* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2018 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 "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"
+#include "relational.h"
+#include "compiler.h"
-namespace GiNaC {
+#include <iostream>
+#include <limits>
+#include <stdexcept>
+#include <vector>
+#include <algorithm>
-GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
+namespace GiNaC {
-typedef std::vector<int> intvector;
+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).
+ print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
//////////
-// default ctor, dtor, copy ctor assignment operator and helpers
+// default constructor
//////////
-// public
-
-power::power() : basic(TINFO_power)
-{
- debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
-}
-
-// 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() { }
//////////
-// other ctors
+// other constructors
//////////
-// public
-
-power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
-{
- debugmsg("power ctor 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 ctor 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)
+void power::read_archive(const archive_node &n, lst &sym_lst)
{
- debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
+ inherited::read_archive(n, 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);
}
//////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
//////////
// public
-void power::print(std::ostream & os, unsigned upper_precedence) const
+void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) 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 << ")";
- }
+ // 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::printraw(std::ostream & os) const
+void power::do_print_dflt(const print_dflt & c, unsigned level) const
{
- debugmsg("power printraw",LOGLEVEL_PRINT);
+ if (exponent.is_equal(_ex1_2)) {
- os << "power(";
- basis.printraw(os);
- os << ",";
- exponent.printraw(os);
- os << ",hash=" << hashvalue << ",flags=" << flags << ")";
+ // Square roots are printed in a special way
+ c.s << "sqrt(";
+ basis.print(c);
+ c.s << ')';
+
+ } else
+ print_power(c, "^", "", "", level);
}
-void power::printtree(std::ostream & os, unsigned indent) const
+void power::do_print_latex(const print_latex & c, unsigned level) const
{
- debugmsg("power printtree",LOGLEVEL_PRINT);
+ 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 << '}';
- os << std::string(indent,' ') << "power: "
- << "hash=" << hashvalue
- << " (0x" << std::hex << hashvalue << std::dec << ")"
- << ", flags=" << flags << std::endl;
- basis.printtree(os, indent+delta_indent);
- exponent.printtree(os, indent+delta_indent);
+ } else
+ print_power(c, "^", "{", "}", level);
}
-static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
+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 hack is really necessary.
+ // C.f. ISO/IEC 14882:2011, section 1.9 [intro execution], paragraph 15
+ // to learn why such a parenthesation is really necessary.
if (exp == 1) {
- x.printcsrc(os, type, 0);
+ x.print(c);
} else if (exp == 2) {
- x.printcsrc(os, type, 0);
- os << "*";
- x.printcsrc(os, type, 0);
+ x.print(c);
+ c.s << "*";
+ x.print(c);
} else if (exp & 1) {
- x.printcsrc(os, 0);
- os << "*";
- print_sym_pow(os, type, x, exp-1);
+ x.print(c);
+ c.s << "*";
+ print_sym_pow(c, x, exp-1);
} else {
- os << "(";
- print_sym_pow(os, type, x, exp >> 1);
- os << ")*(";
- print_sym_pow(os, type, x, exp >> 1);
- os << ")";
+ c.s << "(";
+ print_sym_pow(c, x, exp >> 1);
+ c.s << ")*(";
+ print_sym_pow(c, x, exp >> 1);
+ c.s << ")";
}
}
-void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
+void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
+{
+ if (exponent.is_equal(_ex_1)) {
+ c.s << "recip(";
+ basis.print(c);
+ c.s << ')';
+ return;
+ }
+ c.s << "expt(";
+ basis.print(c);
+ c.s << ", ";
+ exponent.print(c);
+ c.s << ')';
+}
+
+void power::do_print_csrc(const print_csrc & c, unsigned level) const
{
- 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 (exponent.info(info_flags::integer) &&
+ (is_a<symbol>(basis) || is_a<constant>(basis))) {
+ int exp = ex_to<numeric>(exponent).to_int();
if (exp > 0)
- os << "(";
+ c.s << '(';
else {
exp = -exp;
- if (type == csrc_types::ctype_cl_N)
- os << "recip(";
- else
- os << "1.0/(";
+ c.s << "1.0/(";
}
- print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
- os << ")";
+ 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) {
- if (type == csrc_types::ctype_cl_N)
- os << "recip(";
- else
- os << "1.0/(";
- basis.bp->printcsrc(os, type, 0);
- os << ")";
+ } else if (exponent.is_equal(_ex_1)) {
+ c.s << "1.0/(";
+ basis.print(c);
+ c.s << ')';
- // Otherwise, use the pow() or expt() (CLN) functions
+ // Otherwise, use the pow() function
} 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 << ")";
+ 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 basis.info(inf) && 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 basis.info(inf) && exponent.info(info_flags::integer);
+ case info_flags::real:
+ return basis.info(inf) && exponent.info(info_flags::integer);
+ case info_flags::expanded:
+ return (flags & status_flags::expanded);
+ case info_flags::positive:
+ return basis.info(info_flags::positive) && exponent.info(info_flags::real);
+ case info_flags::nonnegative:
+ return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
+ (basis.info(info_flags::real) && exponent.info(info_flags::even));
+ case info_flags::has_indices: {
+ if (flags & status_flags::has_indices)
+ return true;
+ else if (flags & status_flags::has_no_indices)
+ return false;
+ else if (basis.info(info_flags::has_indices)) {
+ setflag(status_flags::has_indices);
+ clearflag(status_flags::has_no_indices);
+ return true;
+ } else {
+ clearflag(status_flags::has_indices);
+ setflag(status_flags::has_no_indices);
+ return false;
+ }
+ }
}
return inherited::info(inf);
}
-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;
}
-int power::degree(const symbol & s) const
+ex power::map(map_function & f) const
{
- if (is_exactly_of_type(*exponent.bp,numeric)) {
- if ((*basis.bp).compare(s)==0) {
- if (ex_to_numeric(exponent).is_integer())
- return ex_to_numeric(exponent).to_int();
- else
- return 0;
- } else
- return basis.degree(s) * ex_to_numeric(exponent).to_int();
- }
- return 0;
+ 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 dynallocate<power>(mapped_basis, mapped_exponent);
+ else
+ return *this;
}
-int power::ldegree(const symbol & s) const
+bool power::is_polynomial(const ex & var) const
{
- if (is_exactly_of_type(*exponent.bp,numeric)) {
- if ((*basis.bp).compare(s)==0) {
- if (ex_to_numeric(exponent).is_integer())
- return ex_to_numeric(exponent).to_int();
- else
- return 0;
- } else
- return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
+ if (basis.is_polynomial(var)) {
+ if (basis.has(var))
+ // basis is non-constant polynomial in var
+ return exponent.info(info_flags::nonnegint);
+ else
+ // basis is constant in var
+ return !exponent.has(var);
}
- return 0;
+ // basis is a non-polynomial function of var
+ return false;
+}
+
+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 symbol & s, int n) const
+ex power::coeff(const ex & s, int n) const
{
- if ((*basis.bp).compare(s)!=0) {
+ 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_exactly_of_type(*exponent.bp, 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();
+ 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;
}
}
}
-ex power::eval(int level) const
+/** 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) if x is positive and c1 is real.
+ * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0), case c1=1 should not happen, see below!)
+ * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
+ * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
+ * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
+ */
+ex power::eval() 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))
+ if (flags & status_flags::evaluated)
return *this;
- else if (level == -max_recursion_level)
- throw(std::runtime_error("max recursion level reached"));
-
- const ex & ebasis = level==1 ? basis : basis.eval(level-1);
- const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
-
- bool basis_is_numerical = 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);
+
+ const numeric *num_basis = nullptr;
+ const numeric *num_exponent = nullptr;
+
+ if (is_exactly_a<numeric>(basis)) {
+ num_basis = &ex_to<numeric>(basis);
}
- if (is_exactly_of_type(*eexponent.bp,numeric)) {
- exponent_is_numerical = 1;
- num_exponent = static_cast<numeric *>(eexponent.bp);
+ if (is_exactly_a<numeric>(exponent)) {
+ num_exponent = &ex_to<numeric>(exponent);
}
- // ^(x,0) -> 1 (0^0 also handled here)
- if (eexponent.is_zero()) {
- if (ebasis.is_zero())
+ // ^(x,0) -> 1 (0^0 also handled here)
+ if (exponent.is_zero()) {
+ if (basis.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()))
- return ebasis;
-
- // ^(0,c1) -> 0 or exception (depending on real value of c1)
- if (ebasis.is_zero() && exponent_is_numerical) {
+ if (exponent.is_equal(_ex1))
+ return basis;
+
+ // ^(0,c1) -> 0 or exception (depending on real value of c1)
+ if (basis.is_zero() && num_exponent) {
if ((num_exponent->real()).is_zero())
throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
else if ((num_exponent->real()).is_negative())
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 (basis_is_numerical && exponent_is_numerical) {
- // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
+ if (basis.is_equal(_ex1))
+ return _ex1;
+
+ // power of a function calculated by separate rules defined for this function
+ if (is_exactly_a<function>(basis))
+ return ex_to<function>(basis).power(exponent);
+
+ // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
+ if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
+ return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
+
+ if ( num_exponent ) {
+
+ // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
// except if c1,c2 are rational, but c1^c2 is not)
- 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 ( num_basis ) {
+ const bool basis_is_crational = num_basis->is_crational();
+ const bool exponent_is_crational = num_exponent->is_crational();
+ if (!basis_is_crational || !exponent_is_crational) {
+ // return a plain float
+ return dynallocate<numeric>(num_basis->power(*num_exponent));
}
- 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_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
+
+ 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 dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
+ if (res_bden.is_integer())
+ return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(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).
+ return pow(basis, r.div(m)) * pow(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 (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,c1),c2) -> ^(x,c1*c2)
+ // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
+ // case c1==1 should not happen, see below!)
+ if (is_exactly_a<power>(basis)) {
+ const power & sub_power = ex_to<power>(basis);
+ 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_p)).is_negative() ||
+ (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
+ return dynallocate<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()) {
- 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);
+ // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
+ if (num_exponent->is_integer() && is_exactly_a<mul>(basis)) {
+ return expand_mul(ex_to<mul>(basis), *num_exponent, false);
+ }
+
+ // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
+ if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
+ numeric icont = basis.integer_content();
+ const numeric lead_coeff =
+ ex_to<numeric>(ex_to<add>(basis).seq.begin()->coeff).div(icont);
+
+ const bool canonicalizable = lead_coeff.is_integer();
+ const bool unit_normal = lead_coeff.is_pos_integer();
+ if (canonicalizable && (! unit_normal))
+ icont = icont.mul(*_num_1_p);
+
+ if (canonicalizable && (icont != *_num1_p)) {
+ const add& addref = ex_to<add>(basis);
+ add & addp = dynallocate<add>(addref);
+ addp.clearflag(status_flags::hash_calculated);
+ addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
+ for (auto & i : addp.seq)
+ i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
+
+ const numeric c = icont.power(*num_exponent);
+ if (likely(c != *_num1_p))
+ return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
+ else
+ return dynallocate<power>(addp, *num_exponent);
+ }
+ }
+
+ // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
+ // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
+ if (is_exactly_a<mul>(basis)) {
+ GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
+ const mul & mulref = ex_to<mul>(basis);
+ 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 = dynallocate<mul>(mulref);
+ mulp.overall_coeff = _ex1;
+ mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
+ return dynallocate<mul>(dynallocate<power>(mulp, exponent),
+ dynallocate<power>(num_coeff, *num_exponent));
+ } else {
+ GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
+ if (!num_coeff.is_equal(*_num_1_p)) {
+ mul & mulp = dynallocate<mul>(mulref);
+ mulp.overall_coeff = _ex_1;
+ mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
+ return dynallocate<mul>(dynallocate<power>(mulp, exponent),
+ dynallocate<power>(abs(num_coeff), *num_exponent));
+ }
}
}
}
}
+
+ // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
+ if (num_exponent->is_pos_integer() &&
+ basis.return_type() != return_types::commutative &&
+ !is_a<matrix>(basis)) {
+ return ncmul(exvector(num_exponent->to_int(), basis));
+ }
}
-
- if (are_ex_trivially_equal(ebasis,basis) &&
- are_ex_trivially_equal(eexponent,exponent)) {
- return this->hold();
- }
- return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
- status_flags::evaluated);
+
+ return this->hold();
}
-ex power::evalf(int level) const
+ex power::evalf() const
{
- debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
-
- ex ebasis;
+ ex ebasis = basis.evalf();
ex eexponent;
- if (level==1) {
- ebasis = basis;
+ if (!is_exactly_a<numeric>(exponent))
+ eexponent = exponent.evalf();
+ else
eexponent = exponent;
- } else if (level == -max_recursion_level) {
- throw(std::runtime_error("max recursion level reached"));
- } else {
- ebasis = basis.evalf(level-1);
- if (!is_ex_exactly_of_type(eexponent,numeric))
- eexponent = exponent.evalf(level-1);
- else
- eexponent = exponent;
+
+ return dynallocate<power>(ebasis, eexponent);
+}
+
+ex power::evalm() const
+{
+ const ex ebasis = basis.evalm();
+ const ex eexponent = exponent.evalm();
+ if (is_a<matrix>(ebasis)) {
+ if (is_exactly_a<numeric>(eexponent)) {
+ return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
+ }
+ }
+ return dynallocate<power>(ebasis, eexponent);
+}
+
+bool power::has(const ex & other, unsigned options) const
+{
+ 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) > ex_to<numeric>(other.op(1)) &&
+ basis.match(other.op(0)))
+ return true;
+ if (exponent.info(info_flags::negint) &&
+ other.op(1).info(info_flags::negint) &&
+ ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
+ basis.match(other.op(0)))
+ return true;
+ return basic::has(other, options);
+}
+
+// from mul.cpp
+extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
+
+ex power::subs(const exmap & m, unsigned options) const
+{
+ const ex &subsed_basis = basis.subs(m, options);
+ const ex &subsed_exponent = exponent.subs(m, options);
+
+ if (!are_ex_trivially_equal(basis, subsed_basis)
+ || !are_ex_trivially_equal(exponent, subsed_exponent))
+ return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
+
+ if (!(options & subs_options::algebraic))
+ return subs_one_level(m, options);
+
+ for (auto & it : m) {
+ int nummatches = std::numeric_limits<int>::max();
+ exmap repls;
+ if (tryfactsubs(*this, it.first, nummatches, repls)) {
+ ex anum = it.second.subs(repls, subs_options::no_pattern);
+ ex aden = it.first.subs(repls, subs_options::no_pattern);
+ ex result = (*this) * pow(anum/aden, nummatches);
+ return (ex_to<basic>(result)).subs_one_level(m, options);
+ }
}
- return power(ebasis,eexponent);
+ return subs_one_level(m, options);
}
-ex power::subs(const lst & ls, const lst & lr) const
+ex power::eval_ncmul(const exvector & v) const
{
- const ex & subsed_basis=basis.subs(ls,lr);
- const ex & subsed_exponent=exponent.subs(ls,lr);
+ return inherited::eval_ncmul(v);
+}
- if (are_ex_trivially_equal(basis,subsed_basis)&&
- are_ex_trivially_equal(exponent,subsed_exponent)) {
+ex power::conjugate() const
+{
+ // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
+ // branch cut which runs along the negative real axis.
+ if (basis.info(info_flags::positive)) {
+ ex newexponent = exponent.conjugate();
+ if (are_ex_trivially_equal(exponent, newexponent)) {
+ return *this;
+ }
+ return dynallocate<power>(basis, newexponent);
+ }
+ if (exponent.info(info_flags::integer)) {
+ ex newbasis = basis.conjugate();
+ if (are_ex_trivially_equal(basis, newbasis)) {
+ return *this;
+ }
+ return dynallocate<power>(newbasis, exponent);
+ }
+ return conjugate_function(*this).hold();
+}
+
+ex power::real_part() const
+{
+ // basis == a+I*b, exponent == c+I*d
+ const ex a = basis.real_part();
+ const ex c = exponent.real_part();
+ if (basis.is_equal(a) && exponent.is_equal(c) &&
+ (a.info(info_flags::nonnegative) || c.info(info_flags::integer))) {
+ // Re(a^c)
return *this;
}
-
- return power(subsed_basis, subsed_exponent);
+
+ const ex b = basis.imag_part();
+ if (exponent.info(info_flags::integer)) {
+ // Re((a+I*b)^c) w/ c ∈ ℤ
+ long N = ex_to<numeric>(c).to_long();
+ // Use real terms in Binomial expansion to construct
+ // Re(expand(pow(a+I*b, N))).
+ long NN = N > 0 ? N : -N;
+ ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
+ ex result = 0;
+ for (long n = 0; n <= NN; n += 2) {
+ ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
+ if (n % 4 == 0) {
+ result += term; // sign: I^n w/ n == 4*m
+ } else {
+ result -= term; // sign: I^n w/ n == 4*m+2
+ }
+ }
+ return result;
+ }
+
+ // Re((a+I*b)^(c+I*d))
+ const ex d = exponent.imag_part();
+ return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
}
-ex power::simplify_ncmul(const exvector & v) const
+ex power::imag_part() const
{
- return inherited::simplify_ncmul(v);
+ // basis == a+I*b, exponent == c+I*d
+ const ex a = basis.real_part();
+ const ex c = exponent.real_part();
+ if (basis.is_equal(a) && exponent.is_equal(c) &&
+ (a.info(info_flags::nonnegative) || c.info(info_flags::integer))) {
+ // Im(a^c)
+ return 0;
+ }
+
+ const ex b = basis.imag_part();
+ if (exponent.info(info_flags::integer)) {
+ // Im((a+I*b)^c) w/ c ∈ ℤ
+ long N = ex_to<numeric>(c).to_long();
+ // Use imaginary terms in Binomial expansion to construct
+ // Im(expand(pow(a+I*b, N))).
+ long p = N > 0 ? 1 : 3; // modulus for positive sign
+ long NN = N > 0 ? N : -N;
+ ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
+ ex result = 0;
+ for (long n = 1; n <= NN; n += 2) {
+ ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
+ if (n % 4 == p) {
+ result += term; // sign: I^n w/ n == 4*m+p
+ } else {
+ result -= term; // sign: I^n w/ n == 4*m+2+p
+ }
+ }
+ return result;
+ }
+
+ // Im((a+I*b)^(c+I*d))
+ const ex d = exponent.imag_part();
+ return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
}
// protected
* @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()));
- return mul(newseq, exponent);
+ const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
+ return dynallocate<mul>(std::move(newseq), exponent);
} else {
// D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
- return mul(*this,
- add(mul(exponent.diff(s), log(basis)),
- mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
+ return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
}
}
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));
+ GINAC_ASSERT(is_exactly_a<power>(other));
+ const power &o = static_cast<const power &>(other);
- int cmpval;
- cmpval=basis.compare(o.basis);
- if (cmpval==0) {
+ int cmpval = basis.compare(o.basis);
+ if (cmpval)
+ return cmpval;
+ else
return exponent.compare(o.exponent);
- }
- return cmpval;
}
-unsigned power::return_type(void) const
+unsigned power::return_type() const
{
return basis.return_type();
}
-
-unsigned power::return_type_tinfo(void) const
+
+return_type_t power::return_type_tinfo() const
{
return basis.return_type_tinfo();
}
ex power::expand(unsigned options) const
{
- if (flags & status_flags::expanded)
+ if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
+ // A special case worth optimizing.
+ setflag(status_flags::expanded);
return *this;
-
- ex expanded_basis = basis.expand(options);
- ex expanded_exponent = exponent.expand(options);
+ }
+
+ // (x*p)^c -> x^c * p^c, if p>0
+ // makes sense before expanding the basis
+ if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
+ const mul &m = ex_to<mul>(basis);
+ exvector prodseq;
+ epvector powseq;
+ prodseq.reserve(m.seq.size() + 1);
+ powseq.reserve(m.seq.size() + 1);
+ bool possign = true;
+
+ // search for positive/negative factors
+ for (auto & cit : m.seq) {
+ ex e=m.recombine_pair_to_ex(cit);
+ if (e.info(info_flags::positive))
+ prodseq.push_back(pow(e, exponent).expand(options));
+ else if (e.info(info_flags::negative)) {
+ prodseq.push_back(pow(-e, exponent).expand(options));
+ possign = !possign;
+ } else
+ powseq.push_back(cit);
+ }
+
+ // take care on the numeric coefficient
+ ex coeff=(possign? _ex1 : _ex_1);
+ if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
+ prodseq.push_back(pow(m.overall_coeff, exponent));
+ else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
+ prodseq.push_back(pow(-m.overall_coeff, exponent));
+ else
+ coeff *= m.overall_coeff;
+
+ // If positive/negative factors are found, then extract them.
+ // In either case we set a flag to avoid the second run on a part
+ // which does not have positive/negative terms.
+ if (prodseq.size() > 0) {
+ ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
+ ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
+ return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
+ } else
+ ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
+ }
+
+ const ex expanded_basis = basis.expand(options);
+ const ex expanded_exponent = exponent.expand(options);
// x^(a+b) -> x^a * x^b
- if (is_ex_exactly_of_type(expanded_exponent, add)) {
- const add &a = ex_to_add(expanded_exponent);
+ 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++;
+ for (auto & cit : a.seq) {
+ distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
}
// Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
- if (ex_to_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 (ex_to<numeric>(a.overall_coeff).is_integer()) {
+ const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
+ long 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, options));
else
- distrseq.push_back(power(expanded_basis, a.overall_coeff));
+ distrseq.push_back(pow(expanded_basis, a.overall_coeff));
} else
- distrseq.push_back(power(expanded_basis, a.overall_coeff));
+ distrseq.push_back(pow(expanded_basis, a.overall_coeff));
// Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
- ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
- return r.expand();
+ ex r = dynallocate<mul>(distrseq);
+ return r.expand(options);
}
- if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
- !ex_to_numeric(expanded_exponent).is_integer()) {
+ 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 | status_flags::expanded);
+ return dynallocate<power>(expanded_basis, expanded_exponent).setflag(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();
+ const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
+ long int_exponent = num_exponent.to_long();
// (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))
return this->hold();
else
- return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
+ return dynallocate<power>(expanded_basis, expanded_exponent).setflag(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
+ex power::expand_add(const add & a, long n, unsigned options)
{
+ // The special case power(+(x,...y;x),2) can be optimized better.
if (n==2)
- return expand_add_2(a);
-
- 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;
+ return expand_add_2(a, options);
+
+ // method:
+ //
+ // Consider base as the sum of all symbolic terms and the overall numeric
+ // coefficient and apply the binomial theorem:
+ // S = power(+(x,...,z;c),n)
+ // = power(+(+(x,...,z;0);c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
+ // The multinomial theorem is computed by an outer loop over all
+ // partitions of the exponent and an inner loop over all compositions of
+ // that partition. This method makes the expansion a combinatorial
+ // problem and allows us to directly construct the expanded sum and also
+ // to re-use the multinomial coefficients (since they depend only on the
+ // partition, not on the composition).
+ //
+ // multinomial power(+(x,y,z;0),3) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
+ // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
+ // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
+ // => (x + y + z)^3 =
+ // x^3 + y^3 + z^3
+ // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
+ // + 6*x*y*z
+ //
+ // multinomial power(+(x,y,z;0),4) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
+ // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
+ // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
+ // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
+ // (no [1,1,1,1] partition since it has too many parts)
+ // => (x + y + z)^4 =
+ // x^4 + y^4 + z^4
+ // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
+ // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
+ // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
+ //
+ // Summary:
+ // r = 0
+ // for k from 0 to n:
+ // f = c^(n-k)*binomial(n,k)
+ // for p in all partitions of n with m parts (including zero parts):
+ // h = f * multinomial coefficient of p
+ // for c in all compositions of p:
+ // t = 1
+ // for e in all elements of c:
+ // t = t * a[e]^e
+ // r = r + h*t
+ // return r
+
+ epvector result;
+ // The number of terms will be the number of combinatorial compositions,
+ // i.e. the number of unordered arrangements of m nonnegative integers
+ // which sum up to n. It is frequently written as C_n(m) and directly
+ // related with binomial coefficients: binomial(n+m-1,m-1).
+ size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
}
-
- 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() ||
- !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])));
- 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() ||
- !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])));
- 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);
-
- /*
- 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--;
+ result.reserve(result_size);
+
+ // Iterate over all terms in binomial expansion of
+ // S = power(+(x,...,z;c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ for (int k = 1; k <= n; ++k) {
+ numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
+ if (a.overall_coeff.is_zero()) {
+ // degenerate case with zero overall_coeff:
+ // apply multinomial theorem directly to power(+(x,...z;0),n)
+ binomial_coefficient = 1;
+ if (k < n) {
+ continue;
+ }
+ } else {
+ binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
}
- 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];
+
+ // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
+ // Iterate over all partitions of k with exactly as many parts as
+ // there are symbolic terms in the basis (including zero parts).
+ partition_with_zero_parts_generator partitions(k, a.seq.size());
+ do {
+ const std::vector<unsigned>& partition = partitions.get();
+ // All monomials of this partition have the same number of terms and the same coefficient.
+ const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
+ const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
+
+ // Iterate over all compositions of the current partition.
+ composition_generator compositions(partition);
+ do {
+ const std::vector<unsigned>& exponent = compositions.get();
+ epvector monomial;
+ monomial.reserve(msize);
+ numeric factor = coeff;
+ for (unsigned i = 0; i < exponent.size(); ++i) {
+ const ex & r = a.seq[i].rest;
+ GINAC_ASSERT(!is_exactly_a<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));
+ GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
+ const numeric & c = ex_to<numeric>(a.seq[i].coeff);
+ if (exponent[i] == 0) {
+ // optimize away
+ } else if (exponent[i] == 1) {
+ // optimized
+ monomial.push_back(expair(r, _ex1));
+ if (c != *_num1_p)
+ factor = factor.mul(c);
+ } else { // general case exponent[i] > 1
+ monomial.push_back(expair(r, exponent[i]));
+ if (c != *_num1_p)
+ factor = factor.mul(c.power(exponent[i]));
+ }
+ }
+ result.push_back(expair(mul(std::move(monomial)).expand(options), factor));
+ } while (compositions.next());
+ } while (partitions.next());
+ }
+
+ GINAC_ASSERT(result.size() == result_size);
+ if (a.overall_coeff.is_zero()) {
+ return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
+ } else {
+ return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
}
- return (new add(sum))->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)
{
- epvector sum;
- unsigned a_nops = a.nops();
- sum.reserve((a_nops*(a_nops+1))/2);
- epvector::const_iterator last = a.seq.end();
-
+ epvector result;
+ size_t result_size = (a.nops() * (a.nops()+1)) / 2;
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
+ }
+ result.reserve(result_size);
+
+ auto 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;
+ for (auto 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));
+ 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 (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)) {
+ result.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()));
+ result.push_back(expair(dynallocate<power>(r, _ex2),
+ _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)) {
+ result.push_back(expair(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())));
+ result.push_back(expair(dynallocate<power>(r, _ex2),
+ 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))));
+
+ for (auto cit1=cit0+1; cit1!=last; ++cit1) {
+ const ex & r1 = cit1->rest;
+ const ex & c1 = cit1->coeff;
+ result.push_back(expair(mul(r,r1).expand(options),
+ _num2_p->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)
+ // second part: add terms coming from overall_coeff (if != 0)
if (!a.overall_coeff.is_zero()) {
- 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()));
+ for (auto & i : a.seq)
+ result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
}
-
- 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_zero())
- 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;
+ GINAC_ASSERT(result.size() == result_size);
+
+ if (a.overall_coeff.is_zero()) {
+ return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
+ } else {
+ return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
}
- 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
+/** Expand factors of m in m^n where m is a mul and n is an integer.
+ * @see power::expand */
+ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
{
- // obsolete
-
- exvector distrseq;
- epvector splitseq;
+ GINAC_ASSERT(n.is_integer());
- const add & addref=static_cast<const add &>(*basis.bp);
+ if (n.is_zero()) {
+ return _ex1;
+ }
- 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)));
+ // do not bother to rename indices if there are no any.
+ if (!(options & expand_options::expand_rename_idx) &&
+ m.info(info_flags::has_indices))
+ options |= expand_options::expand_rename_idx;
+ // Leave it to multiplication since dummy indices have to be renamed
+ if ((options & expand_options::expand_rename_idx) &&
+ (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
+ ex result = m;
+ exvector va = get_all_dummy_indices(m);
+ sort(va.begin(), va.end(), ex_is_less());
+
+ for (int i=1; i < n.to_int(); i++)
+ result *= rename_dummy_indices_uniquely(va, m);
+ return result;
}
- 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);
+ epvector distrseq;
+ distrseq.reserve(m.seq.size());
+ bool need_reexpand = false;
+
+ for (auto & cit : m.seq) {
+ 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);
+ }
- return ex(mul(rest_power,basis),0).
- expand(options | expand_options::internal_do_not_expand_mul_operands);
+ const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
+ if (need_reexpand)
+ return ex(result).expand(options);
+ if (from_expand)
+ return result.setflag(status_flags::expanded);
+ return result;
}
-*/
-//////////
-// static member variables
-//////////
-
-// protected
-
-unsigned power::precedence = 60;
-
-// helper function
-
-ex sqrt(const ex & a)
-{
- return power(a,_ex1_2());
-}
+GINAC_BIND_UNARCHIVER(power);
} // namespace GiNaC