* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2007 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
*
* 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 <vector>
#include <iostream>
#include <stdexcept>
+#include <limits>
#include "power.h"
#include "expairseq.h"
#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 "debugmsg.h"
#include "utils.h"
+#include "relational.h"
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 std::vector<int> intvector;
//////////
-// default ctor, dtor, copy ctor assignment operator and helpers
+// default constructor
//////////
-power::power() : inherited(TINFO_power)
-{
- debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
-}
-
-void power::copy(const power & other)
-{
- inherited::copy(other);
- basis = other.basis;
- exponent = other.exponent;
-}
-
-DEFAULT_DESTROY(power)
+power::power() : inherited(&power::tinfo_static) { }
//////////
-// other ctors
+// other constructors
//////////
// all inlined
// archiving
//////////
-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 ctor from archive_node", LOGLEVEL_CONSTRUCT);
n.find_ex("basis", basis, sym_lst);
n.find_ex("exponent", exponent, sym_lst);
}
// 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<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 parenthisation is really necessary.
+ // to learn why such a parenthesation is really necessary.
if (exp == 1) {
x.print(c);
} else if (exp == 2) {
}
}
-void power::print(const print_context & c, unsigned level) const
+void power::do_print_csrc(const print_csrc & c, unsigned level) const
{
- debugmsg("power print", LOGLEVEL_PRINT);
-
- if (is_a<print_tree>(c)) {
-
- inherited::print(c, level);
-
- } else if (is_a<print_csrc>(c)) {
-
- // Integer powers of symbols are printed in a special, optimized way
- if (exponent.info(info_flags::integer)
- && (is_exactly_a<symbol>(basis) || is_exactly_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.compare(_num_1()) == 0) {
+ // 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/(";
- 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 << ')';
}
+ print_sym_pow(c, ex_to<symbol>(basis), exp);
+ c.s << ')';
- } else {
+ // <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 << ')';
- if (exponent.is_equal(_ex1_2())) {
- if (is_a<print_latex>(c))
- c.s << "\\sqrt{";
- else
- c.s << "sqrt(";
- basis.print(c);
- if (is_a<print_latex>(c))
- c.s << '}';
- else
- c.s << ')';
- } else {
- if (precedence() <= level) {
- if (is_a<print_latex>(c))
- c.s << "{(";
- else
- c.s << "(";
- }
- basis.print(c, precedence());
- c.s << '^';
- if (is_a<print_latex>(c))
- c.s << '{';
- exponent.print(c, precedence());
- if (is_a<print_latex>(c))
- c.s << '}';
- if (precedence() <= level) {
- if (is_a<print_latex>(c))
- c.s << ")}";
- else
- 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 << ')';
}
}
+void power::do_print_python(const print_python & c, unsigned level) const
+{
+ 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
{
switch (inf) {
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);
}
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;
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 (exponent.has(var))
+ return false;
+ if (!exponent.info(info_flags::nonnegint))
+ return false;
+ return basis.is_polynomial(var);
}
int power::degree(const ex & s) const
{
- if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
+ 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();
- }
- return 0;
+ } 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_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
+ 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();
- }
- return 0;
+ } 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 (!basis.is_equal(s)) {
+ 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();
+ return _ex0;
} else {
// basis equal to s
- if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
+ 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();
+ return _ex1;
else
- return _ex0();
+ return _ex0;
} else {
// non-integer exponents are treated as zero
if (n == 0)
return *this;
else
- return _ex0();
+ return _ex0;
}
}
}
* - ^(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) if x is positive and c1 is real.
* - ^(^(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)
* @param level cut-off in recursive evaluation */
ex power::eval(int level) const
{
- debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
-
if ((level==1) && (flags & status_flags::evaluated))
return *this;
else if (level == -max_recursion_level)
const numeric *num_basis;
const numeric *num_exponent;
- if (is_ex_exactly_of_type(ebasis, numeric)) {
+ if (is_exactly_a<numeric>(ebasis)) {
basis_is_numerical = true;
num_basis = &ex_to<numeric>(ebasis);
}
- if (is_ex_exactly_of_type(eexponent, numeric)) {
+ if (is_exactly_a<numeric>(eexponent)) {
exponent_is_numerical = true;
num_exponent = &ex_to<numeric>(eexponent);
}
if (ebasis.is_zero())
throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
else
- return _ex1();
+ return _ex1;
}
// ^(x,1) -> x
- if (eexponent.is_equal(_ex1()))
+ if (eexponent.is_equal(_ex1))
return ebasis;
// ^(0,c1) -> 0 or exception (depending on real value of c1)
else if ((num_exponent->real()).is_negative())
throw (pole_error("power::eval(): division by zero",1));
else
- return _ex0();
+ return _ex0;
}
// ^(1,x) -> 1
- if (ebasis.is_equal(_ex1()))
- return _ex1();
+ if (ebasis.is_equal(_ex1))
+ return _ex1;
+
+ // power of a function calculated by separate rules defined for this function
+ if (is_exactly_a<function>(ebasis))
+ return ex_to<function>(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<power>(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 (exponent_is_numerical) {
// ^(^(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)) {
+ 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_ex_exactly_of_type(sub_exponent,numeric)) {
+ 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())
+ if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).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<mul>(ebasis), *num_exponent);
+ if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
+ return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
}
-
+
+ // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
+ if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
+ const numeric icont = ebasis.integer_content();
+ const numeric& lead_coeff =
+ ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
+
+ const bool canonicalizable = lead_coeff.is_integer();
+ const bool unit_normal = lead_coeff.is_pos_integer();
+
+ if (icont != *_num1_p) {
+ return (new mul(power(ebasis/icont, *num_exponent), power(icont, *num_exponent))
+ )->setflag(status_flags::dynallocated);
+ }
+
+ if (canonicalizable && (! unit_normal)) {
+ if (num_exponent->is_even()) {
+ return power(-ebasis, *num_exponent);
+ } else {
+ return (new mul(power(-ebasis, *num_exponent), *_num_1_p)
+ )->setflag(status_flags::dynallocated);
+ }
+ }
+ }
+
// ^(*(...,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<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())) {
+ 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->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) {
+ 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->overall_coeff = _ex_1;
mulp->clearflag(status_flags::evaluated);
mulp->clearflag(status_flags::hash_calculated);
return (new mul(power(*mulp,exponent),
// ^(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<matrix>(ebasis)) {
return ncmul(exvector(num_exponent->to_int(), ebasis), true);
}
}
ex power::evalf(int level) const
{
- debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
-
ex ebasis;
ex eexponent;
throw(std::runtime_error("max recursion level reached"));
} else {
ebasis = basis.evalf(level-1);
- if (!is_ex_exactly_of_type(eexponent,numeric))
+ if (!is_exactly_a<numeric>(exponent))
eexponent = exponent.evalf(level-1);
else
eexponent = exponent;
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<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);
}
-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<power>(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<numeric>(exponent).to_int()
+ > ex_to<numeric>(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<numeric>(exponent).to_int()
+ < ex_to<numeric>(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 ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
+// from mul.cpp
+extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
+
+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<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);
}
+ex power::conjugate() const
+{
+ ex newbasis = basis.conjugate();
+ ex newexponent = exponent.conjugate();
+ if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
+ return *this;
+ }
+ return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
+}
+
+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::imag_part() const
+{
+ 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
+
// protected
/** Implementation of ex::diff() for a power.
* @see ex::diff */
ex power::derivative(const symbol & s) const
{
- if (exponent.info(info_flags::real)) {
+ if (is_a<numeric>(exponent)) {
// D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
epvector newseq;
newseq.reserve(2);
- newseq.push_back(expair(basis, exponent - _ex1()));
- newseq.push_back(expair(basis.diff(s), _ex1()));
+ 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()))));
+ 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));
+ GINAC_ASSERT(is_exactly_a<power>(other));
const power &o = static_cast<const power &>(other);
int cmpval = basis.compare(o.basis);
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
+
+tinfo_t power::return_type_tinfo() const
{
return basis.return_type_tinfo();
}
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<add>(expanded_exponent)) {
const add &a = ex_to<add>(expanded_exponent);
exvector distrseq;
distrseq.reserve(a.seq.size() + 1);
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_ex_exactly_of_type(expanded_basis, add))
- distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
+ if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
+ distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
else
distrseq.push_back(power(expanded_basis, a.overall_coeff));
} else
// 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<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();
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<add>(expanded_basis), int_exponent);
+ if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
+ return expand_add(ex_to<add>(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<mul>(expanded_basis), num_exponent);
+ if (is_exactly_a<mul>(expanded_basis))
+ return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
// cannot expand further
if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
// 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
+ex power::expand_add(const add & a, int n, unsigned options) const
{
if (n==2)
- return expand_add_2(a);
-
- int m = a.nops();
- exvector sum;
- sum.reserve((n+1)*(m-1));
+ return expand_add_2(a, options);
+
+ 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 arrangements of m nonnegative integers
+ // which sum up to n. It is frequently written as C_n(m) and directly
+ // related with binomial coefficients:
+ result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
intvector k(m-1);
intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
intvector upper_limit(m-1);
int l;
-
- for (int l=0; l<m-1; 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++) {
+ 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) ||
+ 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_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
- if (is_ex_exactly_of_type(b,mul))
- term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
+ !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]), options, true));
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) ||
+ 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_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
- !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
- if (is_ex_exactly_of_type(b,mul))
- term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
+ !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]), options, true));
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++)
+ for (l=1; l<m-1; ++l)
f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
-
+
term.push_back(f);
-
- // TODO: Can we optimize this? Alex seemed to think so...
- sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
-
+
+ result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
+
// increment k[]
l = m-2;
while ((l>=0) && ((++k[l])>upper_limit[l])) {
- k[l] = 0;
+ 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[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 (int i=l+1; i<m-1; i++)
+
+ for (size_t 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 );
+
+ 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();
-
+
// 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) ||
+ 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_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));
+ !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 (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()));
+ if (c.is_equal(_ex1)) {
+ if (is_exactly_a<mul>(r)) {
+ sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ _ex1));
} else {
- sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
- _ex1()));
+ 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())));
+ if (is_exactly_a<mul>(r)) {
+ sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ ex_to<numeric>(c).power_dyn(*_num2_p)));
} else {
- sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
- ex_to<numeric>(c).power_dyn(_num2())));
+ 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_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<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
+ _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
}
}
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())));
+ sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
++i;
}
- sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
+ sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_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
+/** 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
{
- if (n.is_zero())
- return _ex1();
-
+ GINAC_ASSERT(n.is_integer());
+
+ if (n.is_zero()) {
+ return _ex1;
+ }
+
+ // Leave it to multiplication since dummy indices have to be renamed
+ if (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<numeric>((*cit).coeff).mul(n)));
+ expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
+ if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(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<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
-}
-// helper function
-
-ex sqrt(const ex & a)
-{
- return power(a,_ex1_2());
+ const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(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;
}
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff