#define USE_REMEMBER 0
// Set this if you want divide_in_z() to use trial division followed by
-// polynomial interpolation (usually slower except for very large problems)
+// polynomial interpolation (always slower except for completely dense
+// polynomials)
#define USE_TRIAL_DIVISION 0
// Set this to enable some statistical output for the GCD routines
* @param e expression to search
* @param x pointer to first symbol found (returned)
* @return "false" if no symbol was found, "true" otherwise */
-
static bool get_first_symbol(const ex &e, const symbol *&x)
{
if (is_ex_exactly_of_type(e, symbol)) {
* @param a first multivariate polynomial
* @param b second multivariate polynomial
* @param v vector of sym_desc structs (filled in) */
-
static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
{
collect_symbols(a.eval(), v); // eval() to expand assigned symbols
*
* @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, _num1());
*
* @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)) {
*
* @param e expanded polynomial
* @return integer content */
-
numeric ex::integer_content(void) const
{
GINAC_ASSERT(bp!=0);
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return quotient of a and b in Q[x] */
-
ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
{
if (b.is_zero())
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return remainder of a(x) and b(x) in Q[x] */
-
ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
if (b.is_zero())
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return pseudo-remainder of a(x) and b(x) in Z[x] */
-
ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
if (b.is_zero())
}
+/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Z[x].
+ *
+ * @param a first polynomial in x (dividend)
+ * @param b second polynomial in x (divisor)
+ * @param x a and b are polynomials in x
+ * @param check_args check whether a and b are polynomials with rational
+ * coefficients (defaults to "true")
+ * @return sparse pseudo-remainder of a(x) and b(x) in Z[x] */
+
+ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
+{
+ if (b.is_zero())
+ 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 _ex0();
+ else
+ return b;
+ }
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));
+
+ // Polynomial long division
+ ex r = a.expand();
+ ex eb = b.expand();
+ int rdeg = r.degree(x);
+ int bdeg = eb.degree(x);
+ ex blcoeff;
+ if (bdeg <= rdeg) {
+ blcoeff = eb.coeff(x, bdeg);
+ if (bdeg == 0)
+ eb = _ex0();
+ else
+ eb -= blcoeff * power(x, bdeg);
+ } else
+ blcoeff = _ex1();
+
+ 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 = _ex0();
+ else
+ r -= rlcoeff * power(x, rdeg);
+ r = (blcoeff * r).expand() - term;
+ rdeg = r.degree(x);
+ }
+ return r;
+}
+
+
/** Exact polynomial division of a(X) by b(X) in Q[X].
*
* @param a first multivariate polynomial (dividend)
* coefficients (defaults to "true")
* @return "true" when exact division succeeds (quotient returned in q),
* "false" otherwise */
-
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
q = _ex0();
if (b.is_zero())
throw(std::overflow_error("divide: division by zero"));
- if (a.is_zero())
- return true;
+ if (a.is_zero())
+ return true;
if (is_ex_exactly_of_type(b, numeric)) {
q = a / b;
return true;
return true;
}
#endif
- if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ if (check_args && (!a.info(info_flags::rational_polynomial) ||
+ !b.info(info_flags::rational_polynomial)))
throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));
// Find first symbol
* @param x variable in which to compute the primitive part
* @param c previously computed content part
* @return primitive part */
-
ex ex::primpart(const symbol &x, const ex &c) const
{
if (is_zero())
* GCD of multivariate polynomials
*/
-/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
- * (not really suited for multivariate GCDs). This function is only provided
- * for testing purposes.
+/** 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
d = a;
}
+ // Normalize in Q[x]
+ c = c / c.lcoeff(*x);
+ d = d / d.lcoeff(*x);
+
// Euclidean algorithm
ex r;
for (;;) {
//clog << " d = " << d << endl;
r = rem(c, d, *x, false);
if (r.is_zero())
- return d.primpart(*x);
+ return d / d.lcoeff(*x);
c = d;
d = r;
}
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 (;;) {
//clog << " d = " << d << endl;
r = prem(c, d, *x, false);
if (r.is_zero())
- return d.primpart(*x);
+ return d.primpart(*x) * gamma;
c = d;
d = r;
}
c = c.primpart(*x, cont_c);
d = d.primpart(*x, cont_d);
- // First element of subresultant sequence
+ // First element of divisor sequence
ex r, ri = _ex1();
int delta = cdeg - ddeg;
/** Compute GCD of multivariate polynomials using the subresultant PRS
* algorithm. This function is used internally by gcd().
*
- * @param a first multivariate polynomial
- * @param b second multivariate polynomial
- * @param x pointer to symbol (main variable) in which to compute the GCD in
+ * @param a first multivariate polynomial
+ * @param b second multivariate polynomial
+ * @param var iterator to first element of vector of sym_desc structs
* @return the GCD as a new expression
* @see gcd */
-static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
+static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
{
//clog << "sr_gcd(" << a << "," << b << ")\n";
#if STATISTICS
sr_gcd_called++;
#endif
+ // The first symbol is our main variable
+ const symbol &x = *(var->sym);
+
// Sort c and d so that c has higher degree
ex c, d;
- int adeg = a.degree(*x), bdeg = b.degree(*x);
+ int adeg = a.degree(x), bdeg = b.degree(x);
int cdeg, ddeg;
if (adeg >= bdeg) {
c = a;
}
// 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 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);
+ c = c.primpart(x, cont_c);
+ d = d.primpart(x, cont_d);
//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
// First element of subresultant sequence
// Calculate polynomial pseudo-remainder
//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
//clog << " d = " << d << endl;
- r = prem(c, d, *x, false);
+ r = prem(c, d, x, false);
if (r.is_zero())
- return gamma * d.primpart(*x);
+ return gamma * d.primpart(x);
c = d;
cdeg = ddeg;
//clog << " dividing...\n";
- if (!divide(r, ri * pow(psi, delta), d, false))
+ if (!divide_in_z(r, ri * pow(psi, delta), d, var+1))
throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
- ddeg = d.degree(*x);
+ ddeg = d.degree(x);
if (ddeg == 0) {
if (is_ex_exactly_of_type(r, numeric))
return gamma;
else
- return gamma * r.primpart(*x);
+ return gamma * r.primpart(x);
}
// Next element of subresultant sequence
//clog << " calculating next subresultant...\n";
- ri = c.expand().lcoeff(*x);
+ ri = c.expand().lcoeff(x);
if (delta == 1)
psi = ri;
else if (delta)
- divide(pow(ri, delta), pow(psi, delta-1), psi, false);
+ divide_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
delta = cdeg - ddeg;
}
}
* @param e expanded multivariate polynomial
* @return maximum coefficient
* @see heur_gcd */
-
numeric ex::max_coefficient(void) const
{
GINAC_ASSERT(bp!=0);
* @param xi modulus
* @return mapped polynomial
* @see heur_gcd */
-
ex ex::smod(const numeric &xi) const
{
GINAC_ASSERT(bp!=0);
}
+/** xi-adic polynomial interpolation */
+static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x)
+{
+ ex g = _ex0();
+ ex e = gamma;
+ numeric rxi = xi.inverse();
+ for (int i=0; !e.is_zero(); i++) {
+ ex gi = e.smod(xi);
+ g += gi * power(x, i);
+ e = (e - gi) * rxi;
+ }
+ return g;
+}
+
/** Exception thrown by heur_gcd() to signal failure. */
class gcdheu_failed {};
* @return the GCD as a new expression
* @see gcd
* @exception gcdheu_failed() */
-
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";
}
// The first symbol is our main variable
- const symbol *x = var->sym;
+ const symbol &x = *(var->sym);
// Remove integer content
numeric gc = gcd(a.integer_content(), b.integer_content());
numeric rgc = gc.inverse();
ex p = a * rgc;
ex q = b * rgc;
- int maxdeg = max(p.degree(*x), q.degree(*x));
+ int maxdeg = max(p.degree(x), q.degree(x));
// Find evaluation point
numeric mp = p.max_coefficient(), mq = q.max_coefficient();
}
// Apply evaluation homomorphism and calculate GCD
- ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand();
+ ex cp, cq;
+ ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand();
if (!is_ex_exactly_of_type(gamma, fail)) {
// Reconstruct polynomial from GCD of mapped polynomials
- ex g = _ex0();
- numeric rxi = xi.inverse();
- for (int i=0; !gamma.is_zero(); i++) {
- ex gi = gamma.smod(xi);
- g += gi * power(*x, i);
- gamma = (gamma - gi) * rxi;
- }
+ ex g = interpolate(gamma, xi, x);
+
// Remove integer content
g /= g.integer_content();
- // If the calculated polynomial divides both a and b, this is the GCD
+ // If the calculated polynomial divides both p and q, this is the GCD
ex dummy;
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);
+ ex lc = g.lcoeff(x);
if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
return -g;
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_ex_exactly_of_type(lc, numeric) && 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_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
+ return -g;
+ else
+ return g;
+ }
+ }
+#endif
}
// Next evaluation point
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return the GCD as a new expression */
-
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
{
//clog << "gcd(" << a << "," << b << ")\n";
// The symbol with least degree is our main variable
sym_desc_vec::const_iterator var = sym_stats.begin();
- const symbol *x = var->sym;
+ const symbol &x = *(var->sym);
// Cancel trivial common factor
int ldeg_a = var->ldeg_a;
int ldeg_b = var->ldeg_b;
int min_ldeg = min(ldeg_a, ldeg_b);
if (min_ldeg > 0) {
- ex common = power(*x, min_ldeg);
+ ex common = power(x, min_ldeg);
//clog << "trivial common factor " << common << endl;
return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
}
// Try to eliminate variables
if (var->deg_a == 0) {
-//clog << "eliminating variable " << *x << " from b" << endl;
- ex c = bex.content(*x);
+//clog << "eliminating variable " << x << " from b" << endl;
+ ex c = bex.content(x);
ex g = gcd(aex, c, ca, cb, false);
if (cb)
- *cb *= bex.unit(*x) * bex.primpart(*x, c);
+ *cb *= bex.unit(x) * bex.primpart(x, c);
return g;
} else if (var->deg_b == 0) {
-//clog << "eliminating variable " << *x << " from a" << endl;
- ex c = aex.content(*x);
+//clog << "eliminating variable " << x << " from a" << endl;
+ ex c = aex.content(x);
ex g = gcd(c, bex, ca, cb, false);
if (ca)
- *ca *= aex.unit(*x) * aex.primpart(*x, c);
+ *ca *= aex.unit(x) * aex.primpart(x, c);
return g;
}
#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, x);
+// 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 (ca)
for (unsigned i=0; i<repl_lst.nops(); i++)
if (repl_lst.op(i).is_equal(e))
return sym_lst.op(i);
-
+
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't contain symbols from the sym_lst
// because subs() is not recursive
for (unsigned i=0; i<repl_lst.nops(); i++)
if (repl_lst.op(i).op(1).is_equal(e))
return repl_lst.op(i).op(0);
-
+
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't contain symbols from the sym_lst
// because subs() is not recursive
* @see ex::normal */
ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
{
+ if (level == 1)
+ return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
// Normalize and expand children, chop into summands
exvector o;
o.reserve(seq.size()+1);
* @see ex::normal() */
ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
{
+ if (level == 1)
+ return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
// Normalize children, separate into numerator and denominator
ex num = _ex1();
ex den = _ex1();
* @see ex::normal */
ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
{
+ if (level == 1)
+ return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
// Normalize basis
ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
}
-/** Implementation of ex::to_rational() for symbols. This returns the unmodified symbol.
+/** Implementation of ex::to_rational() for symbols. This returns the
+ * unmodified symbol.
* @see ex::to_rational */
ex symbol::to_rational(lst &repl_lst) const
{
}
-/** Implementation of ex::to_rational() for a numeric. It splits complex numbers
- * into re+I*im and replaces I and non-rational real numbers with a temporary
- * symbol.
+/** Implementation of ex::to_rational() for a numeric. It splits complex
+ * numbers into re+I*im and replaces I and non-rational real numbers with a
+ * temporary symbol.
* @see ex::to_rational */
ex numeric::to_rational(lst &repl_lst) const
{
if (is_real()) {
- if (!is_integer())
+ if (!is_rational())
return replace_with_symbol(*this, repl_lst);
} else { // complex
- numeric re = real(), im = imag();
+ numeric re = real();
+ numeric im = imag();
ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
return re_ex + im_ex * replace_with_symbol(I, repl_lst);
}
+/** Implementation of ex::to_rational() for expairseqs.
+ * @see ex::to_rational */
+ex expairseq::to_rational(lst &repl_lst) const
+{
+ epvector s;
+ s.reserve(seq.size());
+ for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
+ s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst)));
+ // s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst),
+ }
+ ex oc = overall_coeff.to_rational(repl_lst);
+ if (oc.info(info_flags::numeric))
+ return thisexpairseq(s, overall_coeff);
+ else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1()));
+ return thisexpairseq(s, default_overall_coeff());
+}
+
+
/** Rationalization of non-rational functions.
* This function converts a general expression to a rational polynomial
* by replacing all non-rational subexpressions (like non-rational numbers,
git://www.ginac.de / ginac.git / blobdiff