* 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
#include <vector>
#include <iostream>
#include <stdexcept>
+#include <limits>
#include "power.h"
#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 "operators.h"
+#include "inifcns.h" // for log() in power::derivative()
+#include "matrix.h"
+#include "indexed.h"
#include "symbol.h"
+#include "lst.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)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
+ print_func<print_dflt>(&power::do_print_dflt).
+ print_func<print_latex>(&power::do_print_latex).
+ print_func<print_csrc>(&power::do_print_csrc).
+ print_func<print_python>(&power::do_print_python).
+ print_func<print_python_repr>(&power::do_print_python_repr))
-typedef vector<int> intvector;
+typedef std::vector<int> intvector;
//////////
-// default constructor, destructor, copy constructor assignment operator and helpers
+// default constructor
//////////
-// 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
-
-void power::copy(const power & other)
-{
- inherited::copy(other);
- basis=other.basis;
- exponent=other.exponent;
-}
-
-void power::destroy(bool call_parent)
-{
- if (call_parent) inherited::destroy(call_parent);
-}
+power::power() : inherited(TINFO_power) { }
//////////
// other constructors
//////////
-// 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)
+power::power(const archive_node &n, 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);
+ 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);
-}
-
-/** 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);
-}
+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<numeric>(exponent) && ex_to<numeric>(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 parenthesation 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::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<symbol>(basis) || is_a<constant>(basis))) {
+ int exp = ex_to<numeric>(exponent).to_int();
+ if (exp > 0)
+ c.s << '(';
+ else {
+ exp = -exp;
+ if (is_a<print_csrc_cl_N>(c))
+ c.s << "recip(";
+ else
+ c.s << "1.0/(";
+ }
+ print_sym_pow(c, ex_to<symbol>(basis), exp);
+ c.s << ')';
+
+ // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
+ } else if (exponent.is_equal(_ex_1)) {
+ if (is_a<print_csrc_cl_N>(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<print_csrc_cl_N>(c))
+ c.s << "expt(";
+ else
+ c.s << "pow(";
+ basis.print(c);
+ c.s << ',';
+ exponent.print(c);
+ c.s << ')';
+ }
+}
-static void print_sym_pow(ostream & os, unsigned type, const symbol &x, int exp)
+void power::do_print_python(const print_python & c, unsigned level) const
{
- // 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<const symbol &>(*basis.bp), exp);
- os << ")";
-
- // <expr>^-1 is printed as "1.0/<expr>" 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 << ")";
- }
+ print_power(c, "**", "", "", level);
+}
+
+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
{
- 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);
- }
-}
-
-unsigned power::nops() const
+ 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);
+}
+
+size_t power::nops() const
+{
+ return 2;
+}
+
+ex power::op(size_t i) const
+{
+ 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<basic>(s)))
+ return 1;
+ else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
+ if (basis.is_equal(s))
+ return ex_to<numeric>(exponent).to_int();
+ else
+ return basis.degree(s) * ex_to<numeric>(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<basic>(s)))
+ return 1;
+ else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
+ if (basis.is_equal(s))
+ return ex_to<numeric>(exponent).to_int();
+ else
+ return basis.ldegree(s) * ex_to<numeric>(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<basic>(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_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
+ // integer exponent
+ int int_exp = ex_to<numeric>(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
{
- return 2;
+ 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_exactly_a<numeric>(ebasis)) {
+ basis_is_numerical = true;
+ num_basis = &ex_to<numeric>(ebasis);
+ }
+ if (is_exactly_a<numeric>(eexponent)) {
+ exponent_is_numerical = true;
+ num_exponent = &ex_to<numeric>(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_exactly_a<power>(ebasis)) {
+ const power & sub_power = ex_to<power>(ebasis);
+ const ex & sub_basis = sub_power.basis;
+ const ex & sub_exponent = sub_power.exponent;
+ if (is_exactly_a<numeric>(sub_exponent)) {
+ 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) - _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_exactly_a<mul>(ebasis)) {
+ 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 (is_exactly_a<mul>(ebasis)) {
+ 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()) {
+ 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_a<matrix>(ebasis)) {
+ 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);
}
-ex & power::let_op(int i)
+ex power::evalf(int level) const
{
- GINAC_ASSERT(i>=0);
- GINAC_ASSERT(i<2);
+ 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<numeric>(exponent))
+ eexponent = exponent.evalf(level-1);
+ else
+ eexponent = exponent;
+ }
- return i==0 ? basis : exponent;
+ return power(ebasis,eexponent);
}
-int power::degree(const symbol & s) const
+ex power::evalm() 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;
+ const ex ebasis = basis.evalm();
+ const ex eexponent = exponent.evalm();
+ if (is_a<matrix>(ebasis)) {
+ if (is_exactly_a<numeric>(eexponent)) {
+ return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
+ }
+ }
+ return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
}
-int power::ldegree(const symbol & s) 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;
-}
+// from mul.cpp
+extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
-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<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
- return _ex1();
- }
-
- return _ex0();
-}
+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);
-ex power::eval(int level) const
-{
- // simplifications: ^(x,0) -> 1 (0^0 handled here)
- // ^(x,1) -> x
- // ^(0,c1) -> 0 or exception (depending on real value of c1)
- // ^(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<numeric *>(ebasis.bp);
- }
- if (is_exactly_of_type(*eexponent.bp,numeric)) {
- exponent_is_numerical = 1;
- num_exponent = static_cast<numeric *>(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,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 (std::overflow_error("power::eval(): division by zero"));
- 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);
-}
+ 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);
-ex power::evalf(int level) 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);
-}
+ if (!(options & subs_options::algebraic))
+ return subs_one_level(m, options);
-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);
+ for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
+ int nummatches = std::numeric_limits<int>::max();
+ lst repls;
+ if (tryfactsubs(*this, it->first, nummatches, repls))
+ return (ex_to<basic>((*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);
+ }
+
+ return subs_one_level(m, options);
}
-ex power::simplify_ncmul(const exvector & v) const
+ex power::eval_ncmul(const exvector & v) const
{
- return inherited::simplify_ncmul(v);
+ return inherited::eval_ncmul(v);
}
// protected
* @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 power &>(const_cast<basic &>(other));
-
- int cmpval;
- cmpval=basis.compare(o.basis);
- if (cmpval==0) {
- return exponent.compare(o.exponent);
- }
- return cmpval;
+ GINAC_ASSERT(is_exactly_a<power>(other));
+ const power &o = static_cast<const power &>(other);
+
+ int cmpval = basis.compare(o.basis);
+ if (cmpval)
+ return cmpval;
+ else
+ return exponent.compare(o.exponent);
}
-unsigned power::return_type(void) const
+unsigned power::return_type() const
{
- return basis.return_type();
+ return basis.return_type();
}
-unsigned power::return_type_tinfo(void) const
+unsigned power::return_type_tinfo() 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_exactly_a<add>(expanded_exponent)) {
+ const add &a = ex_to<add>(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<numeric>(a.overall_coeff).is_integer()) {
+ const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
+ int int_exponent = num_exponent.to_int();
+ if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
+ distrseq.push_back(expand_add(ex_to<add>(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_exactly_a<numeric>(expanded_exponent) ||
+ !ex_to<numeric>(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<numeric>(expanded_exponent);
+ int int_exponent = num_exponent.to_int();
+
+ // (x+y)^n, n>0
+ if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
+ return expand_add(ex_to<add>(expanded_basis), int_exponent);
+
+ // (x*y)^n -> x^n * y^n
+ if (is_exactly_a<mul>(expanded_basis))
+ return expand_mul(ex_to<mul>(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));
}
//////////
// 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; l<m-1; l++) {
- k[l] = 0;
- k_cum[l] = 0;
- upper_limit[l] = n;
- }
-
- while (1) {
- exvector term;
- term.reserve(m+1);
- for (l=0; l<m-1; l++) {
- const ex & b = a.op(l);
- GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
- GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
- !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
- !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
- if (is_ex_exactly_of_type(b,mul)) {
- term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
- } else {
- term.push_back(power(b,k[l]));
- }
- }
-
- const ex & b = a.op(l);
- GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
- GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
