X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpower.cpp;h=c3fe14a132e14f98b79a24f19ec9ad5d0ef95d71;hp=a6decf3cb39804eb1c255dd64d2ecebb317861ad;hb=d56a0f74afa5380a1730599c3b1b21f34be2f061;hpb=1ebd5f62696a5144e8249127d958bd1d3004857f diff --git a/ginac/power.cpp b/ginac/power.cpp index a6decf3c..c3fe14a1 100644 --- a/ginac/power.cpp +++ b/ginac/power.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */ /* - * GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -17,13 +17,9 @@ * * 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 -#include -#include - #include "power.h" #include "expairseq.h" #include "add.h" @@ -31,36 +27,40 @@ #include "ncmul.h" #include "numeric.h" #include "constant.h" +#include "operators.h" #include "inifcns.h" // for log() in power::derivative() #include "matrix.h" +#include "indexed.h" #include "symbol.h" -#include "print.h" +#include "lst.h" #include "archive.h" #include "utils.h" +#include "relational.h" +#include "compiler.h" -namespace GiNaC { +#include +#include +#include +#include -GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic) +namespace GiNaC { -typedef std::vector intvector; +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic, + print_func(&power::do_print_dflt). + 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). + print_func(&power::do_print_csrc_cl_N)) ////////// -// default ctor, dtor, copy ctor, assignment operator and helpers +// default constructor ////////// -power::power() : inherited(TINFO_power) { } - -void power::copy(const power & other) -{ - inherited::copy(other); - basis = other.basis; - exponent = other.exponent; -} - -DEFAULT_DESTROY(power) +power::power() { } ////////// -// other ctors +// other constructors ////////// // all inlined @@ -69,8 +69,9 @@ DEFAULT_DESTROY(power) // archiving ////////// -power::power(const archive_node &n, const 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); } @@ -82,19 +83,64 @@ void power::archive(archive_node &n) const n.add_ex("exponent", exponent); } -DEFAULT_UNARCHIVE(power) - ////////// // functions overriding virtual functions from base classes ////////// // public +void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const +{ + // Ordinary output of powers using '^' or '**' + if (precedence() <= level) + c.s << openbrace << '('; + basis.print(c, precedence()); + c.s << powersymbol; + c.s << openbrace; + exponent.print(c, precedence()); + c.s << closebrace; + if (precedence() <= level) + c.s << ')' << closebrace; +} + +void power::do_print_dflt(const print_dflt & c, unsigned level) const +{ + if (exponent.is_equal(_ex1_2)) { + + // Square roots are printed in a special way + c.s << "sqrt("; + basis.print(c); + c.s << ')'; + + } else + print_power(c, "^", "", "", level); +} + +void power::do_print_latex(const print_latex & c, unsigned level) const +{ + if (is_exactly_a(exponent) && ex_to(exponent).is_negative()) { + + // Powers with negative numeric exponents are printed as fractions + c.s << "\\frac{1}{"; + power(basis, -exponent).eval().print(c); + c.s << '}'; + + } else if (exponent.is_equal(_ex1_2)) { + + // Square roots are printed in a special way + c.s << "\\sqrt{"; + basis.print(c); + c.s << '}'; + + } else + print_power(c, "^", "{", "}", level); +} + 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 - // to learn why such a parenthisation is really necessary. + // to learn why such a parenthesation is really necessary. if (exp == 1) { x.print(c); } else if (exp == 2) { @@ -114,96 +160,64 @@ static void print_sym_pow(const print_context & c, const symbol &x, int exp) } } -void power::print(const print_context & c, unsigned level) const +void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const { - if (is_a(c)) { - - inherited::print(c, level); - - } else if (is_a(c)) { - - // Integer powers of symbols are printed in a special, optimized way - if (exponent.info(info_flags::integer) - && (is_exactly_a(basis) || is_exactly_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/("; - } - 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/("; - basis.print(c); - c.s << ')'; + 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 << ')'; +} - // Otherwise, use the pow() or expt() (CLN) functions - } else { - if (is_a(c)) - c.s << "expt("; - else - c.s << "pow("; - 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))) { + int exp = ex_to(exponent).to_int(); + if (exp > 0) + c.s << '('; + else { + exp = -exp; + c.s << "1.0/("; } + print_sym_pow(c, ex_to(basis), exp); + c.s << ')'; - } else if (is_a(c)) { + // ^-1 is printed as "1.0/" or with the recip() function of CLN + } else if (exponent.is_equal(_ex_1)) { + c.s << "1.0/("; + basis.print(c); + c.s << ')'; - c.s << class_name() << '('; + // Otherwise, use the pow() function + } else { + c.s << "pow("; basis.print(c); c.s << ','; exponent.print(c); c.s << ')'; + } +} - } else { +void power::do_print_python(const print_python & c, unsigned level) const +{ + print_power(c, "**", "", "", level); +} - if (exponent.is_equal(_ex1_2)) { - if (is_a(c)) - c.s << "\\sqrt{"; - else - c.s << "sqrt("; - basis.print(c); - if (is_a(c)) - c.s << '}'; - else - c.s << ')'; - } else { - if (precedence() <= level) { - if (is_a(c)) - c.s << "{("; - else - c.s << "("; - } - basis.print(c, precedence()); - if (is_a(c)) - c.s << "**"; - else - c.s << '^'; - if (is_a(c)) - c.s << '{'; - exponent.print(c, precedence()); - if (is_a(c)) - c.s << '}'; - if (precedence() <= level) { - if (is_a(c)) - c.s << ")}"; - else - c.s << ')'; - } - } - } +void power::do_print_python_repr(const print_python_repr & c, unsigned level) const +{ + c.s << class_name() << '('; + basis.print(c); + c.s << ','; + exponent.print(c); + c.s << ')'; } bool power::info(unsigned inf) const @@ -214,24 +228,47 @@ 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); + case info_flags::expanded: + return (flags & status_flags::expanded); + case info_flags::positive: + return basis.info(info_flags::positive) && exponent.info(info_flags::real); + case info_flags::nonnegative: + return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) || + (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even)); + case info_flags::has_indices: { + if (flags & status_flags::has_indices) + return true; + else if (flags & status_flags::has_no_indices) + return false; + else if (basis.info(info_flags::has_indices)) { + setflag(status_flags::has_indices); + clearflag(status_flags::has_no_indices); + return true; + } else { + clearflag(status_flags::has_indices); + setflag(status_flags::has_no_indices); + return false; + } + } } return inherited::info(inf); } -unsigned power::nops() const +size_t power::nops() const { return 2; } -ex & power::let_op(int i) +ex power::op(size_t i) const { - GINAC_ASSERT(i>=0); GINAC_ASSERT(i<2); return i==0 ? basis : exponent; @@ -239,14 +276,35 @@ ex & power::let_op(int i) 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 (basis.is_polynomial(var)) { + if (basis.has(var)) + // basis is non-constant polynomial in var + return exponent.info(info_flags::nonnegint); + else + // basis is constant in var + return !exponent.has(var); + } + // basis is a non-polynomial function of var + return false; } int power::degree(const ex & s) const { if (is_equal(ex_to(s))) return 1; - else if (is_ex_exactly_of_type(exponent, numeric) && ex_to(exponent).is_integer()) { + else if (is_exactly_a(exponent) && ex_to(exponent).is_integer()) { if (basis.is_equal(s)) return ex_to(exponent).to_int(); else @@ -261,7 +319,7 @@ int power::ldegree(const ex & s) const { if (is_equal(ex_to(s))) return 1; - else if (is_ex_exactly_of_type(exponent, numeric) && ex_to(exponent).is_integer()) { + else if (is_exactly_a(exponent) && ex_to(exponent).is_integer()) { if (basis.is_equal(s)) return ex_to(exponent).to_int(); else @@ -284,7 +342,7 @@ ex power::coeff(const ex & s, int n) const return _ex0; } else { // basis equal to s - if (is_ex_exactly_of_type(exponent, numeric) && ex_to(exponent).is_integer()) { + if (is_exactly_a(exponent) && ex_to(exponent).is_integer()) { // integer exponent int int_exp = ex_to(exponent).to_int(); if (n == int_exp) @@ -309,7 +367,8 @@ 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) @@ -325,17 +384,13 @@ ex power::eval(int level) const const ex & ebasis = level==1 ? basis : basis.eval(level-1); const ex & eexponent = level==1 ? exponent : exponent.eval(level-1); - bool basis_is_numerical = false; - bool exponent_is_numerical = false; - const numeric *num_basis; - const numeric *num_exponent; + const numeric *num_basis = NULL; + const numeric *num_exponent = NULL; - if (is_ex_exactly_of_type(ebasis, numeric)) { - basis_is_numerical = true; + if (is_exactly_a(ebasis)) { num_basis = &ex_to(ebasis); } - if (is_ex_exactly_of_type(eexponent, numeric)) { - exponent_is_numerical = true; + if (is_exactly_a(eexponent)) { num_exponent = &ex_to(eexponent); } @@ -352,7 +407,7 @@ ex power::eval(int level) const return ebasis; // ^(0,c1) -> 0 or exception (depending on real value of c1) - if (ebasis.is_zero() && exponent_is_numerical) { + if ( ebasis.is_zero() && num_exponent ) { if ((num_exponent->real()).is_zero()) throw (std::domain_error("power::eval(): pow(0,I) is undefined")); else if ((num_exponent->real()).is_negative()) @@ -365,11 +420,19 @@ ex power::eval(int level) const if (ebasis.is_equal(_ex1)) return _ex1; - if (exponent_is_numerical) { + // power of a function calculated by separate rules defined for this function + if (is_exactly_a(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 ( 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) { @@ -423,28 +486,58 @@ ex power::eval(int level) const } // ^(^(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_ex_exactly_of_type(ebasis,power)) { + if (is_exactly_a(ebasis)) { const power & sub_power = ex_to(ebasis); const ex & sub_basis = sub_power.basis; const ex & sub_exponent = sub_power.exponent; - if (is_ex_exactly_of_type(sub_exponent,numeric)) { + 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() + || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) { 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_ex_exactly_of_type(ebasis,mul)) { - return expand_mul(ex_to(ebasis), *num_exponent); + if (num_exponent->is_integer() && is_exactly_a(ebasis)) { + return expand_mul(ex_to(ebasis), *num_exponent, 0); } - + + // (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(); + const numeric lead_coeff = + ex_to(ex_to(ebasis).seq.begin()->coeff).div(icont); + + const bool canonicalizable = lead_coeff.is_integer(); + const bool unit_normal = lead_coeff.is_pos_integer(); + if (canonicalizable && (! unit_normal)) + icont = icont.mul(*_num_1_p); + + if (canonicalizable && (icont != *_num1_p)) { + const add& addref = ex_to(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 numeric c = icont.power(*num_exponent); + if (likely(c != *_num1_p)) + return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated); + else + return power(*addp, *num_exponent); + } + } + // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0) // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0) - if (is_ex_exactly_of_type(ebasis,mul)) { + if (is_exactly_a(ebasis)) { GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above const mul & mulref = ex_to(ebasis); if (!mulref.overall_coeff.is_equal(_ex1)) { @@ -453,15 +546,17 @@ ex power::eval(int level) const if (num_coeff.is_positive()) { mul *mulp = new mul(mulref); mulp->overall_coeff = _ex1; + mulp->setflag(status_flags::dynallocated); mulp->clearflag(status_flags::evaluated); mulp->clearflag(status_flags::hash_calculated); return (new mul(power(*mulp,exponent), power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated); } else { - GINAC_ASSERT(num_coeff.compare(_num0)<0); - if (!num_coeff.is_equal(_num_1)) { + GINAC_ASSERT(num_coeff.compare(*_num0_p)<0); + if (!num_coeff.is_equal(*_num_1_p)) { mul *mulp = new mul(mulref); mulp->overall_coeff = _ex_1; + mulp->setflag(status_flags::dynallocated); mulp->clearflag(status_flags::evaluated); mulp->clearflag(status_flags::hash_calculated); return (new mul(power(*mulp,exponent), @@ -475,7 +570,7 @@ 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_ex_of_type(ebasis,matrix)) { + !is_a(ebasis)) { return ncmul(exvector(num_exponent->to_int(), ebasis), true); } } @@ -509,33 +604,145 @@ ex power::evalf(int level) const return power(ebasis,eexponent); } -ex power::evalm(void) const +ex power::evalm() const { const ex ebasis = basis.evalm(); const ex eexponent = exponent.evalm(); - if (is_ex_of_type(ebasis,matrix)) { - if (is_ex_of_type(eexponent,numeric)) { + if (is_a(ebasis)) { + if (is_exactly_a(eexponent)) { return (new matrix(ex_to(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated); } } return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated); } -ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const +bool power::has(const ex & other, unsigned options) const { - const ex &subsed_basis = basis.subs(ls, lr, no_pattern); - const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern); + 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); +} - if (are_ex_trivially_equal(basis, subsed_basis) - && are_ex_trivially_equal(exponent, subsed_exponent)) - return basic::subs(ls, lr, no_pattern); - else - return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern); +// from mul.cpp +extern bool tryfactsubs(const ex &, const ex &, int &, exmap&); + +ex power::subs(const exmap & m, unsigned options) const +{ + const ex &subsed_basis = basis.subs(m, options); + const ex &subsed_exponent = exponent.subs(m, options); + + if (!are_ex_trivially_equal(basis, subsed_basis) + || !are_ex_trivially_equal(exponent, subsed_exponent)) + return power(subsed_basis, subsed_exponent).subs_one_level(m, options); + + if (!(options & subs_options::algebraic)) + return subs_one_level(m, options); + + for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) { + 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); + return (ex_to(result)).subs_one_level(m, options); + } + } + + return subs_one_level(m, options); +} + +ex power::eval_ncmul(const exvector & v) const +{ + return inherited::eval_ncmul(v); +} + +ex power::conjugate() const +{ + // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the + // branch cut which runs along the negative real axis. + if (basis.info(info_flags::positive)) { + ex newexponent = exponent.conjugate(); + if (are_ex_trivially_equal(exponent, newexponent)) { + return *this; + } + return (new power(basis, newexponent))->setflag(status_flags::dynallocated); + } + if (exponent.info(info_flags::integer)) { + ex newbasis = basis.conjugate(); + if (are_ex_trivially_equal(basis, newbasis)) { + return *this; + } + return (new power(newbasis, exponent))->setflag(status_flags::dynallocated); + } + return conjugate_function(*this).hold(); +} + +ex power::real_part() const +{ + 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::simplify_ncmul(const exvector & v) const +ex power::imag_part() const { - return inherited::simplify_ncmul(v); + 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 @@ -544,7 +751,7 @@ ex power::simplify_ncmul(const exvector & v) const * @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); @@ -571,26 +778,74 @@ int power::compare_same_type(const basic & other) const return exponent.compare(o.exponent); } -unsigned power::return_type(void) const +unsigned power::return_type() const { return basis.return_type(); } - -unsigned power::return_type_tinfo(void) 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); + 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); + 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; + } + + // take care on the numeric coefficient + ex coeff=(possign? _ex1 : _ex_1); + if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1) + prodseq.push_back(power(m.overall_coeff, exponent)); + else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1) + prodseq.push_back(power(-m.overall_coeff, exponent)); + else + coeff *= m.overall_coeff; + + // If positive/negative factors are found, then extract them. + // In either case we set a flag to avoid the second run on a part + // which does not have positive/negative terms. + if (prodseq.size() > 0) { + ex newbasis = coeff*mul(powseq); + 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); + } else + ex_to(basis).setflag(status_flags::purely_indefinite); + } + const ex expanded_basis = basis.expand(options); const ex expanded_exponent = exponent.expand(options); // x^(a+b) -> x^a * x^b - if (is_ex_exactly_of_type(expanded_exponent, add)) { + if (is_exactly_a(expanded_exponent)) { const add &a = ex_to(expanded_exponent); exvector distrseq; distrseq.reserve(a.seq.size() + 1); @@ -605,8 +860,8 @@ ex power::expand(unsigned options) const if (ex_to(a.overall_coeff).is_integer()) { const numeric &num_exponent = ex_to(a.overall_coeff); int int_exponent = num_exponent.to_int(); - if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add)) - distrseq.push_back(expand_add(ex_to(expanded_basis), int_exponent)); + 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)); } else @@ -614,10 +869,10 @@ 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_ex_exactly_of_type(expanded_exponent, numeric) || + if (!is_exactly_a(expanded_exponent) || !ex_to(expanded_exponent).is_integer()) { if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) { return this->hold(); @@ -631,12 +886,12 @@ ex power::expand(unsigned options) const int int_exponent = num_exponent.to_int(); // (x+y)^n, n>0 - if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) - return expand_add(ex_to(expanded_basis), int_exponent); + if (int_exponent > 0 && is_exactly_a(expanded_basis)) + return expand_add(ex_to(expanded_basis), int_exponent, options); // (x*y)^n -> x^n * y^n - if (is_ex_exactly_of_type(expanded_basis,mul)) - return expand_mul(ex_to(expanded_basis), num_exponent); + if (is_exactly_a(expanded_basis)) + 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)) @@ -655,99 +910,335 @@ ex power::expand(unsigned options) const // non-virtual functions in this class ////////// -/** 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 -{ - if (n==2) - return expand_add_2(a); +namespace { // anonymous namespace for power::expand_add() helpers - const int 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 - // 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 (int 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_ex_exactly_of_type(b,mul)) - term.push_back(expand_mul(ex_to(b),numeric(k[l]))); - else - term.push_back(power(b,k[l])); +/** Helper class to generate all bounded combinatorial partitions of an integer + * n with exactly m parts (including zero parts) in non-decreaing 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; + } +}; - 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_ex_exactly_of_type(b,mul)) - term.push_back(expand_mul(ex_to(b),numeric(n-k_cum[m-2]))); - else - term.push_back(power(b,n-k_cum[m-2])); - - numeric f = binomial(numeric(n),numeric(k[0])); - for (l=1; l) + 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 = NULL; + 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 != NULL && 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 == NULL && 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 != NULL) { + 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; + } +}; - term.push_back(f); +/** 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; +} - result.push_back((new mul(term))->setflag(status_flags::dynallocated)); +} // anonymous namespace - // increment k[] - l = m-2; - while ((l>=0) && ((++k[l])>upper_limit[l])) { - k[l] = 0; - --l; +/** 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 +{ + // The special case power(+(x,...y;x),2) can be optimized better. + if (n==2) + return expand_add_2(a, options); + + // 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: binomial(n+m-1,m-1). + size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int(); + 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]); + // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k): + // Iterate over all partitions of k with exactly as many parts as + // there are symbolic terms in the basis (including zero parts). + partition_generator partitions(k, a.seq.size()); + do { + const std::vector& partition = partitions.current(); + const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient; + + // Iterate over all compositions of the current partition. + composition_generator compositions(partition); + do { + const std::vector& exponent = compositions.current(); + exvector term; + term.reserve(n); + numeric factor = coeff; + for (unsigned i = 0; i < exponent.size(); ++i) { + const ex & r = a.seq[i].rest; + 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) || + !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)); + if (exponent[i] == 0) { + // optimize away + } else if (exponent[i] == 1) { + // optimized + term.push_back(r); + factor = factor.mul(ex_to(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])); + } + } + result.push_back(a.combine_ex_with_coeff_to_pair(mul(term).expand(options), factor)); + } while (compositions.next()); + } while (partitions.next()); + } - for (int i=l+1; isetflag(status_flags::dynallocated | + status_flags::expanded); + } else { + return (new add(result, ex_to(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated | + status_flags::expanded); } - - return (new add(result))->setflag(status_flags::dynallocated | - 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) const +ex power::expand_add_2(const add & a, unsigned options) const { epvector sum; - unsigned a_nops = a.nops(); + size_t a_nops = a.nops(); sum.reserve((a_nops*(a_nops+1))/2); epvector::const_iterator last = a.seq.end(); @@ -765,42 +1256,42 @@ ex power::expand_add_2(const add & a) const !is_exactly_a(ex_to(r).basis) || !is_exactly_a(ex_to(r).basis)); - if (are_ex_trivially_equal(c,_ex1)) { - if (is_ex_exactly_of_type(r,mul)) { - sum.push_back(expair(expand_mul(ex_to(r),_num2), - _ex1)); + 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)); } else { - sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated), - _ex1)); + sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated), + _ex1)); } } else { - if (is_ex_exactly_of_type(r,mul)) { - sum.push_back(expair(expand_mul(ex_to(r),_num2), - ex_to(c).power_dyn(_num2))); + 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))); } else { - sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated), - ex_to(c).power_dyn(_num2))); + 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))); } } - + for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) { const ex & r1 = cit1->rest; const ex & c1 = cit1->coeff; - sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated), - _num2.mul(ex_to(c)).mul_dyn(ex_to(c1)))); + 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)))); } } 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))); + 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); @@ -808,30 +1299,57 @@ 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; + } + + // do not bother to rename indices if there are no any. + if ((!(options & expand_options::expand_rename_idx)) + && m.info(info_flags::has_indices)) + options |= expand_options::expand_rename_idx; + // Leave it to multiplication since dummy indices have to be renamed + if ((options & expand_options::expand_rename_idx) && + (get_all_dummy_indices(m).size() > 0) && n.is_positive()) { + ex result = m; + exvector va = get_all_dummy_indices(m); + sort(va.begin(), va.end(), ex_is_less()); + + for (int i=1; i < n.to_int(); i++) + result *= rename_dummy_indices_uniquely(va, m); + return result; + } epvector distrseq; distrseq.reserve(m.seq.size()); + bool need_reexpand = false; + epvector::const_iterator last = m.seq.end(); epvector::const_iterator cit = m.seq.begin(); while (cit!=last) { - if (is_ex_exactly_of_type((*cit).rest,numeric)) { - 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; } +GINAC_BIND_UNARCHIVER(power); + } // namespace GiNaC