* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2001 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 "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 "lst.h"
#include "print.h"
#include "archive.h"
-#include "debugmsg.h"
#include "utils.h"
namespace GiNaC {
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(TINFO_power) { }
//////////
-// 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);
}
void power::print(const print_context & c, unsigned level) const
{
- debugmsg("power print", LOGLEVEL_PRINT);
-
if (is_a<print_tree>(c)) {
inherited::print(c, level);
// 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))) {
+ && (is_a<symbol>(basis) || is_a<constant>(basis))) {
int exp = ex_to<numeric>(exponent).to_int();
if (exp > 0)
c.s << '(';
c.s << ')';
// <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
- } else if (exponent.compare(_num_1) == 0) {
+ } else if (exponent.is_equal(_ex_1)) {
if (is_a<print_csrc_cl_N>(c))
c.s << "recip(";
else
c.s << ')';
}
+ } else if (is_a<print_python_repr>(c)) {
+
+ c.s << class_name() << '(';
+ basis.print(c);
+ c.s << ',';
+ exponent.print(c);
+ c.s << ')';
+
} else {
- if (exponent.is_equal(_ex1_2)) {
- if (is_a<print_latex>(c))
- c.s << "\\sqrt{";
- else
- c.s << "sqrt(";
+ bool is_tex = is_a<print_latex>(c);
+
+ if (is_tex && is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
+
+ // Powers with negative numeric exponents are printed as fractions in TeX
+ c.s << "\\frac{1}{";
+ power(basis, -exponent).eval().print(c);
+ c.s << "}";
+
+ } else if (exponent.is_equal(_ex1_2)) {
+
+ // Square roots are printed in a special way
+ c.s << (is_tex ? "\\sqrt{" : "sqrt(");
basis.print(c);
- if (is_a<print_latex>(c))
- c.s << '}';
- else
- c.s << ')';
+ c.s << (is_tex ? '}' : ')');
+
} else {
- if (precedence() <= level) {
- if (is_a<print_latex>(c))
- c.s << "{(";
- else
- c.s << "(";
- }
+
+ // Ordinary output of powers using '^' or '**'
+ if (precedence() <= level)
+ c.s << (is_tex ? "{(" : "(");
basis.print(c, precedence());
- c.s << '^';
- if (is_a<print_latex>(c))
+ if (is_a<print_python>(c))
+ c.s << "**";
+ else
+ c.s << '^';
+ if (is_tex)
c.s << '{';
exponent.print(c, precedence());
- if (is_a<print_latex>(c))
+ if (is_tex)
c.s << '}';
- if (precedence() <= level) {
- if (is_a<print_latex>(c))
- c.s << ")}";
- else
- c.s << ')';
- }
+ if (precedence() <= level)
+ c.s << (is_tex ? ")}" : ")");
}
}
}
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 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;
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)
* @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);
}
// ^(^(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())
}
// ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
- if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
+ 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_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)) {
// ^(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(exponent,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
-{
- const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
- const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
+// from mul.cpp
+extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
- if (are_ex_trivially_equal(basis, subsed_basis)
- && are_ex_trivially_equal(exponent, subsed_exponent))
- return basic::subs(ls, lr, no_pattern);
- else
- return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
+ex power::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);
}
// protected
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
+unsigned 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))
+ 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));
return r.expand();
}
- 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))
+ 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_ex_exactly_of_type(expanded_basis,mul))
+ if (is_exactly_a<mul>(expanded_basis))
return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
// cannot expand further
// 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));
+
+ 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 (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_exactly_a<add>(b));
GINAC_ASSERT(!is_exactly_a<power>(b) ||
!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_ex_exactly_of_type(b,mul))
+ 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<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_ex_exactly_of_type(b,mul))
+ 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++)
+ 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((new mul(term))->setflag(status_flags::dynallocated));
+
// 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);
}
ex power::expand_add_2(const add & a) 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) {
!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)) {
+ if (c.is_equal(_ex1)) {
+ if (is_exactly_a<mul>(r)) {
sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
_ex1));
} else {
_ex1));
}
} else {
- if (is_ex_exactly_of_type(r,mul)) {
- sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
+ 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(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
+ 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;
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 and integer.
* @see power::expand */
ex power::expand_mul(const mul & m, const numeric & n) const
{
+ GINAC_ASSERT(n.is_integer());
+
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));
+ 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)));
+ 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);
+ return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
}
} // namespace GiNaC