* This file implements several functions that work on univariate and
* multivariate polynomials and rational functions.
* These functions include polynomial quotient and remainder, GCD and LCM
- * computation, square-free factorization and rational function normalization.
- */
+ * computation, square-free factorization and rational function normalization. */
/*
* GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
#define FAST_COMPARE 1
// Set this if you want divide_in_z() to use remembering
-#define USE_REMEMBER 1
+#define USE_REMEMBER 0
+
+// Set this if you want divide_in_z() to use trial division followed by
+// polynomial interpolation (usually slower except for very large problems)
+#define USE_TRIAL_DIVISION 0
+
+// Set this to enable some statistical output for the GCD routines
+#define STATISTICS 0
+
+
+#if STATISTICS
+// Statistics variables
+static int gcd_called = 0;
+static int sr_gcd_called = 0;
+static int heur_gcd_called = 0;
+static int heur_gcd_failed = 0;
+
+// Print statistics at end of program
+static struct _stat_print {
+ _stat_print() {}
+ ~_stat_print() {
+ cout << "gcd() called " << gcd_called << " times\n";
+ cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+ cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+ cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
+ }
+} stat_print;
+#endif
/** Return pointer to first symbol found in expression. Due to GiNaC´s
if (bdeg > adeg)
return false;
-#if 1
-
- // Polynomial long division (recursive)
- ex r = a.expand();
- if (r.is_zero())
- return true;
- int rdeg = adeg;
- ex eb = b.expand();
- ex blcoeff = eb.coeff(*x, bdeg);
- while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(*x, rdeg);
- if (!divide_in_z(rcoeff, blcoeff, term, var+1))
- break;
- term = (term * power(*x, rdeg - bdeg)).expand();
- q += term;
- r -= (term * eb).expand();
- if (r.is_zero()) {
-#if USE_REMEMBER
- dr_remember[ex2(a, b)] = exbool(q, true);
-#endif
- return true;
- }
- rdeg = r.degree(*x);
- }
-#if USE_REMEMBER
- dr_remember[ex2(a, b)] = exbool(q, false);
-#endif
- return false;
-
-#else
+#if USE_TRIAL_DIVISION
- // Trial division using polynomial interpolation
+ // Trial division with polynomial interpolation
int i, k;
// Compute values at evaluation points 0..adeg
// Compute inverses
vector<numeric> rcp; rcp.reserve(adeg + 1);
- rcp.push_back(0);
+ rcp.push_back(_num0());
for (k=1; k<=adeg; k++) {
numeric product = alpha[k] - alpha[0];
for (i=1; i<k; i++)
return true;
} else
return false;
+
+#else
+
+ // Polynomial long division (recursive)
+ ex r = a.expand();
+ if (r.is_zero())
+ return true;
+ int rdeg = adeg;
+ ex eb = b.expand();
+ ex blcoeff = eb.coeff(*x, bdeg);
+ while (rdeg >= bdeg) {
+ ex term, rcoeff = r.coeff(*x, rdeg);
+ if (!divide_in_z(rcoeff, blcoeff, term, var+1))
+ break;
+ term = (term * power(*x, rdeg - bdeg)).expand();
+ q += term;
+ r -= (term * eb).expand();
+ if (r.is_zero()) {
+#if USE_REMEMBER
+ dr_remember[ex2(a, b)] = exbool(q, true);
+#endif
+ return true;
+ }
+ rdeg = r.degree(*x);
+ }
+#if USE_REMEMBER
+ dr_remember[ex2(a, b)] = exbool(q, false);
+#endif
+ return false;
+
#endif
}
static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
{
+//clog << "sr_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+ sr_gcd_called++;
+#endif
+
// Sort c and d so that c has higher degree
ex c, d;
int adeg = a.degree(*x), bdeg = b.degree(*x);
return gamma;
c = c.primpart(*x, cont_c);
d = d.primpart(*x, cont_d);
+//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
// First element of subresultant sequence
ex r = _ex0(), ri = _ex1(), psi = _ex1();
for (;;) {
// Calculate polynomial pseudo-remainder
+//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
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))
throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
ddeg = d.degree(*x);
}
// Next element of subresultant sequence
+//clog << " calculating next subresultant...\n";
ri = c.expand().lcoeff(*x);
if (delta == 1)
psi = ri;
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";
+#if STATISTICS
+ heur_gcd_called++;
+#endif
+
+ // GCD of two numeric values -> CLN
if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
numeric rg;
// 6 tries maximum
for (int t=0; t<6; t++) {
- if (xi.int_length() * maxdeg > 50000)
+ if (xi.int_length() * maxdeg > 100000) {
+//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
throw gcdheu_failed();
+ }
// Apply evaluation homomorphism and calculate GCD
ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand();
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) && lc.compare(_ex0()) < 0)
+ if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
return -g;
else
return g;
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
{
+//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)) {
+ 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;
+ }
+
+ // Check arguments
+ if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
+ throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
+ }
+
// 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())
return g;
}
+#if FAST_COMPARE
+ // Input polynomials of the form poly^n are sometimes also trivial
+ if (is_ex_exactly_of_type(a, power)) {
+ ex p = a.op(0);
+ if (is_ex_exactly_of_type(b, power)) {
+ if (p.is_equal(b.op(0))) {
+ // 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();
+ if (cb)
+ *cb = power(p, exp_b - exp_a);
+ return power(p, exp_a);
+ } else {
+ if (ca)
+ *ca = power(p, exp_a - exp_b);
+ if (cb)
+ *cb = _ex1();
+ return power(p, exp_b);
+ }
+ }
+ } else {
+ if (p.is_equal(b)) {
+ // a = p^n, b = p, gcd = p
+ if (ca)
+ *ca = power(p, a.op(1) - 1);
+ if (cb)
+ *cb = _ex1();
+ return p;
+ }
+ }
+ } else if (is_ex_exactly_of_type(b, power)) {
+ ex p = b.op(0);
+ if (p.is_equal(a)) {
+ // a = p, b = p^n, gcd = p
+ if (ca)
+ *ca = _ex1();
+ if (cb)
+ *cb = power(p, b.op(1) - 1);
+ return p;
+ }
+ }
+#endif
+
// Some trivial cases
ex aex = a.expand(), bex = b.expand();
if (aex.is_zero()) {
return a;
}
#endif
- if (is_ex_exactly_of_type(aex, numeric) && is_ex_exactly_of_type(bex, numeric)) {
- numeric g = gcd(ex_to_numeric(aex), ex_to_numeric(bex));
- if (ca)
- *ca = ex_to_numeric(aex) / g;
- if (cb)
- *cb = ex_to_numeric(bex) / g;
- return g;
- }
- if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
- throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
- }
// Gather symbol statistics
sym_desc_vec sym_stats;
}
if (is_ex_exactly_of_type(g, fail)) {
//clog << "heuristics failed" << endl;
+#if STATISTICS
+ heur_gcd_failed++;
+#endif
g = sr_gcd(aex, bex, x);
- if (ca)
- divide(aex, g, *ca, false);
- if (cb)
- divide(bex, g, *cb, false);
- }
- return g;
+ 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;
+ }
}
*/
/** Create a symbol for replacing the expression "e" (or return a previously
- * assigned symbol). The symbol is appended to sym_list and returned, the
- * expression is appended to repl_list.
+ * assigned symbol). The symbol is appended to sym_lst and returned, the
+ * expression is appended to repl_lst.
* @see ex::normal */
static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
{
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 ex::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);
+
+ // 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
+ symbol s;
+ ex es(s);
+ ex e_replaced = e.subs(repl_lst);
+ repl_lst.append(es == e_replaced);
+ return es;
+}
/** Default implementation of ex::normal(). It replaces the object with a
* temporary symbol.
// as defined by get_first_symbol() is made positive)
const symbol *x;
if (get_first_symbol(den, x)) {
- if (den.unit(*x).compare(_ex0()) < 0) {
+ GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric));
+ if (ex_to_numeric(den.unit(*x)).is_negative()) {
num *= _ex_1();
den *= _ex_1();
}
}
// Return result as list
+//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
}
// split it into numerator and denominator
GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
- o.push_back((new lst(overall.numer(), overall.denom()))->setflag(status_flags::dynallocated));
+ o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
} else
o.push_back(n);
it++;
// Determine common denominator
ex den = _ex1();
exvector::const_iterator ait = o.begin(), aitend = o.end();
+//clog << "add::normal uses the following summands:\n";
while (ait != aitend) {
+//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
den = lcm(ait->op(1), den, false);
ait++;
}
+//clog << " common denominator = " << den << endl;
// Add fractions
if (den.is_equal(_ex1())) {
// should not happen
throw(std::runtime_error("invalid expression in add::normal, division failed"));
}
- num_seq.push_back(ait->op(0) * q);
+ num_seq.push_back((ait->op(0) * q).expand());
}
ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
// (a/b)^n -> {a^n, b^n}
return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
- } else if (exponent.info(info_flags::negint)) {
+ } else if (exponent.info(info_flags::negative)) {
// (a/b)^-n -> {b^n, a^n}
return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
}
} else {
+
if (exponent.info(info_flags::positive)) {
- // (a/b)^z -> {sym((a/b)^z), 1}
+ // (a/b)^x -> {sym((a/b)^x), 1}
return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
- } else {
+ } else if (exponent.info(info_flags::negative)) {
if (n.op(1).is_equal(_ex1())) {
} else {
- // (a/b)^-x -> {(b/a)^x, 1}
+ // (a/b)^-x -> {sym((b/a)^x), 1}
return (new lst(replace_with_symbol(power(n.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
+
+ } else { // exponent not numeric
+
+ // (a/b)^x -> {sym((a/b)^x, 1}
+ return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
}
}
new_seq.push_back(expair(it->rest.normal(), it->coeff));
it++;
}
- ex n = pseries(var, point, new_seq);
+ ex n = pseries(relational(var,point), new_seq);
return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
* This function converts an expression to its normal form
* "numerator/denominator", where numerator and denominator are (relatively
* prime) polynomials. Any subexpressions which are not rational functions
- * (like non-rational numbers, non-integer powers or functions like Sin(),
- * Cos() etc.) are replaced by temporary symbols which are re-substituted by
+ * (like non-rational numbers, non-integer powers or functions like sin(),
+ * cos() etc.) are replaced by temporary symbols which are re-substituted by
* the (normalized) subexpressions before normal() returns (this way, any
* expression can be treated as a rational function). normal() is applied
* recursively to arguments of functions etc.
return e.op(1);
}
+
+/** Default implementation of ex::to_rational(). It replaces the object with a
+ * temporary symbol.
+ * @see ex::to_rational */
+ex basic::to_rational(lst &repl_lst) const
+{
+ return replace_with_symbol(*this, repl_lst);
+}
+
+
+/** Implementation of ex::to_rational() for symbols. This returns the unmodified symbol.
+ * @see ex::to_rational */
+ex symbol::to_rational(lst &repl_lst) 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.
+ * @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);
+ } else { // complex
+ numeric re = num.real(), im = num.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 numex;
+}
+
+
+/** Implementation of ex::to_rational() for powers. It replaces non-integer
+ * powers by temporary symbols.
+ * @see ex::to_rational */
+ex power::to_rational(lst &repl_lst) const
+{
+ if (exponent.info(info_flags::integer))
+ return power(basis.to_rational(repl_lst), exponent);
+ else
+ return replace_with_symbol(*this, repl_lst);
+}
+
+
+/** 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,
+ * 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().
+ *
+ * @param repl_lst collects a list of all temporary symbols and their replacements
+ * @return rationalized expression */
+ex ex::to_rational(lst &repl_lst) const
+{
+ return bp->to_rational(repl_lst);
+}
+
+
#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff