* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2023 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
* @param cb pointer to expression that will receive the cofactor of b, or nullptr
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
+ * @param options see GiNaC::gcd_options
* @return the GCD as a new expression */
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args, unsigned options)
{
ex z = w.diff(x);
ex g = gcd(w, z);
if (g.is_zero()) {
- return epvector{};
+ // manifest zero or hidden zero
+ return {};
}
if (g.is_equal(_ex1)) {
- return epvector{expair(a, _ex1)};
+ // w(x) and w'(x) share no factors: w(x) is square-free
+ return {expair(a, _ex1)};
}
- epvector results;
- ex exponent = _ex0;
+
+ epvector factors;
+ ex i = 0; // exponent
do {
w = quo(w, g, x);
if (w.is_zero()) {
- return results;
+ // hidden zero
+ break;
}
z = quo(z, g, x) - w.diff(x);
- exponent = exponent + 1;
+ i += 1;
if (w.is_equal(x)) {
// shortcut for x^n with n ∈ ℕ
- exponent += quo(z, w.diff(x), x);
- results.push_back(expair(w, exponent));
+ i += quo(z, w.diff(x), x);
+ factors.push_back(expair(w, i));
break;
}
g = gcd(w, z);
if (!g.is_equal(_ex1)) {
- results.push_back(expair(g, exponent));
+ factors.push_back(expair(g, i));
}
} while (!z.is_zero());
+
+ // correct for lost factor
+ // (being based on GCDs, Yun's algorithm only finds factors up to a unit)
+ const ex lost_factor = quo(a, mul{factors}, x);
+ if (lost_factor.is_equal(_ex1)) {
+ // trivial lost factor
+ return factors;
+ }
+ if (!factors.empty() && factors[0].coeff.is_equal(1)) {
+ // multiply factor^1 with lost_factor
+ factors[0].rest *= lost_factor;
+ return factors;
+ }
+ // no factor^1: prepend lost_factor^1 to the results
+ epvector results = {expair(lost_factor, 1)};
+ std::move(factors.begin(), factors.end(), std::back_inserter(results));
return results;
}
// convert the argument from something in Q[X] to something in Z[X]
const numeric lcm = lcm_of_coefficients_denominators(a);
- const ex tmp = multiply_lcm(a,lcm);
+ const ex tmp = multiply_lcm(a, lcm);
// find the factors
epvector factors = sqrfree_yun(tmp, x);
+ if (factors.empty()) {
+ // the polynomial was a hidden zero
+ return _ex0;
+ }
// remove symbol x and proceed recursively with the remaining symbols
args.remove_first();
}
// Done with recursion, now construct the final result
- ex result = _ex1;
- for (auto & it : factors)
- result *= pow(it.rest, it.coeff);
-
- // Yun's algorithm does not account for constant factors. (For univariate
- // polynomials it works only in the monic case.) We can correct this by
- // inserting what has been lost back into the result. For completeness
- // we'll also have to recurse down that factor in the remaining variables.
- if (args.nops()>0)
- result *= sqrfree(quo(tmp, result, x), args);
- else
- result *= quo(tmp, result, x);
+ ex result = mul(factors);
// Put in the rational overall factor again and return
- return result * lcm.inverse();
+ return result * lcm.inverse();
}
// Find numerator and denominator
ex nd = numer_denom(a);
ex numer = nd.op(0), denom = nd.op(1);
-//clog << "numer = " << numer << ", denom = " << denom << endl;
+//std::clog << "numer = " << numer << ", denom = " << denom << std::endl;
// Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D)
ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand();
-//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;
+//std::clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << std::endl;
// Factorize denominator and compute cofactors
epvector yun = sqrfree_yun(denom, x);
- size_t yun_max_exponent = yun.empty() ? 0 : ex_to<numeric>(yun.back().coeff).to_int();
exvector factor, cofac;
+ size_t dim = 0;
for (size_t i=0; i<yun.size(); i++) {
numeric i_exponent = ex_to<numeric>(yun[i].coeff);
for (size_t j=0; j<i_exponent; j++) {
factor.push_back(pow(yun[i].rest, j+1));
+ dim += degree(yun[i].rest, x);
ex prod = _ex1;
for (size_t k=0; k<yun.size(); k++) {
if (yun[k].coeff == i_exponent)
cofac.push_back(prod.expand());
}
}
- size_t num_factors = factor.size();
-//clog << "factors : " << exprseq(factor) << endl;
-//clog << "cofactors: " << exprseq(cofac) << endl;
-
- // Construct coefficient matrix for decomposition
- int max_denom_deg = denom.degree(x);
- 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 (size_t j=0; j<num_factors; j++)
- sys(i, j) = cofac[j].coeff(x, i);
- rhs(i, 0) = red_numer.coeff(x, i);
- }
-//clog << "coeffs: " << sys << endl;
-//clog << "rhs : " << rhs << endl;
-
- // Solve resulting linear system
- matrix vars(num_factors, 1);
- for (size_t i=0; i<num_factors; i++)
- vars(i, 0) = symbol();
+//std::clog << "factors : " << exprseq(factor) << std::endl;
+//std::clog << "cofactors: " << exprseq(cofac) << std::endl;
+
+ // Construct linear system for decomposition
+ matrix sys(dim, dim);
+ matrix rhs(dim, 1);
+ matrix vars(dim, 1);
+ for (size_t i=0, n=0, f=0; i<yun.size(); i++) {
+ size_t i_expo = to_int(ex_to<numeric>(yun[i].coeff));
+ for (size_t j=0; j<i_expo; j++) {
+ for (size_t k=0; k<size_t(degree(yun[i].rest, x)); k++) {
+ GINAC_ASSERT(n < dim && f < factor.size());
+
+ // column n of coefficient matrix
+ for (size_t r=0; r+k<dim; r++) {
+ sys(r+k, n) = cofac[f].coeff(x, r);
+ }
+
+ // element n of right hand side vector
+ rhs(n, 0) = red_numer.coeff(x, n);
+
+ // element n of free variables vector
+ vars(n, 0) = symbol();
+
+ n++;
+ }
+ f++;
+ }
+ }
+//std::clog << "coeffs: " << sys << std::endl;
+//std::clog << "rhs : " << rhs << std::endl;
+
+ // Solve resulting linear system and sum up decomposed fractions
matrix sol = sys.solve(vars, rhs);
+//std::clog << "sol : " << sol << std::endl;
+ ex sum = red_poly;
+ for (size_t i=0, n=0, f=0; i<yun.size(); i++) {
+ size_t i_expo = to_int(ex_to<numeric>(yun[i].coeff));
+ for (size_t j=0; j<i_expo; j++) {
+ ex frac_numer = 0;
+ for (size_t k=0; k<size_t(degree(yun[i].rest, x)); k++) {
+ GINAC_ASSERT(n < dim && f < factor.size());
+ frac_numer += sol(n, 0) * pow(x, k);
+ n++;
+ }
+ sum += frac_numer / factor[f];
- // Sum up decomposed fractions
- ex sum = 0;
- for (size_t i=0; i<num_factors; i++)
- sum += sol(i, 0) / factor[i];
+ f++;
+ }
+ }
- return red_poly + sum;
+ return sum;
}
/** Create a symbol for replacing the expression "e" (or return a previously
* assigned symbol). The symbol and expression are appended to repl, for
* a later application of subs().
+ * An entry in the replacement table repl can be changed in some cases.
+ * If it was altered, we need to provide the modifier for the previously build expressions.
+ * The modifier is an (ordered) list, because those substitutions need to be done in the
+ * incremental order.
+ * As an example let us consider a rationalisation of the expression
+ * e = exp(2*x)*cos(exp(2*x)+1)*exp(x)
+ * The first factor GiNaC denotes by something like symbol1 and will record:
+ * e =symbol1*cos(symbol1 + 1)*exp(x)
+ * repl = {symbol1 : exp(2*x)}
+ * Similarly, the second factor would be denoted as symbol2 and we will have
+ * e =symbol1*symbol2*exp(x)
+ * repl = {symbol1 : exp(2*x), symbol2 : cos(symbol1 + 1)}
+ * Denoting the third term as symbol3 GiNaC is willing to re-think exp(2*x) as
+ * symbol3^2 rather than just symbol1. Here are two issues:
+ * 1) The replacement "symbol1 -> symbol3^2" in the previous part of the expression
+ * needs to be done outside of the present routine;
+ * 2) The pair "symbol1 : exp(2*x)" shall be deleted from the replacement table repl.
+ * However, this will create illegal substitution "symbol2 : cos(symbol1 + 1)" with
+ * undefined symbol1.
+ * These both problems are mitigated through the additions of the record
+ * "symbol1==symbol3^2" to the list modifier. Changed length of the modifier signals
+ * to the calling code that the previous portion of the expression needs to be
+ * altered (it solves 1). Thus GiNaC can record now
+ * e =symbol3^2*symbol2*symbol3
+ * repl = {symbol2 : cos(symbol1 + 1), symbol3 : exp(x)}
+ * modifier = {symbol1==symbol3^2}
+ * Then, doing the backward substitutions the list modifier will be used to restore
+ * such iterative substitutions in the right way (this solves 2).
* @see ex::normal */
-static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup)
+static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup, lst & modifier)
{
// Since the repl contains replaced expressions we should search for them
ex e_replaced = e.subs(repl, subs_options::no_pattern);
if (it != rev_lookup.end())
return it->second;
+ // The expression can be the base of substituted power, which requires a more careful search
+ if (! is_a<numeric>(e_replaced))
+ for (auto & it : repl)
+ if (is_a<power>(it.second) && e_replaced.is_equal(it.second.op(0))) {
+ ex degree = pow(it.second.op(1), _ex_1);
+ if (is_a<numeric>(degree) && ex_to<numeric>(degree).is_integer())
+ return pow(it.first, degree);
+ }
+
+ // We treat powers and the exponent functions differently because
+ // they can be rationalised more efficiently
+ if (is_a<function>(e_replaced) && is_ex_the_function(e_replaced, exp)) {
+ for (auto & it : repl) {
+ if (is_a<function>(it.second) && is_ex_the_function(it.second, exp)) {
+ ex ratio = normal(e_replaced.op(0) / it.second.op(0));
+ if (is_a<numeric>(ratio) && ex_to<numeric>(ratio).is_rational()) {
+ // Different exponents can be treated as powers of the same basic equation
+ if (ex_to<numeric>(ratio).is_integer()) {
+ // If ratio is an integer then this is simply the power of the existing symbol.
+ // std::clog << e_replaced << " is a " << ratio << " power of " << it.first << std::endl;
+ return dynallocate<power>(it.first, ratio);
+ } else {
+ // otherwise we need to give the replacement pattern to change
+ // the previous expression...
+ ex es = dynallocate<symbol>();
+ ex Num = numer(ratio);
+ modifier.append(it.first == power(es, denom(ratio)));
+ // std::clog << e_replaced << " is power " << Num << " and "
+ // << it.first << " is power " << denom(ratio) << " of the common base "
+ // << exp(e_replaced.op(0)/Num) << std::endl;
+ // ... and modify the replacement tables
+ rev_lookup.erase(it.second);
+ rev_lookup.insert({exp(e_replaced.op(0)/Num), es});
+ repl.erase(it.first);
+ repl.insert({es, exp(e_replaced.op(0)/Num)});
+ return dynallocate<power>(es, Num);
+ }
+ }
+ }
+ }
+ } else if (is_a<power>(e_replaced) && !is_a<numeric>(e_replaced.op(0)) // We do not replace simple monomials like x^3 or sqrt(2)
+ && ! (is_a<symbol>(e_replaced.op(0))
+ && is_a<numeric>(e_replaced.op(1)) && ex_to<numeric>(e_replaced.op(1)).is_integer())) {
+ for (auto & it : repl) {
+ if (e_replaced.op(0).is_equal(it.second) // The base is an allocated symbol or base of power
+ || (is_a<power>(it.second) && e_replaced.op(0).is_equal(it.second.op(0)))) {
+ ex ratio; // We bind together two above cases
+ if (is_a<power>(it.second))
+ ratio = normal(e_replaced.op(1) / it.second.op(1));
+ else
+ ratio = e_replaced.op(1);
+ if (is_a<numeric>(ratio) && ex_to<numeric>(ratio).is_rational()) {
+ // Different powers can be treated as powers of the same basic equation
+ if (ex_to<numeric>(ratio).is_integer()) {
+ // If ratio is an integer then this is simply the power of the existing symbol.
+ //std::clog << e_replaced << " is a " << ratio << " power of " << it.first << std::endl;
+ return dynallocate<power>(it.first, ratio);
+ } else {
+ // otherwise we need to give the replacement pattern to change
+ // the previous expression...
+ ex es = dynallocate<symbol>();
+ ex Num = numer(ratio);
+ modifier.append(it.first == power(es, denom(ratio)));
+ //std::clog << e_replaced << " is power " << Num << " and "
+ // << it.first << " is power " << denom(ratio) << " of the common base "
+ // << pow(e_replaced.op(0), e_replaced.op(1)/Num) << std::endl;
+ // ... and modify the replacement tables
+ rev_lookup.erase(it.second);
+ rev_lookup.insert({pow(e_replaced.op(0), e_replaced.op(1)/Num), es});
+ repl.erase(it.first);
+ repl.insert({es, pow(e_replaced.op(0), e_replaced.op(1)/Num)});
+ return dynallocate<power>(es, Num);
+ }
+ }
+ }
+ }
+ // There is no existing substitution, thus we are creating a new one.
+ // This needs to be done separately to treat possible occurrences of
+ // b = e_replaced.op(0) elsewhere in the expression as pow(b, 1).
+ ex degree = pow(e_replaced.op(1), _ex_1);
+ if (is_a<numeric>(degree) && ex_to<numeric>(degree).is_integer()) {
+ ex es = dynallocate<symbol>();
+ modifier.append(e_replaced.op(0) == power(es, degree));
+ repl.insert({es, e_replaced});
+ rev_lookup.insert({e_replaced, es});
+ return es;
+ }
+ }
+
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
/** Default implementation of ex::normal(). It normalizes the children and
* replaces the object with a temporary symbol.
* @see ex::normal */
-ex basic::normal(exmap & repl, exmap & rev_lookup) const
+ex basic::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
if (nops() == 0)
- return dynallocate<lst>({replace_with_symbol(*this, repl, rev_lookup), _ex1});
+ return dynallocate<lst>({replace_with_symbol(*this, repl, rev_lookup, modifier), _ex1});
normal_map_function map_normal;
- return dynallocate<lst>({replace_with_symbol(map(map_normal), repl, rev_lookup), _ex1});
+ size_t nmod = modifier.nops(); // To watch new modifiers to the replacement list
+ ex result = replace_with_symbol(map(map_normal), repl, rev_lookup, modifier);
+ for (size_t imod = nmod; imod < modifier.nops(); ++imod) {
+ exmap this_repl;
+ this_repl.insert(std::make_pair(modifier.op(imod).op(0), modifier.op(imod).op(1)));
+ result = result.subs(this_repl, subs_options::no_pattern);
+ }
+
+ // Sometimes we may obtain negative powers, they need to be placed to denominator
+ if (is_a<power>(result) && result.op(1).info(info_flags::negative))
+ return dynallocate<lst>({_ex1, power(result.op(0), -result.op(1))});
+ else
+ return dynallocate<lst>({result, _ex1});
}
/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
* @see ex::normal */
-ex symbol::normal(exmap & repl, exmap & rev_lookup) const
+ex symbol::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
return dynallocate<lst>({*this, _ex1});
}
* into re+I*im and replaces I and non-rational real numbers with a temporary
* symbol.
* @see ex::normal */
-ex numeric::normal(exmap & repl, exmap & rev_lookup) const
+ex numeric::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
numeric num = numer();
ex numex = num;
if (num.is_real()) {
if (!num.is_integer())
- numex = replace_with_symbol(numex, repl, rev_lookup);
+ numex = replace_with_symbol(numex, repl, rev_lookup, modifier);
} else { // complex
numeric re = num.real(), im = num.imag();
- 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);
+ ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl, rev_lookup, modifier);
+ ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl, rev_lookup, modifier);
+ numex = re_ex + im_ex * replace_with_symbol(I, repl, rev_lookup, modifier);
}
// Denominator is always a real integer (see numeric::denom())
/** Implementation of ex::normal() for a sum. It expands terms and performs
* fractional addition.
* @see ex::normal */
-ex add::normal(exmap & repl, exmap & rev_lookup) const
+ex add::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
// Normalize children and split each one into numerator and denominator
exvector nums, dens;
nums.reserve(seq.size()+1);
dens.reserve(seq.size()+1);
+ size_t nmod = modifier.nops(); // To watch new modifiers to the replacement list
for (auto & it : seq) {
- ex n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup);
+ ex n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup, modifier);
nums.push_back(n.op(0));
dens.push_back(n.op(1));
}
- ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup);
+ ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, modifier);
nums.push_back(n.op(0));
dens.push_back(n.op(1));
GINAC_ASSERT(nums.size() == dens.size());
auto num_it = nums.begin(), num_itend = nums.end();
auto den_it = dens.begin(), den_itend = dens.end();
//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
+ for (size_t imod = nmod; imod < modifier.nops(); ++imod) {
+ while (num_it != num_itend) {
+ *num_it = num_it->subs(modifier.op(imod), subs_options::no_pattern);
+ ++num_it;
+ *den_it = den_it->subs(modifier.op(imod), subs_options::no_pattern);
+ ++den_it;
+ }
+ // Reset iterators for the next round
+ num_it = nums.begin();
+ den_it = dens.begin();
+ }
+
ex num = *num_it++, den = *den_it++;
while (num_it != num_itend) {
//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
/** Implementation of ex::normal() for a product. It cancels common factors
* from fractions.
* @see ex::normal() */
-ex mul::normal(exmap & repl, exmap & rev_lookup) const
+ex mul::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
// Normalize children, separate into numerator and denominator
exvector num; num.reserve(seq.size());
exvector den; den.reserve(seq.size());
ex n;
+ size_t nmod = modifier.nops(); // To watch new modifiers to the replacement list
for (auto & it : seq) {
- n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup);
+ n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup, modifier);
num.push_back(n.op(0));
den.push_back(n.op(1));
}
- n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup);
+ n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, modifier);
num.push_back(n.op(0));
den.push_back(n.op(1));
+ auto num_it = num.begin(), num_itend = num.end();
+ auto den_it = den.begin();
+ for (size_t imod = nmod; imod < modifier.nops(); ++imod) {
+ while (num_it != num_itend) {
+ *num_it = num_it->subs(modifier.op(imod), subs_options::no_pattern);
+ ++num_it;
+ *den_it = den_it->subs(modifier.op(imod), subs_options::no_pattern);
+ ++den_it;
+ }
+ num_it = num.begin();
+ den_it = den.begin();
+ }
// Perform fraction cancellation
return frac_cancel(dynallocate<mul>(num), dynallocate<mul>(den));
}
-/** Implementation of ex::normal([B) for powers. It normalizes the basis,
+/** Implementation of ex::normal() 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(exmap & repl, exmap & rev_lookup) const
+ex power::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
// Normalize basis and exponent (exponent gets reassembled)
- ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup);
- ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup);
+ size_t nmod = modifier.nops(); // To watch new modifiers to the replacement list
+ ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup, modifier);
+ for (size_t imod = nmod; imod < modifier.nops(); ++imod)
+ n_basis = n_basis.subs(modifier.op(imod), subs_options::no_pattern);
+
+ nmod = modifier.nops();
+ ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup, modifier);
+ for (size_t imod = nmod; imod < modifier.nops(); ++imod)
+ n_exponent = n_exponent.subs(modifier.op(imod), subs_options::no_pattern);
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 dynallocate<lst>({replace_with_symbol(pow(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1});
+ return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup, modifier), _ex1});
} else if (n_exponent.info(info_flags::negative)) {
if (n_basis.op(1).is_equal(_ex1)) {
// a^-x -> {1, sym(a^x)}
- return dynallocate<lst>({_ex1, replace_with_symbol(pow(n_basis.op(0), -n_exponent), repl, rev_lookup)});
+ return dynallocate<lst>({_ex1, replace_with_symbol(pow(n_basis.op(0), -n_exponent), repl, rev_lookup, modifier)});
} else {
// (a/b)^-x -> {sym((b/a)^x), 1}
- return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(1) / n_basis.op(0), -n_exponent), repl, rev_lookup), _ex1});
+ return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(1) / n_basis.op(0), -n_exponent), repl, rev_lookup, modifier), _ex1});
}
}
}
// (a/b)^x -> {sym((a/b)^x, 1}
- return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1});
+ return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup, modifier), _ex1});
}
/** Implementation of ex::normal() for pseries. It normalizes each coefficient
* and replaces the series by a temporary symbol.
* @see ex::normal */
-ex pseries::normal(exmap & repl, exmap & rev_lookup) const
+ex pseries::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
epvector newseq;
for (auto & it : seq) {
newseq.push_back(expair(restexp, it.coeff));
}
ex n = pseries(relational(var,point), std::move(newseq));
- return dynallocate<lst>({replace_with_symbol(n, repl, rev_lookup), _ex1});
+ return dynallocate<lst>({replace_with_symbol(n, repl, rev_lookup, modifier), _ex1});
}
ex ex::normal() const
{
exmap repl, rev_lookup;
+ lst modifier;
- ex e = bp->normal(repl, rev_lookup);
+ ex e = bp->normal(repl, rev_lookup, modifier);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
- if (!repl.empty())
+ if (!repl.empty()) {
+ for(size_t i=0; i < modifier.nops(); ++i)
+ e = e.subs(modifier.op(i), subs_options::no_pattern);
e = e.subs(repl, subs_options::no_pattern);
+ }
// Convert {numerator, denominator} form back to fraction
return e.op(0) / e.op(1);
ex ex::numer() const
{
exmap repl, rev_lookup;
+ lst modifier;
- ex e = bp->normal(repl, rev_lookup);
+ ex e = bp->normal(repl, rev_lookup, modifier);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (repl.empty())
return e.op(0);
- else
+ else {
+ for(size_t i=0; i < modifier.nops(); ++i)
+ e = e.subs(modifier.op(i), subs_options::no_pattern);
+
return e.op(0).subs(repl, subs_options::no_pattern);
+ }
}
/** Get denominator of an expression. If the expression is not of the normal
ex ex::denom() const
{
exmap repl, rev_lookup;
+ lst modifier;
- ex e = bp->normal(repl, rev_lookup);
+ ex e = bp->normal(repl, rev_lookup, modifier);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (repl.empty())
return e.op(1);
- else
+ else {
+ for(size_t i=0; i < modifier.nops(); ++i)
+ e = e.subs(modifier.op(i), subs_options::no_pattern);
+
return e.op(1).subs(repl, subs_options::no_pattern);
+ }
}
/** Get numerator and denominator of an expression. If the expression is not
ex ex::numer_denom() const
{
exmap repl, rev_lookup;
+ lst modifier;
- ex e = bp->normal(repl, rev_lookup);
+ ex e = bp->normal(repl, rev_lookup, modifier);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (repl.empty())
return e;
- else
+ else {
+ for(size_t i=0; i < modifier.nops(); ++i)
+ e = e.subs(modifier.op(i), subs_options::no_pattern);
+
return e.subs(repl, subs_options::no_pattern);
+ }
}
ex power::to_polynomial(exmap & repl) const
{
if (exponent.info(info_flags::posint))
- return pow(basis.to_rational(repl), exponent);
+ return pow(basis.to_polynomial(repl), exponent);
else if (exponent.info(info_flags::negint))
{
ex basis_pref = collect_common_factors(basis);
x *= f;
}
- if (i == 0)
+ if (gc.is_zero())
gc = x;
else
gc = gcd(gc, x);
if (gc.is_equal(_ex1))
return e;
+ if (gc.is_zero())
+ return _ex0;
+
// The GCD is the factor we pull out
factor *= gc;