* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2001 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
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <algorithm>
#include "numeric.h"
#include "power.h"
#include "relational.h"
+#include "operators.h"
#include "matrix.h"
#include "pseries.h"
#include "symbol.h"
#endif
-/** Return pointer to first symbol found in expression. Due to GiNaCยดs
+/** Return pointer to first symbol found in expression. Due to GiNaC's
* internal ordering of terms, it may not be obvious which symbol this
* function returns for a given expression.
*
* @param e expression to search
- * @param x pointer to first symbol found (returned)
+ * @param x first symbol found (returned)
* @return "false" if no symbol was found, "true" otherwise */
-static bool get_first_symbol(const ex &e, const symbol *&x)
+static bool get_first_symbol(const ex &e, ex &x)
{
- if (is_ex_exactly_of_type(e, symbol)) {
- x = &ex_to<symbol>(e);
+ if (is_a<symbol>(e)) {
+ x = e;
return true;
- } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- for (unsigned i=0; i<e.nops(); i++)
+ } else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
+ for (size_t 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)) {
+ } else if (is_exactly_a<power>(e)) {
if (get_first_symbol(e.op(0), x))
return true;
}
*
* @see get_symbol_stats */
struct sym_desc {
- /** Pointer to symbol */
- const symbol *sym;
+ /** Reference to symbol */
+ ex sym;
/** Highest degree of symbol in polynomial "a" */
int deg_a;
int max_deg;
/** Maximum number of terms of leading coefficient of symbol in both polynomials */
- int max_lcnops;
+ size_t max_lcnops;
/** Commparison operator for sorting */
bool operator<(const sym_desc &x) const
typedef std::vector<sym_desc> sym_desc_vec;
// Add symbol the sym_desc_vec (used internally by get_symbol_stats())
-static void add_symbol(const symbol *s, sym_desc_vec &v)
+static void add_symbol(const ex &s, sym_desc_vec &v)
{
sym_desc_vec::const_iterator it = v.begin(), itend = v.end();
while (it != itend) {
- if (it->sym->compare(*s) == 0) // If it's already in there, don't add it a second time
+ if (it->sym.is_equal(s)) // If it's already in there, don't add it a second time
return;
++it;
}
// Collect all symbols of an expression (used internally by get_symbol_stats())
static void collect_symbols(const ex &e, sym_desc_vec &v)
{
- if (is_ex_exactly_of_type(e, symbol)) {
- add_symbol(&ex_to<symbol>(e), v);
- } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- for (unsigned i=0; i<e.nops(); i++)
+ if (is_a<symbol>(e)) {
+ add_symbol(e, v);
+ } else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
+ for (size_t i=0; i<e.nops(); i++)
collect_symbols(e.op(i), v);
- } else if (is_ex_exactly_of_type(e, power)) {
+ } else if (is_exactly_a<power>(e)) {
collect_symbols(e.op(0), v);
}
}
collect_symbols(b.eval(), v);
sym_desc_vec::iterator it = v.begin(), itend = v.end();
while (it != itend) {
- int deg_a = a.degree(*(it->sym));
- int deg_b = b.degree(*(it->sym));
+ int deg_a = a.degree(it->sym);
+ int deg_b = b.degree(it->sym);
it->deg_a = deg_a;
it->deg_b = deg_b;
it->max_deg = std::max(deg_a, deg_b);
- it->max_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops());
- it->ldeg_a = a.ldegree(*(it->sym));
- it->ldeg_b = b.ldegree(*(it->sym));
+ it->max_lcnops = std::max(a.lcoeff(it->sym).nops(), b.lcoeff(it->sym).nops());
+ it->ldeg_a = a.ldegree(it->sym);
+ it->ldeg_b = b.ldegree(it->sym);
++it;
}
std::sort(v.begin(), v.end());
+
#if 0
std::clog << "Symbols:\n";
it = v.begin(); itend = v.end();
while (it != itend) {
- std::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 << ", max_lcnops=" << it->max_lcnops << endl;
- std::clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
+ std::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 << ", max_lcnops=" << it->max_lcnops << endl;
+ std::clog << " lcoeff_a=" << a.lcoeff(it->sym) << ", lcoeff_b=" << b.lcoeff(it->sym) << endl;
++it;
}
#endif
{
if (e.info(info_flags::rational))
return lcm(ex_to<numeric>(e).denom(), l);
- else if (is_ex_exactly_of_type(e, add)) {
- numeric c = _num1;
- for (unsigned i=0; i<e.nops(); i++)
+ else if (is_exactly_a<add>(e)) {
+ numeric c = *_num1_p;
+ for (size_t 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);
+ } else if (is_exactly_a<mul>(e)) {
+ numeric c = *_num1_p;
+ for (size_t i=0; i<e.nops(); i++)
+ c *= lcmcoeff(e.op(i), *_num1_p);
return lcm(c, l);
- } else if (is_ex_exactly_of_type(e, power)) {
- if (is_ex_exactly_of_type(e.op(0), symbol))
+ } else if (is_exactly_a<power>(e)) {
+ if (is_a<symbol>(e.op(0)))
return l;
else
return pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
* @return LCM of denominators of coefficients */
static numeric lcm_of_coefficients_denominators(const ex &e)
{
- return lcmcoeff(e, _num1);
+ return lcmcoeff(e, *_num1_p);
}
/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
* @param lcm LCM to multiply in */
static ex multiply_lcm(const ex &e, const numeric &lcm)
{
- if (is_ex_exactly_of_type(e, mul)) {
- unsigned num = e.nops();
+ if (is_exactly_a<mul>(e)) {
+ size_t num = e.nops();
exvector v; v.reserve(num + 1);
- numeric lcm_accum = _num1;
- for (unsigned i=0; i<e.nops(); i++) {
- numeric op_lcm = lcmcoeff(e.op(i), _num1);
+ numeric lcm_accum = *_num1_p;
+ for (size_t i=0; i<num; i++) {
+ numeric op_lcm = lcmcoeff(e.op(i), *_num1_p);
v.push_back(multiply_lcm(e.op(i), op_lcm));
lcm_accum *= op_lcm;
}
v.push_back(lcm / lcm_accum);
return (new mul(v))->setflag(status_flags::dynallocated);
- } else if (is_ex_exactly_of_type(e, add)) {
- unsigned num = e.nops();
+ } else if (is_exactly_a<add>(e)) {
+ size_t num = e.nops();
exvector v; v.reserve(num);
- for (unsigned i=0; i<num; i++)
+ for (size_t i=0; i<num; i++)
v.push_back(multiply_lcm(e.op(i), lcm));
return (new add(v))->setflag(status_flags::dynallocated);
- } else if (is_ex_exactly_of_type(e, power)) {
- if (is_ex_exactly_of_type(e.op(0), symbol))
+ } else if (is_exactly_a<power>(e)) {
+ if (is_a<symbol>(e.op(0)))
return e * lcm;
else
return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
/** Compute the integer content (= GCD of all numeric coefficients) of an
- * expanded polynomial.
+ * expanded polynomial. For a polynomial with rational coefficients, this
+ * returns g/l where g is the GCD of the coefficients' numerators and l
+ * is the LCM of the coefficients' denominators.
*
- * @param e expanded polynomial
* @return integer content */
-numeric ex::integer_content(void) const
+numeric ex::integer_content() const
{
- GINAC_ASSERT(bp!=0);
return bp->integer_content();
}
-numeric basic::integer_content(void) const
+numeric basic::integer_content() const
{
- return _num1;
+ return *_num1_p;
}
-numeric numeric::integer_content(void) const
+numeric numeric::integer_content() const
{
return abs(*this);
}
-numeric add::integer_content(void) const
+numeric add::integer_content() const
{
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
- numeric c = _num0;
+ numeric c = *_num0_p, l = *_num1_p;
while (it != itend) {
GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
- c = gcd(ex_to<numeric>(it->coeff), c);
+ c = gcd(ex_to<numeric>(it->coeff).numer(), c);
+ l = lcm(ex_to<numeric>(it->coeff).denom(), l);
it++;
}
GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
- c = gcd(ex_to<numeric>(overall_coeff),c);
- return c;
+ c = gcd(ex_to<numeric>(overall_coeff).numer(), c);
+ l = lcm(ex_to<numeric>(overall_coeff).denom(), l);
+ return c/l;
}
-numeric mul::integer_content(void) const
+numeric mul::integer_content() const
{
#ifdef DO_GINAC_ASSERT
epvector::const_iterator it = seq.begin();
* @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)
+ex quo(const ex &a, const ex &b, const ex &x, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("quo: division by zero"));
- if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
+ if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b))
return a / b;
#if FAST_COMPARE
if (a.is_equal(b))
int bdeg = b.degree(x);
int rdeg = r.degree(x);
ex blcoeff = b.expand().coeff(x, bdeg);
- bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
- exvector v; v.reserve(rdeg - bdeg + 1);
+ bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
+ exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
while (rdeg >= bdeg) {
ex term, rcoeff = r.coeff(x, rdeg);
if (blcoeff_is_numeric)
* @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)
+ex rem(const ex &a, const ex &b, const ex &x, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("rem: division by zero"));
- if (is_ex_exactly_of_type(a, numeric)) {
- if (is_ex_exactly_of_type(b, numeric))
+ if (is_exactly_a<numeric>(a)) {
+ if (is_exactly_a<numeric>(b))
return _ex0;
else
return a;
int bdeg = b.degree(x);
int rdeg = r.degree(x);
ex blcoeff = b.expand().coeff(x, bdeg);
- bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
+ bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
while (rdeg >= bdeg) {
ex term, rcoeff = r.coeff(x, rdeg);
if (blcoeff_is_numeric)
* @param a rational function in x
* @param x a is a function of x
* @return decomposed function. */
-ex decomp_rational(const ex &a, const symbol &x)
+ex decomp_rational(const ex &a, const ex &x)
{
ex nd = numer_denom(a);
ex numer = nd.op(0), denom = nd.op(1);
ex q = quo(numer, denom, x);
- if (is_ex_exactly_of_type(q, fail))
+ if (is_exactly_a<fail>(q))
return a;
else
return q + rem(numer, denom, x) / denom;
}
-/** Pseudo-remainder of polynomials a(x) and b(x) in Z[x].
+/** Pseudo-remainder of polynomials a(x) and b(x) in Q[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 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)
+ * @return pseudo-remainder of a(x) and b(x) in Q[x] */
+ex prem(const ex &a, const ex &b, const ex &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))
+ if (is_exactly_a<numeric>(a)) {
+ if (is_exactly_a<numeric>(b))
return _ex0;
else
return b;
}
-/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Z[x].
+/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Q[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)
+ * @return sparse pseudo-remainder of a(x) and b(x) in Q[x] */
+ex sprem(const ex &a, const ex &b, const ex &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))
+ if (is_exactly_a<numeric>(a)) {
+ if (is_exactly_a<numeric>(b))
return _ex0;
else
return b;
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return "true" when exact division succeeds (quotient returned in q),
- * "false" otherwise */
+ * "false" otherwise (q left untouched) */
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())
+ if (a.is_zero()) {
+ q = _ex0;
return true;
- if (is_ex_exactly_of_type(b, numeric)) {
+ }
+ if (is_exactly_a<numeric>(b)) {
q = a / b;
return true;
- } else if (is_ex_exactly_of_type(a, numeric))
+ } else if (is_exactly_a<numeric>(a))
return false;
#if FAST_COMPARE
if (a.is_equal(b)) {
throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));
// Find first symbol
- const symbol *x;
+ ex x;
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())
+ if (r.is_zero()) {
+ q = _ex0;
return true;
- int bdeg = b.degree(*x);
- int rdeg = r.degree(*x);
- ex blcoeff = b.expand().coeff(*x, bdeg);
- bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
- exvector v; v.reserve(rdeg - bdeg + 1);
+ }
+ int bdeg = b.degree(x);
+ int rdeg = r.degree(x);
+ ex blcoeff = b.expand().coeff(x, bdeg);
+ bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
+ exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(*x, rdeg);
+ ex term, rcoeff = r.coeff(x, rdeg);
if (blcoeff_is_numeric)
term = rcoeff / blcoeff;
else
if (!divide(rcoeff, blcoeff, term, false))
return false;
- term *= power(*x, rdeg - bdeg);
+ term *= power(x, rdeg - bdeg);
v.push_back(term);
r -= (term * b).expand();
if (r.is_zero()) {
q = (new add(v))->setflag(status_flags::dynallocated);
return true;
}
- rdeg = r.degree(*x);
+ rdeg = r.degree(x);
}
return false;
}
/** Exact polynomial division of a(X) by b(X) in Z[X].
* This functions works like divide() but the input and output polynomials are
* in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike
- * divide(), it doesnยดt check whether the input polynomials really are integer
+ * divide(), it doesn't check whether the input polynomials really are integer
* polynomials, so be careful of what you pass in. Also, you have to run
* get_symbol_stats() over the input polynomials before calling this function
* and pass an iterator to the first element of the sym_desc vector. This
q = a;
return true;
}
- if (is_ex_exactly_of_type(a, numeric)) {
- if (is_ex_exactly_of_type(b, numeric)) {
+ if (is_exactly_a<numeric>(a)) {
+ if (is_exactly_a<numeric>(b)) {
q = a / b;
return q.info(info_flags::integer);
} else
}
#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 symbol *x = var->sym;
+ const ex &x = var->sym;
// Compare degrees
- int adeg = a.degree(*x), bdeg = b.degree(*x);
+ int adeg = a.degree(x), bdeg = b.degree(x);
if (bdeg > adeg)
return false;
// Compute values at evaluation points 0..adeg
vector<numeric> alpha; alpha.reserve(adeg + 1);
exvector u; u.reserve(adeg + 1);
- numeric point = _num0;
+ numeric point = *_num0_p;
ex c;
for (i=0; i<=adeg; i++) {
- ex bs = b.subs(*x == point);
+ ex bs = b.subs(x == point, subs_options::no_pattern);
while (bs.is_zero()) {
- point += _num1;
- bs = b.subs(*x == point);
+ point += *_num1_p;
+ bs = b.subs(x == point, subs_options::no_pattern);
}
- if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
+ if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1))
return false;
alpha.push_back(point);
u.push_back(c);
- point += _num1;
+ point += *_num1_p;
}
// Compute inverses
vector<numeric> rcp; rcp.reserve(adeg + 1);
- rcp.push_back(_num0);
+ rcp.push_back(*_num0_p);
for (k=1; k<=adeg; k++) {
numeric product = alpha[k] - alpha[0];
for (i=1; i<k; i++)
// Convert from Newton form to standard form
c = v[adeg];
for (k=adeg-1; k>=0; k--)
- c = c * (*x - alpha[k]) + v[k];
+ c = c * (x - alpha[k]) + v[k];
- if (c.degree(*x) == (adeg - bdeg)) {
+ if (c.degree(x) == (adeg - bdeg)) {
q = c.expand();
return true;
} else
return true;
int rdeg = adeg;
ex eb = b.expand();
- ex blcoeff = eb.coeff(*x, bdeg);
- exvector v; v.reserve(rdeg - bdeg + 1);
+ ex blcoeff = eb.coeff(x, bdeg);
+ exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(*x, rdeg);
+ ex term, rcoeff = r.coeff(x, rdeg);
if (!divide_in_z(rcoeff, blcoeff, term, var+1))
break;
- term = (term * power(*x, rdeg - bdeg)).expand();
+ term = (term * power(x, rdeg - bdeg)).expand();
v.push_back(term);
r -= (term * eb).expand();
if (r.is_zero()) {
#endif
return true;
}
- rdeg = r.degree(*x);
+ rdeg = r.degree(x);
}
#if USE_REMEMBER
dr_remember[ex2(a, b)] = exbool(q, false);
*/
/** Compute unit part (= sign of leading coefficient) of a multivariate
- * polynomial in Z[x]. The product of unit part, content part, and primitive
+ * polynomial in Q[x]. The product of unit part, content part, and primitive
* part is the polynomial itself.
*
- * @param x variable in which to compute the unit part
+ * @param x main variable
* @return unit part
- * @see ex::content, ex::primpart */
-ex ex::unit(const symbol &x) const
+ * @see ex::content, ex::primpart, ex::unitcontprim */
+ex ex::unit(const ex &x) const
{
ex c = expand().lcoeff(x);
- if (is_ex_exactly_of_type(c, numeric))
- return c < _ex0 ? _ex_1 : _ex1;
+ if (is_exactly_a<numeric>(c))
+ return c.info(info_flags::negative) ?_ex_1 : _ex1;
else {
- const symbol *y;
+ ex y;
if (get_first_symbol(c, y))
- return c.unit(*y);
+ return c.unit(y);
else
throw(std::invalid_argument("invalid expression in unit()"));
}
/** Compute content part (= unit normal GCD of all coefficients) of a
- * multivariate polynomial in Z[x]. The product of unit part, content part,
+ * multivariate polynomial in Q[x]. The product of unit part, content part,
* and primitive part is the polynomial itself.
*
- * @param x variable in which to compute the content part
+ * @param x main variable
* @return content part
- * @see ex::unit, ex::primpart */
-ex ex::content(const symbol &x) const
+ * @see ex::unit, ex::primpart, ex::unitcontprim */
+ex ex::content(const ex &x) const
{
- if (is_zero())
- return _ex0;
- if (is_ex_exactly_of_type(*this, numeric))
+ if (is_exactly_a<numeric>(*this))
return info(info_flags::negative) ? -*this : *this;
+
ex e = expand();
if (e.is_zero())
return _ex0;
- // First, try the integer content
+ // First, divide out the integer content (which we can calculate very efficiently).
+ // If the leading coefficient of the quotient is an integer, we are done.
ex c = e.integer_content();
ex r = e / c;
- ex lcoeff = r.lcoeff(x);
+ int deg = r.degree(x);
+ ex lcoeff = r.coeff(x, deg);
if (lcoeff.info(info_flags::integer))
return c;
// GCD of all coefficients
- int deg = e.degree(x);
- int ldeg = e.ldegree(x);
+ int ldeg = r.ldegree(x);
if (deg == ldeg)
- return e.lcoeff(x) / e.unit(x);
- c = _ex0;
+ return lcoeff * c / lcoeff.unit(x);
+ ex cont = _ex0;
for (int i=ldeg; i<=deg; i++)
- c = gcd(e.coeff(x, i), c, NULL, NULL, false);
- return c;
+ cont = gcd(r.coeff(x, i), cont, NULL, NULL, false);
+ return cont * c;
}
-/** Compute primitive part of a multivariate polynomial in Z[x].
- * The product of unit part, content part, and primitive part is the
- * polynomial itself.
+/** Compute primitive part of a multivariate polynomial in Q[x]. The result
+ * will be a unit-normal polynomial with a content part of 1. The product
+ * of unit part, content part, and primitive part is the polynomial itself.
*
- * @param x variable in which to compute the primitive part
+ * @param x main variable
* @return primitive part
- * @see ex::unit, ex::content */
-ex ex::primpart(const symbol &x) const
+ * @see ex::unit, ex::content, ex::unitcontprim */
+ex ex::primpart(const ex &x) const
{
- if (is_zero())
- return _ex0;
- if (is_ex_exactly_of_type(*this, numeric))
- return _ex1;
-
- ex c = content(x);
- if (c.is_zero())
- return _ex0;
- ex u = unit(x);
- if (is_ex_exactly_of_type(c, numeric))
- return *this / (c * u);
- else
- return quo(*this, c * u, x, false);
+ // We need to compute the unit and content anyway, so call unitcontprim()
+ ex u, c, p;
+ unitcontprim(x, u, c, p);
+ return p;
}
-/** Compute primitive part of a multivariate polynomial in Z[x] when the
+/** Compute primitive part of a multivariate polynomial in Q[x] when the
* content part is already known. This function is faster in computing the
* primitive part than the previous function.
*
- * @param x variable in which to compute the primitive part
+ * @param x main variable
* @param c previously computed content part
* @return primitive part */
-ex ex::primpart(const symbol &x, const ex &c) const
+ex ex::primpart(const ex &x, const ex &c) const
{
- if (is_zero())
+ if (is_zero() || c.is_zero())
return _ex0;
- if (c.is_zero())
- return _ex0;
- if (is_ex_exactly_of_type(*this, numeric))
+ if (is_exactly_a<numeric>(*this))
return _ex1;
+ // Divide by unit and content to get primitive part
ex u = unit(x);
- if (is_ex_exactly_of_type(c, numeric))
+ if (is_exactly_a<numeric>(c))
return *this / (c * u);
else
return quo(*this, c * u, x, false);
}
-/*
- * 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.
