* @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)) {
/** Lowest degree of symbol in polynomial "b" */
int ldeg_b;
- /** Minimum of ldeg_a and ldeg_b (Used for sorting) */
- int min_deg;
+ /** Maximum of deg_a and deg_b (Used for sorting) */
+ int max_deg;
/** Commparison operator for sorting */
- bool operator<(const sym_desc &x) const {return min_deg < x.min_deg;}
+ bool operator<(const sym_desc &x) const {return max_deg < x.max_deg;}
};
// Vector of sym_desc structures
* @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
int deg_b = b.degree(*(it->sym));
it->deg_a = deg_a;
it->deg_b = deg_b;
- it->min_deg = min(deg_a, deg_b);
+ it->max_deg = max(deg_a, deg_b);
it->ldeg_a = a.ldegree(*(it->sym));
it->ldeg_b = b.ldegree(*(it->sym));
it++;
}
sort(v.begin(), v.end());
+#if 0
+ clog << "Symbols:\n";
+ it = v.begin(); itend = v.end();
+ while (it != itend) {
+ clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << endl;
+ clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
+ it++;
+ }
+#endif
}
*
* @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 (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 subresultant PRS
- * algorithm. This function is used internally gy gcd().
+/** Compute GCD of multivariate polynomials using the Euclidean PRS 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)
+{
+//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;
+ }
+
+ // Euclidean algorithm
+ ex r;
+ for (;;) {
+//clog << " d = " << d << endl;
+ r = rem(c, d, *x, false);
+ if (r.is_zero())
+ return d.primpart(*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)
+{
+//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;
+ }
+
+ // 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);
+ 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
* @return the GCD as a new expression
* @see gcd */
+static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//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 (;;) {
+//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)
+{
+//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 subresultant sequence
+ ex r, ri = _ex1();
+ int delta = cdeg - ddeg;
+
+ for (;;) {
+ // Calculate polynomial pseudo-remainder
+//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_ex_exactly_of_type(r, numeric))
+ 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().
+ *
+ * @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 sr_gcd(const ex &a, const ex &b, const symbol *x)
{
//clog << "sr_gcd(" << a << "," << b << ")\n";
for (;;) {
// Calculate polynomial pseudo-remainder
//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
+//clog << " d = " << d << endl;
r = prem(c, d, *x, false);
if (r.is_zero())
return gamma * d.primpart(*x);
c = d;
cdeg = ddeg;
//clog << " dividing...\n";
- if (!divide(r, ri * power(psi, delta), d, false))
+ if (!divide(r, ri * pow(psi, delta), d, false))
throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
ddeg = d.degree(*x);
if (ddeg == 0) {
if (delta == 1)
psi = ri;
else if (delta)
- divide(power(ri, delta), power(psi, delta-1), psi, false);
+ divide(pow(ri, delta), pow(psi, delta-1), psi, false);
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);
* @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";
* @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";
return g;
}
- // Try heuristic algorithm first, fall back to PRS if that failed
ex g;
+#if 1
+ // Try heuristic algorithm first, fall back to PRS if that failed
try {
g = heur_gcd(aex, bex, ca, cb, var);
} catch (gcdheu_failed) {
#if STATISTICS
heur_gcd_failed++;
#endif
- g = sr_gcd(aex, bex, x);
+#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);
if (g.is_equal(_ex1())) {
// Keep cofactors factored if possible
if (ca)
if (cb)
divide(bex, g, *cb, false);
}
+#if 1
} else {
if (g.is_equal(_ex1())) {
// Keep cofactors factored if possible
if (cb)
*cb = b;
}
- return g;
}
+#endif
+ return g;
}
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
{
- numeric num = numer();
- ex numex = num;
-
- if (num.is_real()) {
- if (!num.is_integer())
- numex = replace_with_symbol(numex, repl_lst);
+ if (is_real()) {
+ if (!is_rational())
+ return replace_with_symbol(*this, repl_lst);
} else { // complex
- numeric re = num.real(), im = num.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);
- numex = re_ex + im_ex * replace_with_symbol(I, repl_lst);
+ return re_ex + im_ex * replace_with_symbol(I, repl_lst);
}
- return numex;
+ return *this;
}
}
+/** 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