* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2002 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 "constant.h"
#include "inifcns.h" // for log() in power::derivative()
#include "matrix.h"
+#include "indexed.h"
#include "symbol.h"
#include "print.h"
#include "archive.h"
// 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 << '(';
} 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());
if (is_a<print_python>(c))
c.s << "**";
else
c.s << '^';
- if (is_a<print_latex>(c))
+ 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 ? ")}" : ")");
}
}
}
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_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
if (basis.is_equal(s))
return ex_to<numeric>(exponent).to_int();
else
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_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
if (basis.is_equal(s))
return ex_to<numeric>(exponent).to_int();
else
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;
// 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 int m = a.nops();
+ exvector result;
+ // The number of terms will be the number of combinatorial compositions,
+ // i.e. the number of unordered arrangement of m nonnegative integers
+ // which sum up to n. It is frequently written as C_n(m) and directly
+ // related with binomial coefficients:
+ result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
intvector k(m-1);
intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
intvector upper_limit(m-1);
int l;
-
- for (int l=0; l<m-1; l++) {
+
+ for (int 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) ||
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) ||
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 (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++)
+
+ for (int i=l+1; i<m-1; ++i)
upper_limit[i] = n-k_cum[i-1];
}
- return (new add(sum))->setflag(status_flags::dynallocated |
- status_flags::expanded );
+
+ return (new add(result))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
}
unsigned a_nops = a.nops();
sum.reserve((a_nops*(a_nops+1))/2);
epvector::const_iterator last = a.seq.end();
-
+
// power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
// first part: ignore overall_coeff and expand other terms
for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
}
} else {
if (is_ex_exactly_of_type(r,mul)) {
- sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
+ 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)));
}
}
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_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)));
+ 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
git://www.ginac.de / ginac.git / blobdiff