+/** Compute unit part, content part, and primitive part of a multivariate
+ * polynomial in Q[x]. The product of the three parts is the polynomial
+ * itself.
*
- * @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)
+ * @param x main variable
+ * @param u unit part (returned)
+ * @param c content part (returned)
+ * @param p primitive part (returned)
+ * @see ex::unit, ex::content, ex::primpart */
+void ex::unitcontprim(const ex &x, ex &u, ex &c, ex &p) const
{
-//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;
+ // Quick check for zero (avoid expanding)
+ if (is_zero()) {
+ u = _ex1;
+ c = p = _ex0;
+ return;
+ }
+
+ // Special case: input is a number
+ if (is_exactly_a<numeric>(*this)) {
+ if (info(info_flags::negative)) {
+ u = _ex_1;
+ c = abs(ex_to<numeric>(*this));
+ } else {
+ u = _ex1;
+ c = *this;
+ }
+ p = _ex1;
+ return;
}
-}
-
-
-/** 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;
+ // Expand input polynomial
+ ex e = expand();
+ if (e.is_zero()) {
+ u = _ex1;
+ c = p = _ex0;
+ return;
}
- // 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);
+ // Compute unit and content
+ u = unit(x);
+ c = content(x);
- // 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);
+ // Divide by unit and content to get primitive part
+ if (c.is_zero()) {
+ p = _ex0;
+ return;
}
+ if (is_exactly_a<numeric>(c))
+ p = *this / (c * u);
+ else
+ p = quo(e, c * u, x, false);
}
-/** 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_ex_exactly_of_type(r, numeric))
- return gamma;
- else
- return gamma * r.primpart(*x);
- }
-
- ri = c.expand().lcoeff(*x);
- delta = cdeg - ddeg;
- }
-}
-
+/*
+ * GCD of multivariate polynomials
+ */
/** 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
// The first symbol is our main variable
- const symbol &x = *(var->sym);
+ const ex &x = var->sym;
// Sort c and d so that c has higher degree
ex c, d;
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);
if (ddeg == 0) {
- if (is_ex_exactly_of_type(r, numeric))
+ if (is_exactly_a<numeric>(r))
return gamma;
else
return gamma * r.primpart(x);
}
// Next element of subresultant sequence
-//std::clog << " calculating next subresultant...\n";
ri = c.expand().lcoeff(x);
if (delta == 1)
psi = ri;
/** Return maximum (absolute value) coefficient of a polynomial.
* This function is used internally by heur_gcd().
*
- * @param e expanded multivariate polynomial
* @return maximum coefficient
* @see heur_gcd */
-numeric ex::max_coefficient(void) const
+numeric ex::max_coefficient() const
{
- GINAC_ASSERT(bp!=0);
return bp->max_coefficient();
}
/** Implementation ex::max_coefficient().
* @see heur_gcd */
-numeric basic::max_coefficient(void) const
+numeric basic::max_coefficient() const
{
- return _num1;
+ return *_num1_p;
}
-numeric numeric::max_coefficient(void) const
+numeric numeric::max_coefficient() const
{
return abs(*this);
}
-numeric add::max_coefficient(void) const
+numeric add::max_coefficient() const
{
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
return cur_max;
}
-numeric mul::max_coefficient(void) const
+numeric mul::max_coefficient() const
{
#ifdef DO_GINAC_ASSERT
epvector::const_iterator it = seq.begin();
/** xi-adic polynomial interpolation */
-static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x, int degree_hint = 1)
+static ex interpolate(const ex &gamma, const numeric &xi, const ex &x, int degree_hint = 1)
{
exvector g; g.reserve(degree_hint);
ex e = gamma;
* 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)
{
-//std::clog << "heur_gcd(" << a << "," << b << ")\n";
#if STATISTICS
heur_gcd_called++;
#endif
// 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_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+ if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
if (ca)
*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
- const symbol &x = *(var->sym);
+ const ex &x = var->sym;
// Remove integer content
numeric gc = gcd(a.integer_content(), b.integer_content());
numeric mq = q.max_coefficient();
numeric xi;
if (mp > mq)
- xi = mq * _num2 + _num2;
+ xi = mq * (*_num2_p) + (*_num2_p);
else
- xi = mp * _num2 + _num2;
+ xi = mp * (*_num2_p) + (*_num2_p);
// 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();
}
// Apply evaluation homomorphism and calculate GCD
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)) {
-
+ 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;
- 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;
- }
+ res = g;
+ return true;
}
-#endif
}
// 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;
}
/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
- * and b(X) in Z[X].
+ * and b(X) in Z[X]. Optionally also compute the cofactors of a and b,
+ * defined by a = ca * gcd(a, b) and b = cb * gcd(a, b).
*
* @param a first multivariate polynomial
* @param b second multivariate polynomial
+ * @param ca pointer to expression that will receive the cofactor of a, or NULL
+ * @param cb pointer to expression that will receive the cofactor of b, or NULL
* @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)
{
-//std::clog << "gcd(" << a << "," << b << ")\n";
#if STATISTICS
gcd_called++;
#endif
// GCD of numerics -> CLN
- if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+ if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
if (ca || cb) {
if (g.is_zero()) {
}
// 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())
+ if (is_exactly_a<mul>(a)) {
+ if (is_exactly_a<mul>(b) && b.nops() > a.nops())
goto factored_b;
factored_a:
- unsigned num = a.nops();
+ size_t num = a.nops();
exvector g; g.reserve(num);
exvector acc_ca; acc_ca.reserve(num);
ex part_b = b;
- for (unsigned i=0; i<num; i++) {
+ for (size_t i=0; i<num; i++) {
ex part_ca, part_cb;
g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
acc_ca.push_back(part_ca);
if (cb)
*cb = part_b;
return (new mul(g))->setflag(status_flags::dynallocated);
- } else if (is_ex_exactly_of_type(b, mul)) {
- if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
+ } else if (is_exactly_a<mul>(b)) {
+ if (is_exactly_a<mul>(a) && a.nops() > b.nops())
goto factored_a;
factored_b:
- unsigned num = b.nops();
+ size_t num = b.nops();
exvector g; g.reserve(num);
exvector acc_cb; acc_cb.reserve(num);
ex part_a = a;
- for (unsigned i=0; i<num; i++) {
+ for (size_t i=0; i<num; i++) {
ex part_ca, part_cb;
g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
acc_cb.push_back(part_cb);
#if FAST_COMPARE
// Input polynomials of the form poly^n are sometimes also trivial
- if (is_ex_exactly_of_type(a, power)) {
+ if (is_exactly_a<power>(a)) {
ex p = a.op(0);
- if (is_ex_exactly_of_type(b, power)) {
- if (p.is_equal(b.op(0))) {
+ const ex& exp_a = a.op(1);
+ if (is_exactly_a<power>(b)) {
+ ex pb = b.op(0);
+ const ex& exp_b = b.op(1);
+ if (p.is_equal(pb)) {
// a = p^n, b = p^m, gcd = p^min(n, m)
- ex exp_a = a.op(1), exp_b = b.op(1);
if (exp_a < exp_b) {
if (ca)
*ca = _ex1;
*cb = _ex1;
return power(p, exp_b);
}
- }
+ } else {
+ ex p_co, pb_co;
+ ex p_gcd = gcd(p, pb, &p_co, &pb_co, check_args);
+ if (p_gcd.is_equal(_ex1)) {
+ // a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==>
+ // gcd(a,b) = 1
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ // XXX: do I need to check for p_gcd = -1?
+ } else {
+ // there are common factors:
+ // a(x) = g(x)^n A(x)^n, b(x) = g(x)^m B(x)^m ==>
+ // gcd(a, b) = g(x)^n gcd(A(x)^n, g(x)^(n-m) B(x)^m
+ if (exp_a < exp_b) {
+ return power(p_gcd, exp_a)*
+ gcd(power(p_co, exp_a), power(p_gcd, exp_b-exp_a)*power(pb_co, exp_b), ca, cb, false);
+ } else {
+ return power(p_gcd, exp_b)*
+ gcd(power(p_gcd, exp_a - exp_b)*power(p_co, exp_a), power(pb_co, exp_b), ca, cb, false);
+ }
+ } // p_gcd.is_equal(_ex1)
+ } // p.is_equal(pb)
+
} else {
if (p.is_equal(b)) {
// a = p^n, b = p, gcd = p
if (cb)
*cb = _ex1;
return p;
+ }
+
+ ex p_co, bpart_co;
+ ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);
+
+ if (p_gcd.is_equal(_ex1)) {
+ // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ } else {
+ // a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
+ return p_gcd*gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false);
}
- }
- } else if (is_ex_exactly_of_type(b, power)) {
+ } // is_exactly_a<power>(b)
+
+ } else if (is_exactly_a<power>(b)) {
ex p = b.op(0);
if (p.is_equal(a)) {
// a = p, b = p^n, gcd = p
*cb = power(p, b.op(1) - 1);
return p;
}
+
+ ex p_co, apart_co;
+ const ex& exp_b(b.op(1));
+ ex p_gcd = gcd(a, p, &apart_co, &p_co, false);
+ if (p_gcd.is_equal(_ex1)) {
+ // b=p(x)^n, gcd(a, p) = 1 ==> gcd(a, b) == 1
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ } else {
+ // there are common factors:
+ // a(x) = g(x) A(x), b(x) = g(x)^n B(x)^n ==> gcd = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
+
+ 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
}
#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 is our main variable
+ // 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 symbol &x = *(var->sym);
+ const ex &x = var->sym;
// Cancel trivial common factor
int ldeg_a = var->ldeg_a;
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 (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.unit(x) * bex.primpart(x, c);
+ *cb *= bex_u * bex_p;
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);
+ } 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.unit(x) * aex.primpart(x, c);
+ *ca *= aex_u * aex_p;
return g;
}
- 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) {
- g = fail();
- }
- if (is_ex_exactly_of_type(g, fail)) {
-//std::clog << "heuristics failed" << std::endl;
+ ex g;
+ bool found = heur_gcd(g, aex, bex, ca, cb, var);
+ if (!found) {
#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;
}
* @return the LCM as a new expression */
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))
+ if (is_exactly_a<numeric>(a) && is_exactly_a<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"));
*/
/** Compute square-free factorization of multivariate polynomial a(x) using
- * Yunยดs algorithm. Used internally by sqrfree().
+ * Yun's algorithm. Used internally by sqrfree().
*
* @param a multivariate polynomial over Z[X], treated here as univariate
* polynomial in x.
return res;
}
-/** Compute square-free factorization of multivariate polynomial in Q[X].
+
+/** Compute a square-free factorization of a multivariate polynomial in Q[X].
*
* @param a multivariate polynomial over Q[X]
- * @param x lst of variables to factor in, may be left empty for autodetection
- * @return polynomial a in square-free factored form. */
+ * @param l lst of variables to factor in, may be left empty for autodetection
+ * @return a square-free factorization of \p a.
+ *
+ * \note
+ * A polynomial \f$p(X) \in C[X]\f$ is said <EM>square-free</EM>
+ * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$
+ * are such that
+ * \f[
+ * p(X) = q(X)^2 r(X),
+ * \f]
+ * we have \f$q(X) \in C\f$.
+ * This means that \f$p(X)\f$ has no repeated factors, apart
+ * eventually from constants.
+ * Given a polynomial \f$p(X) \in C[X]\f$, we say that the
+ * decomposition
+ * \f[
+ * p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r}
+ * \f]
+ * is a <EM>square-free factorization</EM> of \f$p(X)\f$ if the
+ * following conditions hold:
+ * -# \f$b \in C\f$ and \f$b \neq 0\f$;
+ * -# \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$;
+ * -# the degree of the polynomial \f$p_i\f$ is strictly positive
+ * for \f$i = 1, \ldots, r\f$;
+ * -# the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free.
+ *
+ * Square-free factorizations need not be unique. For example, if
+ * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$
+ * into \f$-p_i(X)\f$.
+ * Observe also that the factors \f$p_i(X)\f$ need not be irreducible
+ * polynomials.
+ */
ex sqrfree(const ex &a, const lst &l)
{
- if (is_a<numeric>(a) || // algorithm does not trap a==0
+ if (is_exactly_a<numeric>(a) || // algorithm does not trap a==0
is_a<symbol>(a)) // shortcut
return a;
get_symbol_stats(a, _ex0, sdv);
sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end();
while (it != itend) {
- args.append(*it->sym);
+ args.append(it->sym);
++it;
}
} else {
}
// Find the symbol to factor in at this stage
- if (!is_ex_of_type(args.op(0), symbol))
+ if (!is_a<symbol>(args.op(0)))
throw (std::runtime_error("sqrfree(): invalid factorization variable"));
const symbol &x = ex_to<symbol>(args.op(0));
const ex tmp = multiply_lcm(a,lcm);
// find the factors
- exvector factors = sqrfree_yun(tmp,x);
+ exvector factors = sqrfree_yun(tmp, x);
// construct the next list of symbols with the first element popped
lst newargs = args;
return result * lcm.inverse();
}
+
/** Compute square-free partial fraction decomposition of rational function
* a(x).
*
// Factorize denominator and compute cofactors
exvector yun = sqrfree_yun(denom, x);
//clog << "yun factors: " << exprseq(yun) << endl;
- unsigned num_yun = yun.size();
+ size_t num_yun = yun.size();
exvector factor; factor.reserve(num_yun);
exvector cofac; cofac.reserve(num_yun);
- for (unsigned i=0; i<num_yun; i++) {
+ for (size_t i=0; i<num_yun; i++) {
if (!yun[i].is_equal(_ex1)) {
- for (unsigned j=0; j<=i; j++) {
+ for (size_t j=0; j<=i; j++) {
factor.push_back(pow(yun[i], j+1));
ex prod = _ex1;
- for (unsigned k=0; k<num_yun; k++) {
+ for (size_t k=0; k<num_yun; k++) {
if (k == i)
prod *= pow(yun[k], i-j);
else
}
}
}
- unsigned num_factors = factor.size();
+ size_t num_factors = factor.size();
//clog << "factors : " << exprseq(factor) << endl;
//clog << "cofactors: " << exprseq(cofac) << endl;
matrix sys(max_denom_deg + 1, num_factors);
matrix rhs(max_denom_deg + 1, 1);
for (int i=0; i<=max_denom_deg; i++) {
- for (unsigned j=0; j<num_factors; j++)
+ for (size_t j=0; j<num_factors; j++)
sys(i, j) = cofac[j].coeff(x, i);
rhs(i, 0) = red_numer.coeff(x, i);
}
// Solve resulting linear system
matrix vars(num_factors, 1);
- for (unsigned i=0; i<num_factors; i++)
+ for (size_t i=0; i<num_factors; i++)
vars(i, 0) = symbol();
matrix sol = sys.solve(vars, rhs);
// Sum up decomposed fractions
ex sum = 0;
- for (unsigned i=0; i<num_factors; i++)
+ for (size_t i=0; i<num_factors; i++)
sum += sol(i, 0) / factor[i];
return red_poly + sum;
/** Create a symbol for replacing the expression "e" (or return a previously
- * assigned symbol). The symbol is appended to sym_lst and returned, the
- * expression is appended to repl_lst.
+ * assigned symbol). The symbol and expression are appended to repl, for
+ * a later application of subs().
* @see ex::normal */
-static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
+static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup)
{
- // Expression already in repl_lst? Then return the assigned symbol
- for (unsigned i=0; i<repl_lst.nops(); i++)
- if (repl_lst.op(i).is_equal(e))
- return sym_lst.op(i);
+ // Expression already replaced? Then return the assigned symbol
+ exmap::const_iterator it = rev_lookup.find(e);
+ if (it != rev_lookup.end())
+ return it->second;
// Otherwise create new symbol and add to list, taking care that the
- // replacement expression doesn't contain symbols from the sym_lst
+ // replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
- symbol s;
- ex es(s);
- ex e_replaced = e.subs(sym_lst, repl_lst);
- sym_lst.append(es);
- repl_lst.append(e_replaced);
+ ex es = (new symbol)->setflag(status_flags::dynallocated);
+ ex e_replaced = e.subs(repl, subs_options::no_pattern);
+ repl.insert(std::make_pair(es, e_replaced));
+ rev_lookup.insert(std::make_pair(e_replaced, es));
return es;
}
/** Create a symbol for replacing the expression "e" (or return a previously
- * assigned symbol). An expression of the form "symbol == expression" is added
- * to repl_lst and the symbol is returned.
- * @see basic::to_rational */
-static ex replace_with_symbol(const ex &e, lst &repl_lst)
-{
- // Expression already in repl_lst? Then return the assigned symbol
- 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);
+ * assigned symbol). The symbol and expression are appended to repl, and the
+ * symbol is returned.
+ * @see basic::to_rational
+ * @see basic::to_polynomial */
+static ex replace_with_symbol(const ex & e, exmap & repl)
+{
+ // Expression already replaced? Then return the assigned symbol
+ for (exmap::const_iterator it = repl.begin(); it != repl.end(); ++it)
+ if (it->second.is_equal(e))
+ return it->first;
// Otherwise create new symbol and add to list, taking care that the
- // replacement expression doesn't contain symbols from the sym_lst
+ // replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
- symbol s;
- ex es(s);
- ex e_replaced = e.subs(repl_lst);
- repl_lst.append(es == e_replaced);
+ ex es = (new symbol)->setflag(status_flags::dynallocated);
+ ex e_replaced = e.subs(repl, subs_options::no_pattern);
+ repl.insert(std::make_pair(es, e_replaced));
return es;
}
/** Default implementation of ex::normal(). It normalizes the children and
* replaces the object with a temporary symbol.
* @see ex::normal */
-ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex basic::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (nops() == 0)
- return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
else {
if (level == 1)
- return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
else if (level == -max_recursion_level)
throw(std::runtime_error("max recursion level reached"));
else {
normal_map_function map_normal(level - 1);
- return (new lst(replace_with_symbol(map(map_normal), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(map(map_normal), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
}
}
/** 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
+ex symbol::normal(exmap & repl, exmap & rev_lookup, int level) const
{
return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
}
* into re+I*im and replaces I and non-rational real numbers with a temporary
* symbol.
* @see ex::normal */
-ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex numeric::normal(exmap & repl, exmap & rev_lookup, int level) const
{
numeric num = numer();
ex numex = num;
if (num.is_real()) {
if (!num.is_integer())
- numex = replace_with_symbol(numex, sym_lst, repl_lst);
+ numex = replace_with_symbol(numex, repl, rev_lookup);
} 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);
+ ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl, rev_lookup);
+ ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl, rev_lookup);
+ numex = re_ex + im_ex * replace_with_symbol(I, repl, rev_lookup);
}
// Denominator is always a real integer (see numeric::denom())
{
ex num = n;
ex den = d;
- numeric pre_factor = _num1;
+ numeric pre_factor = *_num1_p;
//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
// Make denominator unit normal (i.e. coefficient of first symbol
// as defined by get_first_symbol() is made positive)
- const symbol *x;
- if (get_first_symbol(den, x)) {
- GINAC_ASSERT(is_exactly_a<numeric>(den.unit(*x)));
- if (ex_to<numeric>(den.unit(*x)).is_negative()) {
+ if (is_exactly_a<numeric>(den)) {
+ if (ex_to<numeric>(den).is_negative()) {
num *= _ex_1;
den *= _ex_1;
}
+ } else {
+ ex x;
+ if (get_first_symbol(den, x)) {
+ GINAC_ASSERT(is_exactly_a<numeric>(den.unit(x)));
+ if (ex_to<numeric>(den.unit(x)).is_negative()) {
+ num *= _ex_1;
+ den *= _ex_1;
+ }
+ }
}
// Return result as list
/** Implementation of ex::normal() for a sum. It expands terms and performs
* fractional addition.
* @see ex::normal */
-ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex add::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (level == 1)
- return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
else if (level == -max_recursion_level)
throw(std::runtime_error("max recursion level reached"));
dens.reserve(seq.size()+1);
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
- ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+ ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(repl, rev_lookup, level-1);
nums.push_back(n.op(0));
dens.push_back(n.op(1));
it++;
}
- ex n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+ ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, level-1);
nums.push_back(n.op(0));
dens.push_back(n.op(1));
GINAC_ASSERT(nums.size() == dens.size());
/** Implementation of ex::normal() for a product. It cancels common factors
* from fractions.
* @see ex::normal() */
-ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex mul::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (level == 1)
- return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
else if (level == -max_recursion_level)
throw(std::runtime_error("max recursion level reached"));
ex n;
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
- n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+ n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(repl, rev_lookup, level-1);
num.push_back(n.op(0));
den.push_back(n.op(1));
it++;
}
- n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+ n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, level-1);
num.push_back(n.op(0));
den.push_back(n.op(1));
}
-/** Implementation of ex::normal() for powers. It normalizes the basis,
+/** Implementation of ex::normal([B) for powers. It normalizes the basis,
* distributes integer exponents to numerator and denominator, and replaces
* non-integer powers by temporary symbols.
* @see ex::normal */
-ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex power::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (level == 1)
- return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
else if (level == -max_recursion_level)
throw(std::runtime_error("max recursion level reached"));
// Normalize basis and exponent (exponent gets reassembled)
- ex n_basis = ex_to<basic>(basis).normal(sym_lst, repl_lst, level-1);
- ex n_exponent = ex_to<basic>(exponent).normal(sym_lst, repl_lst, level-1);
+ ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup, level-1);
+ ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup, level-1);
n_exponent = n_exponent.op(0) / n_exponent.op(1);
if (n_exponent.info(info_flags::integer)) {
if (n_exponent.info(info_flags::positive)) {
// (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);
+ return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
} else if (n_exponent.info(info_flags::negative)) {
if (n_basis.op(1).is_equal(_ex1)) {
// a^-x -> {1, sym(a^x)}
- return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+ return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), repl, rev_lookup)))->setflag(status_flags::dynallocated);
} else {
// (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);
+ return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), repl, rev_lookup), _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), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
/** Implementation of ex::normal() for pseries. It normalizes each coefficient
* and replaces the series by a temporary symbol.
* @see ex::normal */
-ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const
{
epvector newseq;
epvector::const_iterator i = seq.begin(), end = seq.end();
++i;
}
ex n = pseries(relational(var,point), newseq);
- return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ return (new lst(replace_with_symbol(n, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
* @return normalized expression */
ex ex::normal(int level) const
{
- lst sym_lst, repl_lst;
+ exmap repl, rev_lookup;
- ex e = bp->normal(sym_lst, repl_lst, level);
+ ex e = bp->normal(repl, rev_lookup, level);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
- if (sym_lst.nops() > 0)
- e = e.subs(sym_lst, repl_lst);
+ if (!repl.empty())
+ e = e.subs(repl, subs_options::no_pattern);
// Convert {numerator, denominator} form back to fraction
return e.op(0) / e.op(1);
*
* @see ex::normal
* @return numerator */
-ex ex::numer(void) const
+ex ex::numer() const
{
- lst sym_lst, repl_lst;
+ exmap repl, rev_lookup;
- ex e = bp->normal(sym_lst, repl_lst, 0);
+ ex e = bp->normal(repl, rev_lookup, 0);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
- if (sym_lst.nops() > 0)
- return e.op(0).subs(sym_lst, repl_lst);
- else
+ if (repl.empty())
return e.op(0);
+ else
+ return e.op(0).subs(repl, subs_options::no_pattern);
}
/** Get denominator of an expression. If the expression is not of the normal
*
* @see ex::normal
* @return denominator */
-ex ex::denom(void) const
+ex ex::denom() const
{
- lst sym_lst, repl_lst;
+ exmap repl, rev_lookup;
- ex e = bp->normal(sym_lst, repl_lst, 0);
+ ex e = bp->normal(repl, rev_lookup, 0);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
- if (sym_lst.nops() > 0)
- return e.op(1).subs(sym_lst, repl_lst);
- else
+ if (repl.empty())
return e.op(1);
+ else
+ return e.op(1).subs(repl, subs_options::no_pattern);
}
/** Get numerator and denominator of an expression. If the expresison is not
*
* @see ex::normal
* @return a list [numerator, denominator] */
-ex ex::numer_denom(void) const
+ex ex::numer_denom() const
{
- lst sym_lst, repl_lst;
+ exmap repl, rev_lookup;
- ex e = bp->normal(sym_lst, repl_lst, 0);
+ ex e = bp->normal(repl, rev_lookup, 0);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
- if (sym_lst.nops() > 0)
- return e.subs(sym_lst, repl_lst);
- else
+ if (repl.empty())
return e;
+ else
+ return e.subs(repl, subs_options::no_pattern);
}
/** Rationalization of non-rational functions.
- * This function converts a general expression to a rational polynomial
+ * This function converts a general expression to a rational function
* by replacing all non-rational subexpressions (like non-rational numbers,
* non-integer powers or functions like sin(), cos() etc.) to temporary
* symbols. This makes it possible to use functions like gcd() and divide()
* on non-rational functions by applying to_rational() on the arguments,
* calling the desired function and re-substituting the temporary symbols
* in the result. To make the last step possible, all temporary symbols and
- * their associated expressions are collected in the list specified by the
- * repl_lst parameter in the form {symbol == expression}, ready to be passed
- * as an argument to ex::subs().
+ * their associated expressions are collected in the map specified by the
+ * repl parameter, ready to be passed as an argument to ex::subs().
