X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpower.cpp;h=0730e3c8bf7fc385c5d4d1c1a97b6c565997cfab;hp=32038209f51e69e53c0d93afab773c3ca03b7750;hb=dbd9c306a74f1cb258c0d15a346b973b39deaad2;hpb=f2ec274c38bc881d4c52a0e5eb215fd78c730b4d;ds=sidebyside diff --git a/ginac/power.cpp b/ginac/power.cpp index 32038209..0730e3c8 100644 --- a/ginac/power.cpp +++ b/ginac/power.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */ /* - * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2003 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 @@ -28,492 +28,515 @@ #include "expairseq.h" #include "add.h" #include "mul.h" +#include "ncmul.h" #include "numeric.h" -#include "inifcns.h" -#include "relational.h" +#include "constant.h" +#include "inifcns.h" // for log() in power::derivative() +#include "matrix.h" +#include "indexed.h" #include "symbol.h" +#include "print.h" #include "archive.h" -#include "debugmsg.h" #include "utils.h" -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic) -typedef vector intvector; +typedef std::vector intvector; ////////// -// default constructor, destructor, copy constructor assignment operator and helpers +// default ctor, dtor, copy ctor, assignment operator and helpers ////////// -// public - -power::power() : basic(TINFO_power) -{ - debugmsg("power default constructor",LOGLEVEL_CONSTRUCT); -} - -power::~power() -{ - debugmsg("power destructor",LOGLEVEL_DESTRUCT); - destroy(0); -} - -power::power(const power & other) -{ - debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT); - copy(other); -} - -const power & power::operator=(const power & other) -{ - debugmsg("power operator=",LOGLEVEL_ASSIGNMENT); - if (this != &other) { - destroy(1); - copy(other); - } - return *this; -} - -// protected +power::power() : inherited(TINFO_power) { } void power::copy(const power & other) { - inherited::copy(other); - basis=other.basis; - exponent=other.exponent; + inherited::copy(other); + basis = other.basis; + exponent = other.exponent; } -void power::destroy(bool call_parent) -{ - if (call_parent) inherited::destroy(call_parent); -} +DEFAULT_DESTROY(power) ////////// -// other constructors +// other ctors ////////// -// public - -power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh) -{ - debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT); - GINAC_ASSERT(basis.return_type()==return_types::commutative); -} - -power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh) -{ - debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT); - GINAC_ASSERT(basis.return_type()==return_types::commutative); -} +// all inlined ////////// // archiving ////////// -/** Construct object from archive_node. */ power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst) { - debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT); - n.find_ex("basis", basis, sym_lst); - n.find_ex("exponent", exponent, sym_lst); -} - -/** Unarchive the object. */ -ex power::unarchive(const archive_node &n, const lst &sym_lst) -{ - return (new power(n, sym_lst))->setflag(status_flags::dynallocated); + n.find_ex("basis", basis, sym_lst); + n.find_ex("exponent", exponent, sym_lst); } -/** Archive the object. */ void power::archive(archive_node &n) const { - inherited::archive(n); - n.add_ex("basis", basis); - n.add_ex("exponent", exponent); + inherited::archive(n); + n.add_ex("basis", basis); + n.add_ex("exponent", exponent); } +DEFAULT_UNARCHIVE(power) + ////////// -// functions overriding virtual functions from bases classes +// functions overriding virtual functions from base classes ////////// // public -basic * power::duplicate() const -{ - debugmsg("power duplicate",LOGLEVEL_DUPLICATE); - return new power(*this); -} - -void power::print(ostream & os, unsigned upper_precedence) const -{ - debugmsg("power print",LOGLEVEL_PRINT); - if (exponent.is_equal(_ex1_2())) { - os << "sqrt(" << basis << ")"; - } else { - if (precedence<=upper_precedence) os << "("; - basis.print(os,precedence); - os << "^"; - exponent.print(os,precedence); - if (precedence<=upper_precedence) os << ")"; - } -} - -void power::printraw(ostream & os) const -{ - debugmsg("power printraw",LOGLEVEL_PRINT); - - os << "power("; - basis.printraw(os); - os << ","; - exponent.printraw(os); - os << ",hash=" << hashvalue << ",flags=" << flags << ")"; -} - -void power::printtree(ostream & os, unsigned indent) const -{ - debugmsg("power printtree",LOGLEVEL_PRINT); - - os << string(indent,' ') << "power: " - << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")" - << ", flags=" << flags << endl; - basis.printtree(os,indent+delta_indent); - exponent.printtree(os,indent+delta_indent); -} - -static void print_sym_pow(ostream & os, unsigned type, const symbol &x, int exp) -{ - // Optimal output of integer powers of symbols to aid compiler CSE - if (exp == 1) { - x.printcsrc(os, type, 0); - } else if (exp == 2) { - x.printcsrc(os, type, 0); - os << "*"; - x.printcsrc(os, type, 0); - } else if (exp & 1) { - x.printcsrc(os, 0); - os << "*"; - print_sym_pow(os, type, x, exp-1); - } else { - os << "("; - print_sym_pow(os, type, x, exp >> 1); - os << ")*("; - print_sym_pow(os, type, x, exp >> 1); - os << ")"; - } -} - -void power::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const -{ - debugmsg("power print csrc", LOGLEVEL_PRINT); - - // Integer powers of symbols are printed in a special, optimized way - if (exponent.info(info_flags::integer) && - (is_ex_exactly_of_type(basis, symbol) || - is_ex_exactly_of_type(basis, constant))) { - int exp = ex_to_numeric(exponent).to_int(); - if (exp > 0) - os << "("; - else { - exp = -exp; - if (type == csrc_types::ctype_cl_N) - os << "recip("; - else - os << "1.0/("; - } - print_sym_pow(os, type, static_cast(*basis.bp), exp); - os << ")"; - - // ^-1 is printed as "1.0/" or with the recip() function of CLN - } else if (exponent.compare(_num_1()) == 0) { - if (type == csrc_types::ctype_cl_N) - os << "recip("; - else - os << "1.0/("; - basis.bp->printcsrc(os, type, 0); - os << ")"; - - // Otherwise, use the pow() or expt() (CLN) functions - } else { - if (type == csrc_types::ctype_cl_N) - os << "expt("; - else - os << "pow("; - basis.bp->printcsrc(os, type, 0); - os << ","; - exponent.bp->printcsrc(os, type, 0); - os << ")"; - } +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. + if (exp == 1) { + x.print(c); + } else if (exp == 2) { + x.print(c); + c.s << "*"; + x.print(c); + } else if (exp & 1) { + x.print(c); + c.s << "*"; + print_sym_pow(c, x, exp-1); + } else { + c.s << "("; + print_sym_pow(c, x, exp >> 1); + c.s << ")*("; + print_sym_pow(c, x, exp >> 1); + c.s << ")"; + } +} + +void power::print(const print_context & 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_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/("; + } + 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 << ')'; + + // 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 << ')'; + } + + } else if (is_a(c)) { + + c.s << class_name() << '('; + basis.print(c); + c.s << ','; + exponent.print(c); + c.s << ')'; + + } else { + + bool is_tex = is_a(c); + + if (is_tex && is_exactly_a(exponent) && ex_to(exponent).is_negative()) { + + // Powers with negative numeric exponents are printed as fractions in TeX + 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 << (is_tex ? "\\sqrt{" : "sqrt("); + basis.print(c); + c.s << (is_tex ? '}' : ')'); + + } else { + + // Ordinary output of powers using '^' or '**' + if (precedence() <= level) + c.s << (is_tex ? "{(" : "("); + basis.print(c, precedence()); + if (is_a(c)) + c.s << "**"; + else + c.s << '^'; + if (is_tex) + c.s << '{'; + exponent.print(c, precedence()); + if (is_tex) + c.s << '}'; + if (precedence() <= level) + c.s << (is_tex ? ")}" : ")"); + } + } } bool power::info(unsigned inf) const { - if (inf==info_flags::polynomial || - inf==info_flags::integer_polynomial || - inf==info_flags::cinteger_polynomial || - inf==info_flags::rational_polynomial || - inf==info_flags::crational_polynomial) { - return exponent.info(info_flags::nonnegint); - } else if (inf==info_flags::rational_function) { - return exponent.info(info_flags::integer); - } else { - return inherited::info(inf); - } + switch (inf) { + case info_flags::polynomial: + case info_flags::integer_polynomial: + case info_flags::cinteger_polynomial: + case info_flags::rational_polynomial: + case info_flags::crational_polynomial: + return exponent.info(info_flags::nonnegint); + case info_flags::rational_function: + return exponent.info(info_flags::integer); + case info_flags::algebraic: + return (!exponent.info(info_flags::integer) || + basis.info(inf)); + } + return inherited::info(inf); } unsigned power::nops() const { - return 2; + return 2; } ex & power::let_op(int i) { - GINAC_ASSERT(i>=0); - GINAC_ASSERT(i<2); - - return i==0 ? basis : exponent; -} - -int power::degree(const symbol & s) const + GINAC_ASSERT(i>=0); + GINAC_ASSERT(i<2); + + return i==0 ? basis : exponent; +} + +ex power::map(map_function & f) const +{ + return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated); +} + +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()) { + if (basis.is_equal(s)) + return ex_to(exponent).to_int(); + else + return basis.degree(s) * ex_to(exponent).to_int(); + } else if (basis.has(s)) + throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent")); + else + return 0; +} + +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()) { + if (basis.is_equal(s)) + return ex_to(exponent).to_int(); + else + return basis.ldegree(s) * ex_to(exponent).to_int(); + } else if (basis.has(s)) + throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent")); + else + return 0; +} + +ex power::coeff(const ex & s, int n) const +{ + if (is_equal(ex_to(s))) + return n==1 ? _ex1 : _ex0; + else if (!basis.is_equal(s)) { + // basis not equal to s + if (n == 0) + return *this; + else + return _ex0; + } else { + // basis equal to s + if (is_ex_exactly_of_type(exponent, numeric) && ex_to(exponent).is_integer()) { + // integer exponent + int int_exp = ex_to(exponent).to_int(); + if (n == int_exp) + return _ex1; + else + return _ex0; + } else { + // non-integer exponents are treated as zero + if (n == 0) + return *this; + else + return _ex0; + } + } +} + +/** Perform automatic term rewriting rules in this class. In the following + * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2... + * stand for such expressions that contain a plain number. + * - ^(x,0) -> 1 (also handles ^(0,0)) + * - ^(x,1) -> x + * - ^(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,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 { - if (is_exactly_of_type(*exponent.bp,numeric)) { - if ((*basis.bp).compare(s)==0) - return ex_to_numeric(exponent).to_int(); - else - return basis.degree(s) * ex_to_numeric(exponent).to_int(); - } - return 0; + if ((level==1) && (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_ex_exactly_of_type(ebasis, numeric)) { + basis_is_numerical = true; + num_basis = &ex_to(ebasis); + } + if (is_ex_exactly_of_type(eexponent, numeric)) { + exponent_is_numerical = true; + num_exponent = &ex_to(eexponent); + } + + // ^(x,0) -> 1 (0^0 also handled here) + if (eexponent.is_zero()) { + if (ebasis.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; + + // ^(0,c1) -> 0 or exception (depending on real value of c1) + if (ebasis.is_zero() && exponent_is_numerical) { + if ((num_exponent->real()).is_zero()) + throw (std::domain_error("power::eval(): pow(0,I) is undefined")); + else if ((num_exponent->real()).is_negative()) + throw (pole_error("power::eval(): division by zero",1)); + else + return _ex0; + } + + // ^(1,x) -> 1 + if (ebasis.is_equal(_ex1)) + return _ex1; + + if (exponent_is_numerical) { + + // ^(c1,c2) -> c1^c2 (c1, c2 numeric(), + // except if c1,c2 are rational, but c1^c2 is not) + if (basis_is_numerical) { + 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); + } + + const numeric res = num_basis->power(*num_exponent); + if (res.is_crational()) { + return res; + } + GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now + + // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer + if (basis_is_crational && exponent_is_crational + && num_exponent->is_real() + && !num_exponent->is_integer()) { + const numeric n = num_exponent->numer(); + const numeric m = num_exponent->denom(); + numeric r; + numeric q = iquo(n, m, r); + if (r.is_negative()) { + r += m; + --q; + } + if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1 + if (num_basis->is_rational() && !num_basis->is_integer()) { + // try it for numerator and denominator separately, in order to + // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3) + const numeric bnum = num_basis->numer(); + const numeric bden = num_basis->denom(); + 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); + if (res_bden.is_integer()) + return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated); + } + return this->hold(); + } else { + // assemble resulting product, but allowing for a re-evaluation, + // 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); + } + } + } + + // ^(^(x,c1),c2) -> ^(x,c1*c2) + // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, + // case c1==1 should not happen, see below!) + if (is_ex_exactly_of_type(ebasis,power)) { + 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)) { + 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()) + 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); + } + + // ^(*(...,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)) { + GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above + const mul & mulref = ex_to(ebasis); + 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); + } else { + GINAC_ASSERT(num_coeff.compare(_num0)<0); + if (!num_coeff.is_equal(_num_1)) { + 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); + } + } + } + } + } + + // ^(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)) { + return ncmul(exvector(num_exponent->to_int(), ebasis), true); + } + } + + 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); } -int power::ldegree(const symbol & s) const +ex power::evalf(int level) const { - if (is_exactly_of_type(*exponent.bp,numeric)) { - if ((*basis.bp).compare(s)==0) - return ex_to_numeric(exponent).to_int(); - else - return basis.ldegree(s) * ex_to_numeric(exponent).to_int(); - } - return 0; -} + ex ebasis; + ex eexponent; + + if (level==1) { + ebasis = basis; + 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; + } -ex power::coeff(const symbol & s, int n) const -{ - if ((*basis.bp).compare(s)!=0) { - // basis not equal to s - if (n==0) { - return *this; - } else { - return _ex0(); - } - } else if (is_exactly_of_type(*exponent.bp,numeric)&& - (static_cast(*exponent.bp).compare(numeric(n))==0)) { - return _ex1(); - } - - return _ex0(); + return power(ebasis,eexponent); } -ex power::eval(int level) const +ex power::evalm(void) const { - // simplifications: ^(x,0) -> 1 (0^0 handled here) - // ^(x,1) -> x - // ^(0,x) -> 0 (except if the realpart of x is non-positive, in which case an exception is thrown) - // ^(1,x) -> 1 - // ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not) - // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!) - // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer) - // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0) - // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0) - - debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION); - - if ((level==1) && (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 = 0; - bool exponent_is_numerical = 0; - numeric * num_basis; - numeric * num_exponent; - - if (is_exactly_of_type(*ebasis.bp,numeric)) { - basis_is_numerical = 1; - num_basis = static_cast(ebasis.bp); - } - if (is_exactly_of_type(*eexponent.bp,numeric)) { - exponent_is_numerical = 1; - num_exponent = static_cast(eexponent.bp); - } - - // ^(x,0) -> 1 (0^0 also handled here) - if (eexponent.is_zero()) - if (ebasis.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; - - // ^(0,x) -> 0 (except if the realpart of x is non-positive) - if (ebasis.is_zero()) { - if (exponent_is_numerical) { - if ((num_exponent->real()).is_zero()) - throw (std::domain_error("power::eval(): pow(0,I) is undefined")); - else if ((num_exponent->real()).is_negative()) - throw (std::overflow_error("power::eval(): division by zero")); - else - return _ex0(); - } else - return _ex0(); - } - - // ^(1,x) -> 1 - if (ebasis.is_equal(_ex1())) - return _ex1(); - - if (basis_is_numerical && exponent_is_numerical) { - // ^(c1,c2) -> c1^c2 (c1, c2 numeric(), - // except if c1,c2 are rational, but c1^c2 is not) - bool basis_is_crational = num_basis->is_crational(); - bool exponent_is_crational = num_exponent->is_crational(); - numeric res = (*num_basis).power(*num_exponent); - - if ((!basis_is_crational || !exponent_is_crational) - || res.is_crational()) { - return res; - } - GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now - // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer - if (basis_is_crational && exponent_is_crational - && num_exponent->is_real() - && !num_exponent->is_integer()) { - numeric n = num_exponent->numer(); - numeric m = num_exponent->denom(); - numeric r; - numeric q = iquo(n, m, r); - if (r.is_negative()) { - r = r.add(m); - q = q.sub(_num1()); - } - if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1 - return this->hold(); - else { - epvector res; - res.push_back(expair(ebasis,r.div(m))); - return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated); - } - } - } - - // ^(^(x,c1),c2) -> ^(x,c1*c2) - // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, - // case c1==1 should not happen, see below!) - if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) { - const power & sub_power = ex_to_power(ebasis); - const ex & sub_basis = sub_power.basis; - const ex & sub_exponent = sub_power.exponent; - if (is_ex_exactly_of_type(sub_exponent,numeric)) { - const numeric & num_sub_exponent = ex_to_numeric(sub_exponent); - GINAC_ASSERT(num_sub_exponent!=numeric(1)); - if (num_exponent->is_integer() || abs(num_sub_exponent)<1) { - return power(sub_basis,num_sub_exponent.mul(*num_exponent)); - } - } - } - - // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer) - if (exponent_is_numerical && num_exponent->is_integer() && - is_ex_exactly_of_type(ebasis,mul)) { - return expand_mul(ex_to_mul(ebasis), *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 (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) { - GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above - const mul & mulref=ex_to_mul(ebasis); - if (!mulref.overall_coeff.is_equal(_ex1())) { - const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff); - if (num_coeff.is_real()) { - if (num_coeff.is_positive()>0) { - 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); - } else { - GINAC_ASSERT(num_coeff.compare(_num0())<0); - if (num_coeff.compare(_num_1())!=0) { - 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); - } - } - } - } - } - - 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); + const ex ebasis = basis.evalm(); + const ex eexponent = exponent.evalm(); + if (is_ex_of_type(ebasis,matrix)) { + if (is_ex_of_type(eexponent,numeric)) { + return (new matrix(ex_to(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated); + } + } + return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated); } -ex power::evalf(int level) const +ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const { - debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION); - - ex ebasis; - ex eexponent; - - if (level==1) { - ebasis = basis; - eexponent = exponent; - } else if (level == -max_recursion_level) { - throw(std::runtime_error("max recursion level reached")); - } else { - ebasis = basis.evalf(level-1); - if (!is_ex_exactly_of_type(eexponent,numeric)) - eexponent = exponent.evalf(level-1); - else - eexponent = exponent; - } - - return power(ebasis,eexponent); -} + const ex &subsed_basis = basis.subs(ls, lr, no_pattern); + const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern); -ex power::subs(const lst & ls, const lst & lr) const -{ - const ex & subsed_basis=basis.subs(ls,lr); - const ex & subsed_exponent=exponent.subs(ls,lr); - - if (are_ex_trivially_equal(basis,subsed_basis)&& - are_ex_trivially_equal(exponent,subsed_exponent)) { - return *this; - } - - return power(subsed_basis, subsed_exponent); + 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); } ex power::simplify_ncmul(const exvector & v) const { - return inherited::simplify_ncmul(v); + return inherited::simplify_ncmul(v); } // protected @@ -522,78 +545,105 @@ ex power::simplify_ncmul(const exvector & v) const * @see ex::diff */ ex power::derivative(const symbol & s) const { - if (exponent.info(info_flags::real)) { - // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below) - return mul(mul(exponent, power(basis, exponent - _ex1())), basis.diff(s)); - } else { - // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b) - return mul(power(basis, exponent), - add(mul(exponent.diff(s), log(basis)), - mul(mul(exponent, basis.diff(s)), power(basis, -1)))); - } + if (exponent.info(info_flags::real)) { + // 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); + } 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)))); + } } int power::compare_same_type(const basic & other) const { - GINAC_ASSERT(is_exactly_of_type(other, power)); - const power & o=static_cast(const_cast(other)); - - int cmpval; - cmpval=basis.compare(o.basis); - if (cmpval==0) { - return exponent.compare(o.exponent); - } - return cmpval; + GINAC_ASSERT(is_exactly_a(other)); + const power &o = static_cast(other); + + int cmpval = basis.compare(o.basis); + if (cmpval) + return cmpval; + else + return exponent.compare(o.exponent); } unsigned power::return_type(void) const { - return basis.return_type(); + return basis.return_type(); } unsigned power::return_type_tinfo(void) const { - return basis.return_type_tinfo(); + return basis.return_type_tinfo(); } ex power::expand(unsigned options) const { - if (flags & status_flags::expanded) - return *this; - - ex expanded_basis = basis.expand(options); - - if (!is_ex_exactly_of_type(exponent,numeric) || - !ex_to_numeric(exponent).is_integer()) { - if (are_ex_trivially_equal(basis,expanded_basis)) { - return this->hold(); - } else { - return (new power(expanded_basis,exponent))-> - setflag(status_flags::dynallocated | - status_flags::expanded); - } - } - - // integer numeric exponent - const numeric & num_exponent = ex_to_numeric(exponent); - int int_exponent = num_exponent.to_int(); - - if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) { - return expand_add(ex_to_add(expanded_basis), int_exponent); - } - - if (is_ex_exactly_of_type(expanded_basis,mul)) { - return expand_mul(ex_to_mul(expanded_basis), num_exponent); - } - - // cannot expand further - if (are_ex_trivially_equal(basis,expanded_basis)) { - return this->hold(); - } else { - return (new power(expanded_basis,exponent))-> - setflag(status_flags::dynallocated | - status_flags::expanded); - } + if (options == 0 && (flags & status_flags::expanded)) + return *this; + + 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)) { + 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; + } + + // 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(); + if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add)) + distrseq.push_back(expand_add(ex_to(expanded_basis), int_exponent)); + else + distrseq.push_back(power(expanded_basis, a.overall_coeff)); + } else + distrseq.push_back(power(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); + return r.expand(); + } + + if (!is_ex_exactly_of_type(expanded_exponent, numeric) || + !ex_to(expanded_exponent).is_integer()) { + 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)); + } + } + + // integer numeric exponent + const numeric & num_exponent = ex_to(expanded_exponent); + 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); + + // (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); + + // cannot expand further + 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)); } ////////// @@ -606,100 +656,90 @@ ex power::expand(unsigned options) const // non-virtual functions in this class ////////// -/** expand a^n where a is an add and n is an integer. +/** 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); - - int m = a.nops(); - exvector sum; - sum.reserve((n+1)*(m-1)); - 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; lsetflag(status_flags::dynallocated)); - - // increment k[] - l=m-2; - while ((l>=0)&&((++k[l])>upper_limit[l])) { - k[l]=0; - l--; - } - if (l<0) break; - - // recalc k_cum[] and upper_limit[] - if (l==0) { - k_cum[0]=k[0]; - } else { - k_cum[l]=k_cum[l-1]+k[l]; - } - for (int i=l+1; isetflag(status_flags::dynallocated | - status_flags::expanded ); + if (n==2) + return expand_add_2(a); + + 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])); + } + + 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; lsetflag(status_flags::dynallocated)); + + // increment k[] + l = m-2; + while ((l>=0) && ((++k[l])>upper_limit[l])) { + k[l] = 0; + --l; + } + if (l<0) break; + + // recalc k_cum[] and upper_limit[] + k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]); + + for (int i=l+1; isetflag(status_flags::dynallocated | + status_flags::expanded); } @@ -707,153 +747,92 @@ ex power::expand_add(const add & a, int n) const * @see power::expand_add */ ex power::expand_add_2(const add & a) const { - epvector sum; - unsigned a_nops=a.nops(); - sum.reserve((a_nops*(a_nops+1))/2); - epvector::const_iterator 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) { - const ex & r=(*cit0).rest; - const ex & c=(*cit0).coeff; - - GINAC_ASSERT(!is_ex_exactly_of_type(r,add)); - GINAC_ASSERT(!is_ex_exactly_of_type(r,power)|| - !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)|| - !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()|| - !is_ex_exactly_of_type(ex_to_power(r).basis,add)|| - !is_ex_exactly_of_type(ex_to_power(r).basis,mul)|| - !is_ex_exactly_of_type(ex_to_power(r).basis,power)); - - if (are_ex_trivially_equal(c,_ex1())) { - if (is_ex_exactly_of_type(r,mul)) { - sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),_ex1())); - } else { - sum.push_back(expair((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_mul(r),_num2()), - ex_to_numeric(c).power_dyn(_num2()))); - } else { - sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated), - ex_to_numeric(c).power_dyn(_num2()))); - } - } - - for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) { - const ex & r1=(*cit1).rest; - const ex & c1=(*cit1).coeff; - sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated), - _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1)))); - } - } - - GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2); - - // second part: add terms coming from overall_factor (if != 0) - if (!a.overall_coeff.is_equal(_ex0())) { - for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) { - sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2()))); - } - sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1())); - } - - 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 + epvector sum; + unsigned a_nops = a.nops(); + sum.reserve((a_nops*(a_nops+1))/2); + epvector::const_iterator 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) { + const ex & r = cit0->rest; + const ex & c = cit0->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 (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)); + } else { + sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated), + _ex1)); + } + } else { + if (is_ex_exactly_of_type(r,mul)) { + sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to(r),_num2), + ex_to(c).power_dyn(_num2))); + } 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))); + } + } + + 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)))); + } + } + + GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2); + + // second part: add terms coming from overall_factor (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))); + ++i; + } + sum.push_back(expair(ex_to(a.overall_coeff).power_dyn(_num2),_ex1)); + } + + 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. * @see power::expand */ ex power::expand_mul(const mul & m, const numeric & n) const { - if (n.is_equal(_num0())) - return _ex1(); - - epvector distrseq; - distrseq.reserve(m.seq.size()); - 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_numeric((*cit).coeff).mul(n))); - } - ++cit; - } - return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n))) - ->setflag(status_flags::dynallocated); -} + GINAC_ASSERT(n.is_integer()); -/* -ex power::expand_commutative_3(const ex & basis, const numeric & exponent, - unsigned options) const -{ - // obsolete - - exvector distrseq; - epvector splitseq; - - const add & addref=static_cast(*basis.bp); - - splitseq=addref.seq; - splitseq.pop_back(); - ex first_operands=add(splitseq); - ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1)); - - int n=exponent.to_int(); - for (int k=0; k<=n; k++) { - distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))* - power(last_operand,numeric(n-k))); - } - return ex((new add(distrseq))->setflag(status_flags::expanded | - status_flags::dynallocated )). - expand(options); -} -*/ - -/* -ex power::expand_noncommutative(const ex & basis, const numeric & exponent, - unsigned options) const -{ - ex rest_power=ex(power(basis,exponent.add(_num_1()))). - expand(options | expand_options::internal_do_not_expand_power_operands); - - return ex(mul(rest_power,basis),0). - expand(options | expand_options::internal_do_not_expand_mul_operands); -} -*/ - -////////// -// static member variables -////////// - -// protected - -unsigned power::precedence = 60; - -////////// -// global constants -////////// - -const power some_power; -const type_info & typeid_power=typeid(some_power); + if (n.is_zero()) + return _ex1; -// helper function - -ex sqrt(const ex & a) -{ - return power(a,_ex1_2()); + epvector distrseq; + distrseq.reserve(m.seq.size()); + 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))); + } + ++cit; + } + return (new mul(distrseq, ex_to(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated); } -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC