis)) { - basis_is_numerical = true; - num_basis = &ex_to(ebasis); + + const numeric *num_basis = nullptr; + const numeric *num_exponent = nullptr; + + if (is_exactly_a(basis)) { + num_basis = &ex_to(basis); } - if (is_exactly_a(eexponent)) { - exponent_is_numerical = true; - 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() && exponent_is_numerical) { + 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()) @@ -387,25 +409,27 @@ 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); - if (exponent_is_numerical) { + // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real. + 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 ) { // ^(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) { // 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); @@ -435,9 +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 { @@ -445,56 +469,82 @@ 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); } } } // ^(^(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(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)) { 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_p)).is_negative()) - return power(sub_basis,num_sub_exponent.mul(*num_exponent)); + if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() || + (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) { + 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(basis)) { + numeric icont = basis.integer_content(); + const numeric lead_coeff = + 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(); + if (canonicalizable && (! unit_normal)) + icont = icont.mul(*_num_1_p); + + if (canonicalizable && (icont != *_num1_p)) { + 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 dynallocate(dynallocate(addp, *num_exponent), c); + else + 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->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->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)); } } } @@ -503,39 +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 @@ -544,10 +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 @@ -556,26 +593,24 @@ bool power::has(const ex & other, unsigned options) const 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)) + 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))) + if (exponent.info(info_flags::posint) && + other.op(1).info(info_flags::posint) && + ex_to(exponent) > ex_to(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(exponent).to_int() - < ex_to(other.op(1)).to_int() - && basis.match(other.op(0))) + if (exponent.info(info_flags::negint) && + other.op(1).info(info_flags::negint) && + ex_to(exponent) < ex_to(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 { @@ -589,11 +624,15 @@ 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(); - lst repls; - if (tryfactsubs(*this, it->first, nummatches, repls)) - return (ex_to((*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) * pow(anum/aden, nummatches); + return (ex_to(result)).subs_one_level(m, options); + } } return subs_one_level(m, options); @@ -606,12 +645,94 @@ ex power::eval_ncmul(const exvector & v) const 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 dynallocate(basis, newexponent); + } + if (exponent.info(info_flags::integer)) { + ex newbasis = basis.conjugate(); + if (are_ex_trivially_equal(basis, newbasis)) { + return *this; + } + 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)) { + // 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(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; + } + + // 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)) { + // 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(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; + } + + // 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 @@ -622,16 +743,11 @@ 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)); } } @@ -652,16 +768,61 @@ unsigned power::return_type() const return basis.return_type(); } -tinfo_t 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(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(basis) && !basis.info(info_flags::indefinite)) { + const mul &m = ex_to(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(pow(m.overall_coeff, exponent)); + else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1) + prodseq.push_back(pow(-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 = dynallocate(std::move(powseq), coeff); + ex_to(newbasis).setflag(status_flags::purely_indefinite); + return dynallocate(std::move(prodseq)) * pow(newbasis, exponent); + } else + ex_to(basis).setflag(status_flags::purely_indefinite); + } + const ex expanded_basis = basis.expand(options); const ex expanded_exponent = exponent.expand(options); @@ -670,26 +831,23 @@ 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); } @@ -698,13 +856,13 @@ 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)) @@ -718,7 +876,7 @@ 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); } ////////// @@ -731,100 +889,342 @@ 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), 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 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; + 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) { + // 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 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(b)); - GINAC_ASSERT(!is_exactly_a(b) || - !is_exactly_a(ex_to(b).exponent) || - !ex_to(ex_to(b).exponent).is_pos_integer() || - !is_exactly_a(ex_to(b).basis) || - !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]), options, true)); - else - term.push_back(power(b,k[l])); - } - - const ex & b = a.op(l); - GINAC_ASSERT(!is_exactly_a(b)); - GINAC_ASSERT(!is_exactly_a(b) || - !is_exactly_a(ex_to(b).exponent) || - !ex_to(ex_to(b).exponent).is_pos_integer() || - !is_exactly_a(ex_to(b).basis) || - !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]), options, true)); - else - term.push_back(power(b,n-k_cum[m-2])); - - numeric f = binomial(numeric(n),numeric(k[0])); - for (l=1; lsetflag(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(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]); - - for (size_t i=l+1; i& partition = partitions.current(); + // 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(); + epvector monomial; + monomial.reserve(msize); + numeric factor = coeff; + for (unsigned i = 0; i < exponent.size(); ++i) { + const ex & r = a.seq[i].rest; + GINAC_ASSERT(!is_exactly_a(r)); + GINAC_ASSERT(!is_exactly_a(r) || + !is_exactly_a(ex_to(r).exponent) || + !ex_to(ex_to(r).exponent).is_pos_integer() || + !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 + monomial.push_back(expair(r, _ex1)); + if (c != *_num1_p) + factor = factor.mul(c); + } else { // general case exponent[i] > 1 + monomial.push_back(expair(r, exponent[i])); + if (c != *_num1_p) + factor = factor.mul(c.power(exponent[i])); + } + } + result.push_back(expair(mul(monomial).expand(options), factor)); + } while (compositions.next()); + } while (partitions.next()); } - return (new add(result))->setflag(status_flags::dynallocated | - status_flags::expanded); + 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(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 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 @@ -843,50 +1243,48 @@ ex power::expand_add_2(const add & a, unsigned options) const if (c.is_equal(_ex1)) { if (is_exactly_a(r)) { - sum.push_back(expair(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(expair((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) { 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_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_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(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 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 +ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) { GINAC_ASSERT(n.is_integer()); @@ -894,8 +1292,13 @@ ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool fr 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 (get_all_dummy_indices(m).size() > 0 && n.is_positive()) { + 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()); @@ -909,26 +1312,17 @@ 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) { - 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 - numeric new_coeff = ex_to(cit->coeff).mul(n); - if (from_expand && is_exactly_a(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(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; } - ++cit; + distrseq.push_back(p); } - 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) @@ -936,4 +1330,6 @@ ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool fr return result; } +GINAC_BIND_UNARCHIVER(power); + } // namespace GiNaC