- !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
- !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
- if (is_ex_exactly_of_type(b,mul)) {
- term.push_back(expand_mul(ex_to_mul(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<m-1; l++) {
- f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
- }
- term.push_back(f);
-
- /*
- cout << "begin term" << endl;
- for (int i=0; i<m-1; i++) {
- cout << "k[" << i << "]=" << k[i] << endl;
- cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
- cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
- }
- for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
- cout << *cit << endl;
- }
- cout << "end term" << endl;
- */
-
- // TODO: optimize this
- sum.push_back((new mul(term))->setflag(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; i<m-1; i++) {
- k_cum[i]=k_cum[i-1]+k[i];
- }
-
- for (int i=l+1; i<m-1; i++) {
- upper_limit[i]=n-k_cum[i-1];
- }
- }
- return (new add(sum))->setflag(status_flags::dynallocated |
- status_flags::expanded );
+ if (n==2)
+ return expand_add_2(a);
+
+ const size_t m = a.nops();
+ exvector result;
+ // The number of terms will be the number of combinatorial compositions,
+ // i.e. the number of unordered arrangement of m nonnegative integers
+ // 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<m-1; ++l) {
+ k[l] = 0;
+ k_cum[l] = 0;
+ upper_limit[l] = n;
+ }
+
+ while (true) {
+ exvector term;
+ term.reserve(m+1);
+ for (l=0; l<m-1; ++l) {
+ const ex & b = a.op(l);
+ GINAC_ASSERT(!is_exactly_a<add>(b));
+ GINAC_ASSERT(!is_exactly_a<power>(b) ||
+ !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
+ !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
+ !is_exactly_a<add>(ex_to<power>(b).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(b).basis) ||
+ !is_exactly_a<power>(ex_to<power>(b).basis));
+ if (is_exactly_a<mul>(b))
+ term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
+ else
+ term.push_back(power(b,k[l]));
+ }
+
+ const ex & b = a.op(l);
+ GINAC_ASSERT(!is_exactly_a<add>(b));
+ GINAC_ASSERT(!is_exactly_a<power>(b) ||
+ !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
+ !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
+ !is_exactly_a<add>(ex_to<power>(b).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(b).basis) ||
+ !is_exactly_a<power>(ex_to<power>(b).basis));
+ if (is_exactly_a<mul>(b))
+ term.push_back(expand_mul(ex_to<mul>(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<m-1; ++l)
+ f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
+
+ term.push_back(f);
+
+ result.push_back((new mul(term))->setflag(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 (size_t i=l+1; i<m-1; ++i)
+ k_cum[i] = k_cum[i-1]+k[i];
+
+ for (size_t i=l+1; i<m-1; ++i)
+ upper_limit[i] = n-k_cum[i-1];
+ }
+
+ return (new add(result))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
}
* @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;
+ size_t 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<add>(r));
+ GINAC_ASSERT(!is_exactly_a<power>(r) ||
+ !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
+ !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
+ !is_exactly_a<add>(ex_to<power>(r).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(r).basis) ||
+ !is_exactly_a<power>(ex_to<power>(r).basis));
+
+ if (c.is_equal(_ex1)) {
+ if (is_exactly_a<mul>(r)) {
+ 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_exactly_a<mul>(r)) {
+ sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
+ ex_to<numeric>(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<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_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<numeric>(a.overall_coeff).mul_dyn(_num2)));
+ ++i;
+ }
+ 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.
* @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);
-}
-
-/*
-ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
- unsigned options) const
-{
- // obsolete
-
- exvector distrseq;
- epvector splitseq;
-
- const add & addref=static_cast<const add &>(*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);
-}
-*/
+ GINAC_ASSERT(n.is_integer());
-//////////
-// static member variables
-//////////
-
-// protected
-
-unsigned power::precedence = 60;
+ if (n.is_zero())
+ return _ex1;
-//////////
-// global constants
-//////////
-
-const power some_power;
-const type_info & typeid_power=typeid(some_power);
-
-// 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_exactly_a<numeric>(cit->rest)) {
+ distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
+ } else {
+ // it is safe not to call mul::combine_pair_with_coeff_to_pair()
+ // since n is an integer
+ distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
+ }
+ ++cit;
+ }
+ return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
}
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff