* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2008 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
if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
throw(std::invalid_argument("invalid expression in divide()"));
+ // Try to avoid expanding partially factored expressions.
+ if (is_exactly_a<mul>(b)) {
+ // Divide sequentially by each term
+ ex rem_new, rem_old = a;
+ for (size_t i=0; i < b.nops(); i++) {
+ if (! divide(rem_old, b.op(i), rem_new, false))
+ return false;
+ rem_old = rem_new;
+ }
+ q = rem_new;
+ return true;
+ } else if (is_exactly_a<power>(b)) {
+ const ex& bb(b.op(0));
+ int exp_b = ex_to<numeric>(b.op(1)).to_int();
+ ex rem_new, rem_old = a;
+ for (int i=exp_b; i>0; i--) {
+ if (! divide(rem_old, bb, rem_new, false))
+ return false;
+ rem_old = rem_new;
+ }
+ q = rem_new;
+ return true;
+ }
+
+ if (is_exactly_a<mul>(a)) {
+ // Divide sequentially each term. If some term in a is divisible
+ // by b we are done... and if not, we can't really say anything.
+ size_t i;
+ ex rem_i;
+ bool divisible_p = false;
+ for (i=0; i < a.nops(); ++i) {
+ if (divide(a.op(i), b, rem_i, false)) {
+ divisible_p = true;
+ break;
+ }
+ }
+ if (divisible_p) {
+ exvector resv;
+ resv.reserve(a.nops());
+ for (size_t j=0; j < a.nops(); j++) {
+ if (j==i)
+ resv.push_back(rem_i);
+ else
+ resv.push_back(a.op(j));
+ }
+ q = (new mul(resv))->setflag(status_flags::dynallocated);
+ return true;
+ }
+ } else if (is_exactly_a<power>(a)) {
+ // The base itself might be divisible by b, in that case we don't
+ // need to expand a
+ const ex& ab(a.op(0));
+ int a_exp = ex_to<numeric>(a.op(1)).to_int();
+ ex rem_i;
+ if (divide(ab, b, rem_i, false)) {
+ q = rem_i*power(ab, a_exp - 1);
+ return true;
+ }
+ for (int i=2; i < a_exp; i++) {
+ if (divide(power(ab, i), b, rem_i, false)) {
+ q = rem_i*power(ab, a_exp - i);
+ return true;
+ }
+ } // ... so we *really* need to expand expression.
+ }
+
// Polynomial long division (recursive)
ex r = a.expand();
if (r.is_zero()) {
}
#endif
+ if (is_exactly_a<power>(b)) {
+ const ex& bb(b.op(0));
+ ex qbar = a;
+ int exp_b = ex_to<numeric>(b.op(1)).to_int();
+ for (int i=exp_b; i>0; i--) {
+ if (!divide_in_z(qbar, bb, q, var))
+ return false;
+ qbar = q;
+ }
+ return true;
+ }
+
+ if (is_exactly_a<mul>(b)) {
+ ex qbar = a;
+ for (const_iterator itrb = b.begin(); itrb != b.end(); ++itrb) {
+ sym_desc_vec sym_stats;
+ get_symbol_stats(a, *itrb, sym_stats);
+ if (!divide_in_z(qbar, *itrb, q, sym_stats.begin()))
+ return false;
+
+ qbar = q;
+ }
+ return true;
+ }
+
// Main symbol
const ex &x = var->sym;
* polynomials and an iterator to the first element of the sym_desc vector
* passed in. This function is used internally by gcd().
*
- * @param a first multivariate polynomial (expanded)
- * @param b second multivariate polynomial (expanded)
+ * @param a first integer multivariate polynomial (expanded)
+ * @param b second integer multivariate polynomial (expanded)
* @param ca cofactor of polynomial a (returned), NULL to suppress
* calculation of cofactor
* @param cb cofactor of polynomial b (returned), NULL to suppress
* calculation of cofactor
* @param var iterator to first element of vector of sym_desc structs
- * @return the GCD as a new expression
+ * @param res the GCD (returned)
+ * @return true if GCD was computed, false otherwise.
* @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)
+static bool heur_gcd_z(ex& res, const ex &a, const ex &b, ex *ca, ex *cb,
+ sym_desc_vec::const_iterator var)
{
#if STATISTICS
heur_gcd_called++;
// Algorithm only works for non-vanishing input polynomials
if (a.is_zero() || b.is_zero())
- return (new fail())->setflag(status_flags::dynallocated);
+ return false;
// GCD of two numeric values -> CLN
if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
*ca = ex_to<numeric>(a) / g;
if (cb)
*cb = ex_to<numeric>(b) / g;
- return g;
+ res = g;
+ return true;
}
// The first symbol is our main variable
// Apply evaluation homomorphism and calculate GCD
ex cp, cq;
- ex gamma = heur_gcd(p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1).expand();
- if (!is_exactly_a<fail>(gamma)) {
-
+ ex gamma;
+ bool found = heur_gcd_z(gamma,
+ p.subs(x == xi, subs_options::no_pattern),
+ q.subs(x == xi, subs_options::no_pattern),
+ &cp, &cq, var+1);
+ if (found) {
+ gamma = gamma.expand();
// Reconstruct polynomial from GCD of mapped polynomials
ex g = interpolate(gamma, xi, x, maxdeg);
ex dummy;
if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
g *= gc;
- return g;
+ res = g;
+ return true;
}
}
// Next evaluation point
xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
}
- return (new fail())->setflag(status_flags::dynallocated);
+ return false;
}
+/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
+ * get_symbol_stats() must have been called previously with the input
+ * polynomials and an iterator to the first element of the sym_desc vector
+ * passed in. This function is used internally by gcd().
+ *
+ * @param a first rational multivariate polynomial (expanded)
+ * @param b second rational multivariate polynomial (expanded)
+ * @param ca cofactor of polynomial a (returned), NULL to suppress
+ * calculation of cofactor
+ * @param cb cofactor of polynomial b (returned), NULL to suppress
+ * calculation of cofactor
+ * @param var iterator to first element of vector of sym_desc structs
+ * @param res the GCD (returned)
+ * @return true if GCD was computed, false otherwise.
+ * @see heur_gcd_z
+ * @see gcd
+ */
+static bool heur_gcd(ex& res, const ex& a, const ex& b, ex *ca, ex *cb,
+ sym_desc_vec::const_iterator var)
+{
+ if (a.info(info_flags::integer_polynomial) &&
+ b.info(info_flags::integer_polynomial)) {
+ try {
+ return heur_gcd_z(res, a, b, ca, cb, var);
+ } catch (gcdheu_failed) {
+ return false;
+ }
+ }
+
+ // convert polynomials to Z[X]
+ const numeric a_lcm = lcm_of_coefficients_denominators(a);
+ const numeric ab_lcm = lcmcoeff(b, a_lcm);
+
+ const ex ai = a*ab_lcm;
+ const ex bi = b*ab_lcm;
+ if (!ai.info(info_flags::integer_polynomial))
+ throw std::logic_error("heur_gcd: not an integer polynomial [1]");
+
+ if (!bi.info(info_flags::integer_polynomial))
+ throw std::logic_error("heur_gcd: not an integer polynomial [2]");
+
+ bool found = false;
+ try {
+ found = heur_gcd_z(res, ai, bi, ca, cb, var);
+ } catch (gcdheu_failed) {
+ return false;
+ }
+
+ // GCD is not unique, it's defined up to a unit (i.e. invertible
+ // element). If the coefficient ring is a field, every its element is
+ // invertible, so one can multiply the polynomial GCD with any element
+ // of the coefficient field. We use this ambiguity to make cofactors
+ // integer polynomials.
+ if (found)
+ res /= ab_lcm;
+ return found;
+}
+
+
+// gcd helper to handle partially factored polynomials (to avoid expanding
+// large expressions). At least one of the arguments should be a power.
+static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb, bool check_args);
/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
* and b(X) in Z[X]. Optionally also compute the cofactors of a and b,
* @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)
+ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args, unsigned options)
{
#if STATISTICS
gcd_called++;
}
#if FAST_COMPARE
- // Input polynomials of the form poly^n are sometimes also trivial
+ if (is_exactly_a<power>(a) || is_exactly_a<power>(b))
+ return gcd_pf_pow(a, b, ca, cb, check_args);
+#endif
+
+ // Some trivial cases
+ ex aex = a.expand(), bex = b.expand();
+ if (aex.is_zero()) {
+ if (ca)
+ *ca = _ex0;
+ if (cb)
+ *cb = _ex1;
+ return b;
+ }
+ if (bex.is_zero()) {
+ if (ca)
+ *ca = _ex1;
+ if (cb)
+ *cb = _ex0;
+ return a;
+ }
+ if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+#if FAST_COMPARE
+ if (a.is_equal(b)) {
+ if (ca)
+ *ca = _ex1;
+ if (cb)
+ *cb = _ex1;
+ return a;
+ }
+#endif
+
+ if (is_a<symbol>(aex)) {
+ if (! bex.subs(aex==_ex0, subs_options::no_pattern).is_zero()) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ }
+
+ if (is_a<symbol>(bex)) {
+ if (! aex.subs(bex==_ex0, subs_options::no_pattern).is_zero()) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ }
+
+ if (is_exactly_a<numeric>(aex)) {
+ numeric bcont = bex.integer_content();
+ numeric g = gcd(ex_to<numeric>(aex), bcont);
+ if (ca)
+ *ca = ex_to<numeric>(aex)/g;
+ if (cb)
+ *cb = bex/g;
+ return g;
+ }
+
+ if (is_exactly_a<numeric>(bex)) {
+ numeric acont = aex.integer_content();
+ numeric g = gcd(ex_to<numeric>(bex), acont);
+ if (ca)
+ *ca = aex/g;
+ if (cb)
+ *cb = ex_to<numeric>(bex)/g;
+ return g;
+ }
+
+ // Gather symbol statistics
+ sym_desc_vec sym_stats;
+ get_symbol_stats(a, b, sym_stats);
+
+ // The symbol with least degree which is contained in both polynomials
+ // is our main variable
+ sym_desc_vec::iterator vari = sym_stats.begin();
+ while ((vari != sym_stats.end()) &&
+ (((vari->ldeg_b == 0) && (vari->deg_b == 0)) ||
+ ((vari->ldeg_a == 0) && (vari->deg_a == 0))))
+ vari++;
+
+ // No common symbols at all, just return 1:
+ if (vari == sym_stats.end()) {
+ // N.B: keep cofactors factored
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ // move symbols which contained only in one of the polynomials
+ // to the end:
+ rotate(sym_stats.begin(), vari, sym_stats.end());
+
+ sym_desc_vec::const_iterator var = sym_stats.begin();
+ const ex &x = var->sym;
+
+ // Cancel trivial common factor
+ int ldeg_a = var->ldeg_a;
+ int ldeg_b = var->ldeg_b;
+ int min_ldeg = std::min(ldeg_a,ldeg_b);
+ if (min_ldeg > 0) {
+ ex common = power(x, min_ldeg);
+ return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
+ }
+
+ // Try to eliminate variables
+ if (var->deg_a == 0 && var->deg_b != 0 ) {
+ ex bex_u, bex_c, bex_p;
+ bex.unitcontprim(x, bex_u, bex_c, bex_p);
+ ex g = gcd(aex, bex_c, ca, cb, false);
+ if (cb)
+ *cb *= bex_u * bex_p;
+ return g;
+ } else if (var->deg_b == 0 && var->deg_a != 0) {
+ ex aex_u, aex_c, aex_p;
+ aex.unitcontprim(x, aex_u, aex_c, aex_p);
+ ex g = gcd(aex_c, bex, ca, cb, false);
+ if (ca)
+ *ca *= aex_u * aex_p;
+ return g;
+ }
+
+ // Try heuristic algorithm first, fall back to PRS if that failed
+ ex g;
+ bool found = heur_gcd(g, aex, bex, ca, cb, var);
+ if (!found) {
+#if STATISTICS
+ heur_gcd_failed++;
+#endif
+ g = sr_gcd(aex, bex, var);
+ if (g.is_equal(_ex1)) {
+ // Keep cofactors factored if possible
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ } else {
+ if (ca)
+ divide(aex, g, *ca, false);
+ if (cb)
+ divide(bex, g, *cb, false);
+ }
+ } else {
+ if (g.is_equal(_ex1)) {
+ // Keep cofactors factored if possible
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ }
+ }
+
+ return g;
+}
+
+static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb, bool check_args)
+{
if (is_exactly_a<power>(a)) {
ex p = a.op(0);
const ex& exp_a = a.op(1);
return p_gcd*gcd(apart_co, power(p_gcd, exp_b-1)*power(p_co, exp_b), ca, cb, false);
} // p_gcd.is_equal(_ex1)
}
-#endif
-
- // Some trivial cases
- ex aex = a.expand(), bex = b.expand();
- if (aex.is_zero()) {
- if (ca)
- *ca = _ex0;
- if (cb)
- *cb = _ex1;
- return b;
- }
- if (bex.is_zero()) {
- if (ca)
- *ca = _ex1;
- if (cb)
- *cb = _ex0;
- return a;
- }
- if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) {
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- return _ex1;
- }
-#if FAST_COMPARE
- if (a.is_equal(b)) {
- if (ca)
- *ca = _ex1;
- if (cb)
- *cb = _ex1;
- return a;
- }
-#endif
-
- // Gather symbol statistics
- sym_desc_vec sym_stats;
- get_symbol_stats(a, b, sym_stats);
-
- // The symbol with least degree is our main variable
- sym_desc_vec::const_iterator var = sym_stats.begin();
- const ex &x = var->sym;
-
- // Cancel trivial common factor
- int ldeg_a = var->ldeg_a;
- int ldeg_b = var->ldeg_b;
- int min_ldeg = std::min(ldeg_a,ldeg_b);
- if (min_ldeg > 0) {
- ex common = power(x, min_ldeg);
- return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
- }
-
- // Try to eliminate variables
- if (var->deg_a == 0) {
- ex bex_u, bex_c, bex_p;
- bex.unitcontprim(x, bex_u, bex_c, bex_p);
- ex g = gcd(aex, bex_c, ca, cb, false);
- if (cb)
- *cb *= bex_u * bex_p;
- return g;
- } else if (var->deg_b == 0) {
- ex aex_u, aex_c, aex_p;
- aex.unitcontprim(x, aex_u, aex_c, aex_p);
- ex g = gcd(aex_c, bex, ca, cb, false);
- if (ca)
- *ca *= aex_u * aex_p;
- return g;
- }
-
- // 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)) {
-#if STATISTICS
- heur_gcd_failed++;
-#endif
- g = sr_gcd(aex, bex, var);
- if (g.is_equal(_ex1)) {
- // Keep cofactors factored if possible
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- } else {
- if (ca)
- divide(aex, g, *ca, false);
- if (cb)
- divide(bex, g, *cb, false);
- }
- } else {
- if (g.is_equal(_ex1)) {
- // Keep cofactors factored if possible
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- }
- }
-
- return g;
}
-
/** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
*
* @param a first multivariate polynomial
{
if (exponent.info(info_flags::posint))
return power(basis.to_rational(repl), exponent);
+ else if (exponent.info(info_flags::negint))
+ {
+ ex basis_pref = collect_common_factors(basis);
+ if (is_exactly_a<mul>(basis_pref) || is_exactly_a<power>(basis_pref)) {
+ // (A*B)^n will be automagically transformed to A^n*B^n
+ ex t = power(basis_pref, exponent);
+ return t.to_polynomial(repl);
+ }
+ else
+ return power(replace_with_symbol(power(basis, _ex_1), repl), -exponent);
+ }
else
return replace_with_symbol(*this, repl);
}
for (size_t i=0; i<num; i++) {
ex x = e.op(i).to_polynomial(repl);
- if (is_exactly_a<add>(x) || is_exactly_a<mul>(x)) {
+ if (is_exactly_a<add>(x) || is_exactly_a<mul>(x) || is_a<power>(x)) {
ex f = 1;
x = find_common_factor(x, f, repl);
x *= f;
return (new mul(v))->setflag(status_flags::dynallocated);
} else if (is_exactly_a<power>(e)) {
-
- return e.to_polynomial(repl);
+ const ex e_exp(e.op(1));
+ if (e_exp.info(info_flags::integer)) {
+ ex eb = e.op(0).to_polynomial(repl);
+ ex factor_local(_ex1);
+ ex pre_res = find_common_factor(eb, factor_local, repl);
+ factor *= power(factor_local, e_exp);
+ return power(pre_res, e_exp);
+
+ } else
+ return e.to_polynomial(repl);
} else
return e;
* 'a*(b*x+b*y)' to 'a*b*(x+y)'. */
ex collect_common_factors(const ex & e)
{
- if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
+ if (is_exactly_a<add>(e) || is_exactly_a<mul>(e) || is_exactly_a<power>(e)) {
exmap repl;
ex factor = 1;
git://www.ginac.de / ginac.git / blobdiff