*/
/*
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2000 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 "numeric.h"
#include "power.h"
#include "relational.h"
-#include "series.h"
+#include "pseries.h"
#include "symbol.h"
+#include "utils.h"
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
// If comparing expressions (ex::compare()) is fast, you can set this to 1.
// Some routines like quo(), rem() and gcd() will then return a quick answer
x = static_cast<symbol *>(e.bp);
return true;
} else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- for (int i=0; i<e.nops(); i++)
+ for (unsigned i=0; i<e.nops(); i++)
if (get_first_symbol(e.op(i), x))
return true;
} else if (is_ex_exactly_of_type(e, power)) {
if (is_ex_exactly_of_type(e, symbol)) {
add_symbol(static_cast<symbol *>(e.bp), v);
} else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- for (int i=0; i<e.nops(); i++)
+ for (unsigned i=0; i<e.nops(); i++)
collect_symbols(e.op(i), v);
} else if (is_ex_exactly_of_type(e, power)) {
collect_symbols(e.op(0), v);
{
if (e.info(info_flags::rational))
return lcm(ex_to_numeric(e).denom(), l);
- else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- numeric c = numONE();
- for (int i=0; i<e.nops(); i++) {
+ else if (is_ex_exactly_of_type(e, add)) {
+ numeric c = _num1();
+ for (unsigned i=0; i<e.nops(); i++)
c = lcmcoeff(e.op(i), c);
- }
+ return lcm(c, l);
+ } else if (is_ex_exactly_of_type(e, mul)) {
+ numeric c = _num1();
+ for (unsigned i=0; i<e.nops(); i++)
+ c *= lcmcoeff(e.op(i), _num1());
return lcm(c, l);
} else if (is_ex_exactly_of_type(e, power))
- return lcmcoeff(e.op(0), l);
+ return pow(lcmcoeff(e.op(0), l), ex_to_numeric(e.op(1)));
return l;
}
/** Compute LCM of denominators of coefficients of a polynomial.
* Given a polynomial with rational coefficients, this function computes
* the LCM of the denominators of all coefficients. This can be used
- * To bring a polynomial from Q[X] to Z[X].
+ * to bring a polynomial from Q[X] to Z[X].
*
- * @param e multivariate polynomial
+ * @param e multivariate polynomial (need not be expanded)
* @return LCM of denominators of coefficients */
static numeric lcm_of_coefficients_denominators(const ex &e)
{
- return lcmcoeff(e.expand(), numONE());
+ return lcmcoeff(e, _num1());
+}
+
+/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
+ * determined LCM of the coefficient's denominators.
+ *
+ * @param e multivariate polynomial (need not be expanded)
+ * @param lcm LCM to multiply in */
+
+static ex multiply_lcm(const ex &e, const numeric &lcm)
+{
+ if (is_ex_exactly_of_type(e, mul)) {
+ ex c = _ex1();
+ numeric lcm_accum = _num1();
+ for (unsigned i=0; i<e.nops(); i++) {
+ numeric op_lcm = lcmcoeff(e.op(i), _num1());
+ c *= multiply_lcm(e.op(i), op_lcm);
+ lcm_accum *= op_lcm;
+ }
+ c *= lcm / lcm_accum;
+ return c;
+ } else if (is_ex_exactly_of_type(e, add)) {
+ ex c = _ex0();
+ for (unsigned i=0; i<e.nops(); i++)
+ c += multiply_lcm(e.op(i), lcm);
+ return c;
+ } else if (is_ex_exactly_of_type(e, power)) {
+ return pow(multiply_lcm(e.op(0), lcm.power(ex_to_numeric(e.op(1)).inverse())), e.op(1));
+ } else
+ return e * lcm;
}
numeric basic::integer_content(void) const
{
- return numONE();
+ return _num1();
}
numeric numeric::integer_content(void) const
{
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
- numeric c = numZERO();
+ numeric c = _num0();
while (it != itend) {
GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
return a / b;
#if FAST_COMPARE
if (a.is_equal(b))
- return exONE();
+ return _ex1();
#endif
if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));
// Polynomial long division
- ex q = exZERO();
+ ex q = _ex0();
ex r = a.expand();
if (r.is_zero())
return r;
throw(std::overflow_error("rem: division by zero"));
if (is_ex_exactly_of_type(a, numeric)) {
if (is_ex_exactly_of_type(b, numeric))
- return exZERO();
+ return _ex0();
else
return b;
}
#if FAST_COMPARE
if (a.is_equal(b))
- return exZERO();
+ return _ex0();
#endif
if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));
throw(std::overflow_error("prem: division by zero"));
if (is_ex_exactly_of_type(a, numeric)) {
if (is_ex_exactly_of_type(b, numeric))
- return exZERO();
+ return _ex0();
else
return b;
}
if (bdeg <= rdeg) {
blcoeff = eb.coeff(x, bdeg);
if (bdeg == 0)
- eb = exZERO();
+ eb = _ex0();
else
eb -= blcoeff * power(x, bdeg);
} else
- blcoeff = exONE();
+ blcoeff = _ex1();
int delta = rdeg - bdeg + 1, i = 0;
while (rdeg >= bdeg && !r.is_zero()) {
ex rlcoeff = r.coeff(x, rdeg);
ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
if (rdeg == 0)
- r = exZERO();
+ r = _ex0();
else
r -= rlcoeff * power(x, rdeg);
r = (blcoeff * r).expand() - term;
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
- q = exZERO();
+ q = _ex0();
if (b.is_zero())
throw(std::overflow_error("divide: division by zero"));
if (is_ex_exactly_of_type(b, numeric)) {
return false;
#if FAST_COMPARE
if (a.is_equal(b)) {
- q = exONE();
+ q = _ex1();
return true;
}
#endif
* @see get_symbol_stats, heur_gcd */
static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var)
{
- q = exZERO();
+ q = _ex0();
if (b.is_zero())
throw(std::overflow_error("divide_in_z: division by zero"));
- if (b.is_equal(exONE())) {
+ if (b.is_equal(_ex1())) {
q = a;
return true;
}
}
#if FAST_COMPARE
if (a.is_equal(b)) {
- q = exONE();
+ q = _ex1();
return true;
}
#endif
// Compute values at evaluation points 0..adeg
vector<numeric> alpha; alpha.reserve(adeg + 1);
exvector u; u.reserve(adeg + 1);
- numeric point = numZERO();
+ numeric point = _num0();
ex c;
for (i=0; i<=adeg; i++) {
ex bs = b.subs(*x == point);
while (bs.is_zero()) {
- point += numONE();
+ point += _num1();
bs = b.subs(*x == point);
}
if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
return false;
alpha.push_back(point);
u.push_back(c);
- point += numONE();
+ point += _num1();
}
// Compute inverses
{
ex c = expand().lcoeff(x);
if (is_ex_exactly_of_type(c, numeric))
- return c < exZERO() ? exMINUSONE() : exONE();
+ return c < _ex0() ? _ex_1() : _ex1();
else {
const symbol *y;
if (get_first_symbol(c, y))
ex ex::content(const symbol &x) const
{
if (is_zero())
- return exZERO();
+ return _ex0();
if (is_ex_exactly_of_type(*this, numeric))
return info(info_flags::negative) ? -*this : *this;
ex e = expand();
if (e.is_zero())
- return exZERO();
+ return _ex0();
// First, try the integer content
ex c = e.integer_content();
int ldeg = e.ldegree(x);
if (deg == ldeg)
return e.lcoeff(x) / e.unit(x);
- c = exZERO();
+ c = _ex0();
for (int i=ldeg; i<=deg; i++)
c = gcd(e.coeff(x, i), c, NULL, NULL, false);
return c;
ex ex::primpart(const symbol &x) const
{
if (is_zero())
- return exZERO();
+ return _ex0();
if (is_ex_exactly_of_type(*this, numeric))
- return exONE();
+ return _ex1();
ex c = content(x);
if (c.is_zero())
- return exZERO();
+ return _ex0();
ex u = unit(x);
if (is_ex_exactly_of_type(c, numeric))
return *this / (c * u);
ex ex::primpart(const symbol &x, const ex &c) const
{
if (is_zero())
- return exZERO();
+ return _ex0();
if (c.is_zero())
- return exZERO();
+ return _ex0();
if (is_ex_exactly_of_type(*this, numeric))
- return exONE();
+ return _ex1();
ex u = unit(x);
if (is_ex_exactly_of_type(c, numeric))
static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
{
+//clog << "sr_gcd(" << a << "," << b << ")\n";
+
// Sort c and d so that c has higher degree
ex c, d;
int adeg = a.degree(*x), bdeg = b.degree(*x);
d = d.primpart(*x, cont_d);
// First element of subresultant sequence
- ex r = exZERO(), ri = exONE(), psi = exONE();
+ ex r = _ex0(), ri = _ex1(), psi = _ex1();
int delta = cdeg - ddeg;
for (;;) {
numeric basic::max_coefficient(void) const
{
- return numONE();
+ return _num1();
}
numeric numeric::max_coefficient(void) const
ex numeric::smod(const numeric &xi) const
{
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
return GiNaC::smod(*this, xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
return ::smod(*this, xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
}
ex add::smod(const numeric &xi) const
epvector::const_iterator itend = seq.end();
while (it != itend) {
GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
if (!coeff.is_zero())
newseq.push_back(expair(it->rest, coeff));
it++;
}
GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
}
#endif // def DO_GINAC_ASSERT
mul * mulcopyp=new mul(*this);
GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
mulcopyp->clearflag(status_flags::evaluated);
mulcopyp->clearflag(status_flags::hash_calculated);
return mulcopyp->setflag(status_flags::dynallocated);
}
-/** Exception thrown by heur_gcd() to signal failure */
+/** Exception thrown by heur_gcd() to signal failure. */
class gcdheu_failed {};
/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
{
+//clog << "heur_gcd(" << a << "," << b << ")\n";
+
+ // Trivial cases
if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
numeric rg;
numeric mp = p.max_coefficient(), mq = q.max_coefficient();
numeric xi;
if (mp > mq)
- xi = mq * numTWO() + numTWO();
+ xi = mq * _num2() + _num2();
else
- xi = mp * numTWO() + numTWO();
+ xi = mp * _num2() + _num2();
// 6 tries maximum
for (int t=0; t<6; t++) {
- if (xi.int_length() * maxdeg > 50000)
+ if (xi.int_length() * maxdeg > 100000) {
+//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
throw gcdheu_failed();
+ }
// Apply evaluation homomorphism and calculate GCD
ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand();
if (!is_ex_exactly_of_type(gamma, fail)) {
// Reconstruct polynomial from GCD of mapped polynomials
- ex g = exZERO();
+ ex g = _ex0();
numeric rxi = xi.inverse();
for (int i=0; !gamma.is_zero(); i++) {
ex gi = gamma.smod(xi);
if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
g *= gc;
ex lc = g.lcoeff(*x);
- if (is_ex_exactly_of_type(lc, numeric) && lc.compare(exZERO()) < 0)
+ if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
return -g;
else
return g;
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
{
+//clog << "gcd(" << a << "," << b << ")\n";
+
+ // Partially factored cases (to avoid expanding large expressions)
+ if (is_ex_exactly_of_type(a, mul)) {
+ if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
+ goto factored_b;
+factored_a:
+ ex g = _ex1();
+ ex acc_ca = _ex1();
+ ex part_b = b;
+ for (unsigned i=0; i<a.nops(); i++) {
+ ex part_ca, part_cb;
+ g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
+ acc_ca *= part_ca;
+ part_b = part_cb;
+ }
+ if (ca)
+ *ca = acc_ca;
+ if (cb)
+ *cb = part_b;
+ return g;
+ } else if (is_ex_exactly_of_type(b, mul)) {
+ if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
+ goto factored_a;
+factored_b:
+ ex g = _ex1();
+ ex acc_cb = _ex1();
+ ex part_a = a;
+ for (unsigned i=0; i<b.nops(); i++) {
+ ex part_ca, part_cb;
+ g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
+ acc_cb *= part_cb;
+ part_a = part_ca;
+ }
+ if (ca)
+ *ca = part_a;
+ if (cb)
+ *cb = acc_cb;
+ return g;
+ }
+
// Some trivial cases
ex aex = a.expand(), bex = b.expand();
if (aex.is_zero()) {
if (ca)
- *ca = exZERO();
+ *ca = _ex0();
if (cb)
- *cb = exONE();
+ *cb = _ex1();
return b;
}
if (bex.is_zero()) {
if (ca)
- *ca = exONE();
+ *ca = _ex1();
if (cb)
- *cb = exZERO();
+ *cb = _ex0();
return a;
}
- if (aex.is_equal(exONE()) || bex.is_equal(exONE())) {
+ if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) {
if (ca)
*ca = a;
if (cb)
*cb = b;
- return exONE();
+ return _ex1();
}
#if FAST_COMPARE
if (a.is_equal(b)) {
if (ca)
- *ca = exONE();
+ *ca = _ex1();
if (cb)
- *cb = exONE();
+ *cb = _ex1();
return a;
}
#endif
g = *new ex(fail());
}
if (is_ex_exactly_of_type(g, fail)) {
-//clog << "heuristics failed\n";
+//clog << "heuristics failed" << endl;
g = sr_gcd(aex, bex, x);
if (ca)
divide(aex, g, *ca, false);
ex lcm(const ex &a, const ex &b, bool check_args)
{
if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
- return gcd(ex_to_numeric(a), ex_to_numeric(b));
+ return lcm(ex_to_numeric(a), ex_to_numeric(b));
if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
return b;
if (b.is_zero())
return a;
- if (a.is_equal(exONE()) || b.is_equal(exONE()))
- return exONE();
+ if (a.is_equal(_ex1()) || b.is_equal(_ex1()))
+ return _ex1();
if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric))
return gcd(ex_to_numeric(a), ex_to_numeric(b));
if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
ex sqrfree(const ex &a, const symbol &x)
{
int i = 1;
- ex res = exONE();
+ ex res = _ex1();
ex b = a.diff(x);
ex c = univariate_gcd(a, b, x);
ex w;
- if (c.is_equal(exONE())) {
+ if (c.is_equal(_ex1())) {
w = a;
} else {
w = quo(a, c, x);
* Normal form of rational functions
*/
-// Create a symbol for replacing the expression "e" (or return a previously
-// assigned symbol). The symbol is appended to sym_list and returned, the
-// expression is appended to repl_list.
+/*
+ * Note: The internal normal() functions (= basic::normal() and overloaded
+ * functions) all return lists of the form {numerator, denominator}. This
+ * is to get around mul::eval()'s automatic expansion of numeric coefficients.
+ * E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
+ * the information that (a+b) is the numerator and 3 is the denominator.
+ */
+
+/** Create a symbol for replacing the expression "e" (or return a previously
+ * assigned symbol). The symbol is appended to sym_list and returned, the
+ * expression is appended to repl_list.
+ * @see ex::normal */
static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
{
// Expression already in repl_lst? Then return the assigned symbol
- for (int i=0; i<repl_lst.nops(); i++)
+ for (unsigned i=0; i<repl_lst.nops(); i++)
if (repl_lst.op(i).is_equal(e))
return sym_lst.op(i);
* @see ex::normal */
ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- return replace_with_symbol(*this, sym_lst, repl_lst);
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
-/** Implementation of ex::normal() for symbols. This returns the unmodifies symbol.
+/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
* @see ex::normal */
ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- return *this;
+ return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
}
* @see ex::normal */
ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- if (is_real())
- if (is_rational())
- return *this;
- else
- return replace_with_symbol(*this, sym_lst, repl_lst);
- else { // complex
- numeric re = real(), im = imag();
- ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
- ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
- return re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
- }
-}
+ numeric num = numer();
+ ex numex = num;
+
+ if (num.is_real()) {
+ if (!num.is_integer())
+ numex = replace_with_symbol(numex, sym_lst, repl_lst);
+ } else { // complex
+ numeric re = num.real(), im = num.imag();
+ ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
+ ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
+ numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
+ }
+ // Denominator is always a real integer (see numeric::denom())
+ return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
+}
-/*
- * Helper function for fraction cancellation (returns cancelled fraction n/d)
- */
+/** Fraction cancellation.
+ * @param n numerator
+ * @param d denominator
+ * @return cancelled fraction {n, d} as a list */
static ex frac_cancel(const ex &n, const ex &d)
{
ex num = n;
ex den = d;
- ex pre_factor = exONE();
+ numeric pre_factor = _num1();
+
+//clog << "frac_cancel num = " << num << ", den = " << den << endl;
// Handle special cases where numerator or denominator is 0
if (num.is_zero())
- return exZERO();
+ return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
if (den.expand().is_zero())
throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
- // More special cases
- if (is_ex_exactly_of_type(den, numeric))
- return num / den;
- if (num.is_zero())
- return exZERO();
-
// Bring numerator and denominator to Z[X] by multiplying with
// LCM of all coefficients' denominators
- ex num_lcm = lcm_of_coefficients_denominators(num);
- ex den_lcm = lcm_of_coefficients_denominators(den);
- num *= num_lcm;
- den *= den_lcm;
+ numeric num_lcm = lcm_of_coefficients_denominators(num);
+ numeric den_lcm = lcm_of_coefficients_denominators(den);
+ num = multiply_lcm(num, num_lcm);
+ den = multiply_lcm(den, den_lcm);
pre_factor = den_lcm / num_lcm;
// Cancel GCD from numerator and denominator
ex cnum, cden;
- if (gcd(num, den, &cnum, &cden, false) != exONE()) {
+ if (gcd(num, den, &cnum, &cden, false) != _ex1()) {
num = cnum;
den = cden;
}
// as defined by get_first_symbol() is made positive)
const symbol *x;
if (get_first_symbol(den, x)) {
- if (den.unit(*x).compare(exZERO()) < 0) {
- num *= exMINUSONE();
- den *= exMINUSONE();
+ GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric));
+ if (ex_to_numeric(den.unit(*x)).is_negative()) {
+ num *= _ex_1();
+ den *= _ex_1();
}
}
- return pre_factor * num / den;
+
+ // Return result as list
+//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
+ return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
}
* @see ex::normal */
ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- // Normalize and expand children
+ // Normalize and expand children, chop into summands
exvector o;
o.reserve(seq.size()+1);
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
+
+ // Normalize and expand child
ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
- if (is_ex_exactly_of_type(n, add)) {
- epvector::const_iterator bit = (static_cast<add *>(n.bp))->seq.begin(), bitend = (static_cast<add *>(n.bp))->seq.end();
+
+ // If numerator is a sum, chop into summands
+ if (is_ex_exactly_of_type(n.op(0), add)) {
+ epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
while (bit != bitend) {
- o.push_back(recombine_pair_to_ex(*bit));
+ o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
bit++;
}
- o.push_back((static_cast<add *>(n.bp))->overall_coeff);
+
+ // The overall_coeff is already normalized (== rational), we just
+ // split it into numerator and denominator
+ GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
+ numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
+ o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
} else
o.push_back(n);
it++;
}
o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
+ // o is now a vector of {numerator, denominator} lists
+
// Determine common denominator
- ex den = exONE();
+ ex den = _ex1();
exvector::const_iterator ait = o.begin(), aitend = o.end();
+//clog << "add::normal uses the following summands:\n";
while (ait != aitend) {
- den = lcm((*ait).denom(false), den, false);
+//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
+ den = lcm(ait->op(1), den, false);
ait++;
}
+//clog << " common denominator = " << den << endl;
// Add fractions
- if (den.is_equal(exONE()))
- return (new add(o))->setflag(status_flags::dynallocated);
- else {
+ if (den.is_equal(_ex1())) {
+
+ // Common denominator is 1, simply add all numerators
+ exvector num_seq;
+ for (ait=o.begin(); ait!=aitend; ait++) {
+ num_seq.push_back(ait->op(0));
+ }
+ return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ // Perform fractional addition
exvector num_seq;
for (ait=o.begin(); ait!=aitend; ait++) {
ex q;
- if (!divide(den, (*ait).denom(false), q, false)) {
+ if (!divide(den, ait->op(1), q, false)) {
// should not happen
throw(std::runtime_error("invalid expression in add::normal, division failed"));
}
- num_seq.push_back((*ait).numer(false) * q);
+ num_seq.push_back((ait->op(0) * q).expand());
}
- ex num = add(num_seq);
+ ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
// Cancel common factors from num/den
return frac_cancel(num, den);
* @see ex::normal() */
ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- // Normalize children
- exvector o;
- o.reserve(seq.size()+1);
+ // Normalize children, separate into numerator and denominator
+ ex num = _ex1();
+ ex den = _ex1();
+ ex n;
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
- o.push_back(recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1));
+ n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
+ num *= n.op(0);
+ den *= n.op(1);
it++;
}
- o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
- ex n = (new mul(o))->setflag(status_flags::dynallocated);
- return frac_cancel(n.numer(false), n.denom(false));
+ n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
+ num *= n.op(0);
+ den *= n.op(1);
+
+ // Perform fraction cancellation
+ return frac_cancel(num, den);
}
* @see ex::normal */
ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- if (exponent.info(info_flags::integer)) {
- // Integer powers are distributed
- ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
- ex num = n.numer(false);
- ex den = n.denom(false);
- return power(num, exponent) / power(den, exponent);
- } else {
- // Non-integer powers are replaced by temporary symbol (after normalizing basis)
- ex n = power(basis.bp->normal(sym_lst, repl_lst, level-1), exponent);
- return replace_with_symbol(n, sym_lst, repl_lst);
+ // Normalize basis
+ ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+
+ if (exponent.info(info_flags::integer)) {
+
+ if (exponent.info(info_flags::positive)) {
+
+ // (a/b)^n -> {a^n, b^n}
+ return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+
+ } else if (exponent.info(info_flags::negative)) {
+
+ // (a/b)^-n -> {b^n, a^n}
+ return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
+ }
+
+ } else {
+
+ if (exponent.info(info_flags::positive)) {
+
+ // (a/b)^x -> {sym((a/b)^x), 1}
+ return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+
+ } else if (exponent.info(info_flags::negative)) {
+
+ if (n.op(1).is_equal(_ex1())) {
+
+ // a^-x -> {1, sym(a^x)}
+ return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ // (a/b)^-x -> {sym((b/a)^x), 1}
+ return (new lst(replace_with_symbol(power(n.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+ }
+
+ } else { // exponent not numeric
+
+ // (a/b)^x -> {sym((a/b)^x, 1}
+ return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+ }
}
}
-/** Implementation of ex::normal() for series. It normalizes each coefficient and
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient and
* replaces the series by a temporary symbol.
* @see ex::normal */
-ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
{
epvector new_seq;
new_seq.reserve(seq.size());
new_seq.push_back(expair(it->rest.normal(), it->coeff));
it++;
}
-
- ex n = series(var, point, new_seq);
- return replace_with_symbol(n, sym_lst, repl_lst);
+ ex n = pseries(relational(var,point), new_seq);
+ return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
ex ex::normal(int level) const
{
lst sym_lst, repl_lst;
+
ex e = bp->normal(sym_lst, repl_lst, level);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
if (sym_lst.nops() > 0)
- return e.subs(sym_lst, repl_lst);
- else
- return e;
+ e = e.subs(sym_lst, repl_lst);
+
+ // Convert {numerator, denominator} form back to fraction
+ return e.op(0) / e.op(1);
+}
+
+/** Numerator of an expression. If the expression is not of the normal form
+ * "numerator/denominator", it is first converted to this form and then the
+ * numerator is returned.
+ *
+ * @see ex::normal
+ * @return numerator */
+ex ex::numer(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(0).subs(sym_lst, repl_lst);
+ else
+ return e.op(0);
+}
+
+/** Denominator of an expression. If the expression is not of the normal form
+ * "numerator/denominator", it is first converted to this form and then the
+ * denominator is returned.
+ *
+ * @see ex::normal
+ * @return denominator */
+ex ex::denom(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(1).subs(sym_lst, repl_lst);
+ else
+ return e.op(1);
}
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff