++it;
}
std::sort(v.begin(), v.end());
+
#if 0
std::clog << "Symbols:\n";
it = v.begin(); itend = v.end();
* GCD of multivariate polynomials
*/
-/** Compute GCD of polynomials in Q[X] using the Euclidean algorithm (not
- * really suited for multivariate GCDs). This function is only provided for
- * testing purposes.
- *
- * @param a first multivariate polynomial
- * @param b second multivariate polynomial
- * @param x pointer to symbol (main variable) in which to compute the GCD in
- * @return the GCD as a new expression
- * @see gcd */
-
-static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
-{
-//std::clog << "eu_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);
- if (adeg >= bdeg) {
- c = a;
- d = b;
- } else {
- c = b;
- d = a;
- }
-
- // Normalize in Q[x]
- c = c / c.lcoeff(*x);
- d = d / d.lcoeff(*x);
-
- // Euclidean algorithm
- ex r;
- for (;;) {
-//std::clog << " d = " << d << endl;
- r = rem(c, d, *x, false);
- if (r.is_zero())
- return d / d.lcoeff(*x);
- c = d;
- d = r;
- }
-}
-
-
-/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
- * with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]).
- * This function is only provided for testing purposes.
- *
- * @param a first multivariate polynomial
- * @param b second multivariate polynomial
- * @param x pointer to symbol (main variable) in which to compute the GCD in
- * @return the GCD as a new expression
- * @see gcd */
-
-static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
-{
-//std::clog << "euprem_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);
- if (adeg >= bdeg) {
- c = a;
- d = b;
- } else {
- c = b;
- d = a;
- }
-
- // Calculate GCD of contents
- ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false);
-
- // Euclidean algorithm with pseudo-remainders
- ex r;
- for (;;) {
-//std::clog << " d = " << d << endl;
- r = prem(c, d, *x, false);
- if (r.is_zero())
- return d.primpart(*x) * gamma;
- c = d;
- d = r;
- }
-}
-
-
-/** Compute GCD of multivariate polynomials using the primitive Euclidean
- * PRS algorithm (complete content removal at each step). This function is
- * only provided for testing purposes.
- *
- * @param a first multivariate polynomial
- * @param b second multivariate polynomial
- * @param x pointer to symbol (main variable) in which to compute the GCD in
- * @return the GCD as a new expression
- * @see gcd */
-
-static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
-{
-//std::clog << "peu_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);
- int ddeg;
- if (adeg >= bdeg) {
- c = a;
- d = b;
- ddeg = bdeg;
- } else {
- c = b;
- d = a;
- ddeg = adeg;
- }
-
- // Remove content from c and d, to be attached to GCD later
- ex cont_c = c.content(*x);
- ex cont_d = d.content(*x);
- ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
- if (ddeg == 0)
- return gamma;
- c = c.primpart(*x, cont_c);
- d = d.primpart(*x, cont_d);
-
- // Euclidean algorithm with content removal
- ex r;
- for (;;) {
-//std::clog << " d = " << d << endl;
- r = prem(c, d, *x, false);
- if (r.is_zero())
- return gamma * d;
- c = d;
- d = r.primpart(*x);
- }
-}
-
-
-/** Compute GCD of multivariate polynomials using the reduced PRS algorithm.
- * This function is only provided for testing purposes.
- *
- * @param a first multivariate polynomial
- * @param b second multivariate polynomial
- * @param x pointer to symbol (main variable) in which to compute the GCD in
- * @return the GCD as a new expression
- * @see gcd */
-
-static ex red_gcd(const ex &a, const ex &b, const symbol *x)
-{
-//std::clog << "red_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);
- int cdeg, ddeg;
- if (adeg >= bdeg) {
- c = a;
- d = b;
- cdeg = adeg;
- ddeg = bdeg;
- } else {
- c = b;
- d = a;
- cdeg = bdeg;
- ddeg = adeg;
- }
-
- // Remove content from c and d, to be attached to GCD later
- ex cont_c = c.content(*x);
- ex cont_d = d.content(*x);
- ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
- if (ddeg == 0)
- return gamma;
- c = c.primpart(*x, cont_c);
- d = d.primpart(*x, cont_d);
-
- // First element of divisor sequence
- ex r, ri = _ex1;
- int delta = cdeg - ddeg;
-
- for (;;) {
- // Calculate polynomial pseudo-remainder
-//std::clog << " d = " << d << endl;
- r = prem(c, d, *x, false);
- if (r.is_zero())
- return gamma * d.primpart(*x);
- c = d;
- cdeg = ddeg;
-
- if (!divide(r, pow(ri, delta), d, false))
- throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
- ddeg = d.degree(*x);
- if (ddeg == 0) {
- if (is_exactly_a<numeric>(r))
- return gamma;
- else
- return gamma * r.primpart(*x);
- }
-
- ri = c.expand().lcoeff(*x);
- delta = cdeg - ddeg;
- }
-}
-
-
/** Compute GCD of multivariate polynomials using the subresultant PRS
* algorithm. This function is used internally by gcd().
*
static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
{
-//std::clog << "sr_gcd(" << a << "," << b << ")\n";
#if STATISTICS
sr_gcd_called++;
#endif
return gamma;
c = c.primpart(x, cont_c);
d = d.primpart(x, cont_d);
-//std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
// First element of subresultant sequence
ex r = _ex0, ri = _ex1, psi = _ex1;
int delta = cdeg - ddeg;
for (;;) {
+
// Calculate polynomial pseudo-remainder
-//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
-//std::clog << " d = " << d << endl;
r = prem(c, d, x, false);
if (r.is_zero())
return gamma * d.primpart(x);
+
c = d;
cdeg = ddeg;
-//std::clog << " dividing...\n";
if (!divide_in_z(r, ri * pow(psi, delta), d, var))
throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
ddeg = d.degree(x);
}
// Next element of subresultant sequence
-//std::clog << " calculating next subresultant...\n";
ri = c.expand().lcoeff(x);
if (delta == 1)
psi = ri;
* @exception gcdheu_failed() */
static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
{
-//std::clog << "heur_gcd(" << a << "," << b << ")\n";
#if STATISTICS
heur_gcd_called++;
#endif
// 6 tries maximum
for (int t=0; t<6; t++) {
if (xi.int_length() * maxdeg > 100000) {
-//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl;
throw gcdheu_failed();
}
else
return g;
}
-#if 0
- cp = interpolate(cp, xi, x);
- if (divide_in_z(cp, p, g, var)) {
- if (divide_in_z(g, q, cb ? *cb : dummy, var)) {
- g *= gc;
- if (ca)
- *ca = cp;
- ex lc = g.lcoeff(x);
- if (is_exactly_a<numeric>(lc) && ex_to<numeric>(lc).is_negative())
- return -g;
- else
- return g;
- }
- }
- cq = interpolate(cq, xi, x);
- if (divide_in_z(cq, q, g, var)) {
- if (divide_in_z(g, p, ca ? *ca : dummy, var)) {
- g *= gc;
- if (cb)
- *cb = cq;
- ex lc = g.lcoeff(x);
- if (is_exactly_a<numeric>(lc) && ex_to<numeric>(lc).is_negative())
- return -g;
- else
- return g;
- }
- }
-#endif
}
// Next evaluation point
* @return the GCD as a new expression */
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
{
-//std::clog << "gcd(" << a << "," << b << ")\n";
#if STATISTICS
gcd_called++;
#endif
int min_ldeg = std::min(ldeg_a,ldeg_b);
if (min_ldeg > 0) {
ex common = power(x, min_ldeg);
-//std::clog << "trivial common factor " << common << std::endl;
return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
}
// Try to eliminate variables
if (var->deg_a == 0) {
-//std::clog << "eliminating variable " << x << " from b" << std::endl;
ex c = bex.content(x);
ex g = gcd(aex, c, ca, cb, false);
if (cb)
*cb *= bex.unit(x) * bex.primpart(x, c);
return g;
} else if (var->deg_b == 0) {
-//std::clog << "eliminating variable " << x << " from a" << std::endl;
ex c = aex.content(x);
ex g = gcd(c, bex, ca, cb, false);
if (ca)
return g;
}
- ex g;
-#if 1
// Try heuristic algorithm first, fall back to PRS if that failed
+ ex g;
try {
g = heur_gcd(aex, bex, ca, cb, var);
} catch (gcdheu_failed) {
g = fail();
}
if (is_exactly_a<fail>(g)) {
-//std::clog << "heuristics failed" << std::endl;
#if STATISTICS
heur_gcd_failed++;
#endif
-#endif
-// g = heur_gcd(aex, bex, ca, cb, var);
-// g = eu_gcd(aex, bex, &x);
-// g = euprem_gcd(aex, bex, &x);
-// g = peu_gcd(aex, bex, &x);
-// g = red_gcd(aex, bex, &x);
g = sr_gcd(aex, bex, var);
if (g.is_equal(_ex1)) {
// Keep cofactors factored if possible
if (cb)
divide(bex, g, *cb, false);
}
-#if 1
} else {
if (g.is_equal(_ex1)) {
// Keep cofactors factored if possible
*cb = b;
}
}
-#endif
+
return g;
}
// (a/b)^-x -> {sym((b/a)^x), 1}
return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
}
-
- } else { // n_exponent not numeric
-
- // (a/b)^x -> {sym((a/b)^x, 1}
- return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
}
}
+
+ // (a/b)^x -> {sym((a/b)^x, 1}
+ return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
}
projects / ginac.git / commitdiff