*
- * @param repl_lst collects a list of all temporary symbols and their replacements
+ * @param repl collects all temporary symbols and their replacements
* @return rationalized expression */
-ex basic::to_rational(lst &repl_lst) const
+ex ex::to_rational(exmap & repl) const
+{
+ return bp->to_rational(repl);
+}
+
+// GiNaC 1.1 compatibility function
+ex ex::to_rational(lst & repl_lst) const
+{
+ // Convert lst to exmap
+ exmap m;
+ for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it)
+ m.insert(std::make_pair(it->op(0), it->op(1)));
+
+ ex ret = bp->to_rational(m);
+
+ // Convert exmap back to lst
+ repl_lst.remove_all();
+ for (exmap::const_iterator it = m.begin(); it != m.end(); ++it)
+ repl_lst.append(it->first == it->second);
+
+ return ret;
+}
+
+ex ex::to_polynomial(exmap & repl) const
+{
+ return bp->to_polynomial(repl);
+}
+
+// GiNaC 1.1 compatibility function
+ex ex::to_polynomial(lst & repl_lst) const
+{
+ // Convert lst to exmap
+ exmap m;
+ for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it)
+ m.insert(std::make_pair(it->op(0), it->op(1)));
+
+ ex ret = bp->to_polynomial(m);
+
+ // Convert exmap back to lst
+ repl_lst.remove_all();
+ for (exmap::const_iterator it = m.begin(); it != m.end(); ++it)
+ repl_lst.append(it->first == it->second);
+
+ return ret;
+}
+
+/** Default implementation of ex::to_rational(). This replaces the object with
+ * a temporary symbol. */
+ex basic::to_rational(exmap & repl) const
+{
+ return replace_with_symbol(*this, repl);
+}
+
+ex basic::to_polynomial(exmap & repl) const
{
- return replace_with_symbol(*this, repl_lst);
+ return replace_with_symbol(*this, repl);
}
/** Implementation of ex::to_rational() for symbols. This returns the
* unmodified symbol. */
-ex symbol::to_rational(lst &repl_lst) const
+ex symbol::to_rational(exmap & repl) const
+{
+ return *this;
+}
+
+/** Implementation of ex::to_polynomial() for symbols. This returns the
+ * unmodified symbol. */
+ex symbol::to_polynomial(exmap & repl) const
{
return *this;
}
/** 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. */
-ex numeric::to_rational(lst &repl_lst) const
+ex numeric::to_rational(exmap & repl) const
{
if (is_real()) {
if (!is_rational())
- return replace_with_symbol(*this, repl_lst);
+ return replace_with_symbol(*this, repl);
+ } else { // complex
+ numeric re = real();
+ numeric im = imag();
+ ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl);
+ ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl);
+ return re_ex + im_ex * replace_with_symbol(I, repl);
+ }
+ return *this;
+}
+
+/** Implementation of ex::to_polynomial() for a numeric. It splits complex
+ * numbers into re+I*im and replaces I and non-integer real numbers with a
+ * temporary symbol. */
+ex numeric::to_polynomial(exmap & repl) const
+{
+ if (is_real()) {
+ if (!is_integer())
+ return replace_with_symbol(*this, repl);
} else { // complex
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);
+ ex re_ex = re.is_integer() ? re : replace_with_symbol(re, repl);
+ ex im_ex = im.is_integer() ? im : replace_with_symbol(im, repl);
+ return re_ex + im_ex * replace_with_symbol(I, repl);
}
return *this;
}
/** Implementation of ex::to_rational() for powers. It replaces non-integer
* powers by temporary symbols. */
-ex power::to_rational(lst &repl_lst) const
+ex power::to_rational(exmap & repl) const
{
if (exponent.info(info_flags::integer))
- return power(basis.to_rational(repl_lst), exponent);
+ return power(basis.to_rational(repl), exponent);
+ else
+ return replace_with_symbol(*this, repl);
+}
+
+/** Implementation of ex::to_polynomial() for powers. It replaces non-posint
+ * powers by temporary symbols. */
+ex power::to_polynomial(exmap & repl) const
+{
+ 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_lst);
+ return replace_with_symbol(*this, repl);
}
/** Implementation of ex::to_rational() for expairseqs. */
-ex expairseq::to_rational(lst &repl_lst) const
+ex expairseq::to_rational(exmap & repl) const
{
epvector s;
s.reserve(seq.size());
epvector::const_iterator i = seq.begin(), end = seq.end();
while (i != end) {
- s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst)));
+ s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl)));
++i;
}
- ex oc = overall_coeff.to_rational(repl_lst);
+ ex oc = overall_coeff.to_rational(repl);
if (oc.info(info_flags::numeric))
return thisexpairseq(s, overall_coeff);
else
return thisexpairseq(s, default_overall_coeff());
}
+/** Implementation of ex::to_polynomial() for expairseqs. */
+ex expairseq::to_polynomial(exmap & repl) const
+{
+ epvector s;
+ s.reserve(seq.size());
+ epvector::const_iterator i = seq.begin(), end = seq.end();
+ while (i != end) {
+ s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_polynomial(repl)));
+ ++i;
+ }
+ ex oc = overall_coeff.to_polynomial(repl);
+ 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());
+}
+
+
+/** Remove the common factor in the terms of a sum 'e' by calculating the GCD,
+ * and multiply it into the expression 'factor' (which needs to be initialized
+ * to 1, unless you're accumulating factors). */
+static ex find_common_factor(const ex & e, ex & factor, exmap & repl)
+{
+ if (is_exactly_a<add>(e)) {
+
+ size_t num = e.nops();
+ exvector terms; terms.reserve(num);
+ ex gc;
+
+ // Find the common GCD
+ 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) || is_a<power>(x)) {
+ ex f = 1;
+ x = find_common_factor(x, f, repl);
+ x *= f;
+ }
+
+ if (i == 0)
+ gc = x;
+ else
+ gc = gcd(gc, x);
+
+ terms.push_back(x);
+ }
+
+ if (gc.is_equal(_ex1))
+ return e;
+
+ // The GCD is the factor we pull out
+ factor *= gc;
+
+ // Now divide all terms by the GCD
+ for (size_t i=0; i<num; i++) {
+ ex x;
+
+ // Try to avoid divide() because it expands the polynomial
+ ex &t = terms[i];
+ if (is_exactly_a<mul>(t)) {
+ for (size_t j=0; j<t.nops(); j++) {
+ if (t.op(j).is_equal(gc)) {
+ exvector v; v.reserve(t.nops());
+ for (size_t k=0; k<t.nops(); k++) {
+ if (k == j)
+ v.push_back(_ex1);
+ else
+ v.push_back(t.op(k));
+ }
+ t = (new mul(v))->setflag(status_flags::dynallocated);
+ goto term_done;
+ }
+ }
+ }
+
+ divide(t, gc, x);
+ t = x;
+term_done: ;
+ }
+ return (new add(terms))->setflag(status_flags::dynallocated);
+
+ } else if (is_exactly_a<mul>(e)) {
+
+ size_t num = e.nops();
+ exvector v; v.reserve(num);
+
+ for (size_t i=0; i<num; i++)
+ v.push_back(find_common_factor(e.op(i), factor, repl));
+
+ return (new mul(v))->setflag(status_flags::dynallocated);
+
+ } else if (is_exactly_a<power>(e)) {
+ 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;
+}
+
+
+/** Collect common factors in sums. This converts expressions like
+ * '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) || is_exactly_a<power>(e)) {
+
+ exmap repl;
+ ex factor = 1;
+ ex r = find_common_factor(e, factor, repl);
+ return factor.subs(repl, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern);
+
+ } else
+ return e;
+}
+
+
+/** Resultant of two expressions e1,e2 with respect to symbol s.
+ * Method: Compute determinant of Sylvester matrix of e1,e2,s. */
+ex resultant(const ex & e1, const ex & e2, const ex & s)
+{
+ const ex ee1 = e1.expand();
+ const ex ee2 = e2.expand();
+ if (!ee1.info(info_flags::polynomial) ||
+ !ee2.info(info_flags::polynomial))
+ throw(std::runtime_error("resultant(): arguments must be polynomials"));
+
+ const int h1 = ee1.degree(s);
+ const int l1 = ee1.ldegree(s);
+ const int h2 = ee2.degree(s);
+ const int l2 = ee2.ldegree(s);
+
+ const int msize = h1 + h2;
+ matrix m(msize, msize);
+
+ for (int l = h1; l >= l1; --l) {
+ const ex e = ee1.coeff(s, l);
+ for (int k = 0; k < h2; ++k)
+ m(k, k+h1-l) = e;
+ }
+ for (int l = h2; l >= l2; --l) {
+ const ex e = ee2.coeff(s, l);
+ for (int k = 0; k < h1; ++k)
+ m(k+h2, k+h2-l) = e;
+ }
+
+ return m.determinant();
+}
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff