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";
+
// 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())
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))
- return gcd(ex_to_numeric(a), ex_to_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"));
* Normal form of rational functions
*/
-// 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.
+/*
+ * Note: The internal normal() functions (= basic::normal() and overloaded
+ * functions) all return lists of the form {numerator, denominator}. This
+ * is to get around mul::eval()'s automatic expansion of numeric coefficients.
+ * E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
+ * the information that (a+b) is the numerator and 3 is the denominator.
+ */
+
+/** 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.
+ * @see ex::normal */
static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
{
// Expression already in repl_lst? Then return the assigned symbol
* @see ex::normal */
ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- return replace_with_symbol(*this, sym_lst, repl_lst);
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
-/** Implementation of ex::normal() for symbols. This returns the unmodifies symbol.
+/** 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
{
- return *this;
+ return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
}
* @see ex::normal */
ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- if (is_real())
- if (is_rational())
- return *this;
- else
- return replace_with_symbol(*this, sym_lst, repl_lst);
- else { // complex
- numeric re = real(), im = imag();
+ numeric num = numer();
+ ex numex = num;
+
+ if (num.is_real()) {
+ if (!num.is_integer())
+ numex = replace_with_symbol(numex, sym_lst, repl_lst);
+ } 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);
- return re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
+ numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
}
+
+ // Denominator is always a real integer (see numeric::denom())
+ return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
}
/** Fraction cancellation.
* @param n numerator
* @param d denominator
- * @return cancelled fraction n/d */
+ * @return cancelled fraction {n, d} as a list */
static ex frac_cancel(const ex &n, const ex &d)
{
ex num = n;
ex den = d;
numeric pre_factor = _num1();
+//clog << "frac_cancel num = " << num << ", den = " << den << endl;
+
// Handle special cases where numerator or denominator is 0
if (num.is_zero())
- return _ex0();
+ return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
if (den.expand().is_zero())
throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
- // More special cases
- if (is_ex_exactly_of_type(den, numeric))
- return num / den;
-
// Bring numerator and denominator to Z[X] by multiplying with
// LCM of all coefficients' denominators
numeric num_lcm = lcm_of_coefficients_denominators(num);
// 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 pre_factor * num / den;
+
+ // 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);
}
* @see ex::normal */
ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- // Normalize and expand children
+ // Normalize and expand children, chop into summands
exvector o;
o.reserve(seq.size()+1);
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
+
+ // Normalize and expand child
ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
- if (is_ex_exactly_of_type(n, add)) {
- epvector::const_iterator bit = (static_cast<add *>(n.bp))->seq.begin(), bitend = (static_cast<add *>(n.bp))->seq.end();
+
+ // If numerator is a sum, chop into summands
+ if (is_ex_exactly_of_type(n.op(0), add)) {
+ epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
while (bit != bitend) {
- o.push_back(recombine_pair_to_ex(*bit));
+ o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
bit++;
}
- o.push_back((static_cast<add *>(n.bp))->overall_coeff);
+
+ // The overall_coeff is already normalized (== rational), we just
+ // 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() * n.op(1)))->setflag(status_flags::dynallocated));
} else
o.push_back(n);
it++;
}
o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
+ // o is now a vector of {numerator, denominator} lists
+
// 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) {
- den = lcm((*ait).denom(false), den, false);
+//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()))
- return (new add(o))->setflag(status_flags::dynallocated);
- else {
+ if (den.is_equal(_ex1())) {
+
+ // Common denominator is 1, simply add all numerators
+ exvector num_seq;
+ for (ait=o.begin(); ait!=aitend; ait++) {
+ num_seq.push_back(ait->op(0));
+ }
+ return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ // Perform fractional addition
exvector num_seq;
for (ait=o.begin(); ait!=aitend; ait++) {
ex q;
- if (!divide(den, (*ait).denom(false), q, false)) {
+ if (!divide(den, ait->op(1), q, false)) {
// should not happen
throw(std::runtime_error("invalid expression in add::normal, division failed"));
}
- num_seq.push_back((*ait).numer(false) * q);
+ num_seq.push_back((ait->op(0) * q).expand());
}
- ex num = add(num_seq);
+ ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
// Cancel common factors from num/den
return frac_cancel(num, den);
* @see ex::normal() */
ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- // Normalize children
- exvector o;
- o.reserve(seq.size()+1);
+ // Normalize children, separate into numerator and denominator
+ ex num = _ex1();
+ ex den = _ex1();
+ ex n;
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
- o.push_back(recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1));
+ n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
+ num *= n.op(0);
+ den *= n.op(1);
it++;
}
- o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
- ex n = (new mul(o))->setflag(status_flags::dynallocated);
- return frac_cancel(n.numer(false), n.denom(false));
+ n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
+ num *= n.op(0);
+ den *= n.op(1);
+
+ // Perform fraction cancellation
+ return frac_cancel(num, den);
}
* @see ex::normal */
ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- if (exponent.info(info_flags::integer)) {
- // Integer powers are distributed
- ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
- ex num = n.numer(false);
- ex den = n.denom(false);
- return power(num, exponent) / power(den, exponent);
- } else {
- // Non-integer powers are replaced by temporary symbol (after normalizing basis)
- ex n = power(basis.bp->normal(sym_lst, repl_lst, level-1), exponent);
- return replace_with_symbol(n, sym_lst, repl_lst);
+ // Normalize basis
+ ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+
+ if (exponent.info(info_flags::integer)) {
+
+ if (exponent.info(info_flags::positive)) {
+
+ // (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)) {
+
+ // (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}
+ return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ if (n.op(1).is_equal(_ex1())) {
+
+ // a^-x -> {1, sym(a^x)}
+ return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ // (a/b)^-x -> {(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);
+ }
+ }
}
}
new_seq.push_back(expair(it->rest.normal(), it->coeff));
it++;
}
-
ex n = pseries(var, point, new_seq);
- return replace_with_symbol(n, sym_lst, repl_lst);
+ return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}
ex ex::normal(int level) const
{
lst sym_lst, repl_lst;
+
ex e = bp->normal(sym_lst, repl_lst, level);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
if (sym_lst.nops() > 0)
- return e.subs(sym_lst, repl_lst);
- else
- return e;
+ e = e.subs(sym_lst, repl_lst);
+
+ // Convert {numerator, denominator} form back to fraction
+ return e.op(0) / e.op(1);
+}
+
+/** Numerator of an expression. If the expression is not of the normal form
+ * "numerator/denominator", it is first converted to this form and then the
+ * numerator is returned.
+ *
+ * @see ex::normal
+ * @return numerator */
+ex ex::numer(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(0).subs(sym_lst, repl_lst);
+ else
+ return e.op(0);
+}
+
+/** Denominator of an expression. If the expression is not of the normal form
+ * "numerator/denominator", it is first converted to this form and then the
+ * denominator is returned.
+ *
+ * @see ex::normal
+ * @return denominator */
+ex ex::denom(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(1).subs(sym_lst, repl_lst);
+ else
+ return e.op(1);
}
#ifndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff