/** 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;
+ // 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(e_replaced, 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);
+ }
+ }
+ }
+ }
+ }
+
// 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});
+ int nmod = modifier.nops(); // To watch new modifiers to the replacement list
+ lst result = dynallocate<lst>({replace_with_symbol(map(map_normal), repl, rev_lookup, modifier), _ex1});
+ for (int 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 = ex_to<lst>(result.subs(this_repl, subs_options::no_pattern));
+ }
+
+ return result;
}
/** 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);
+ int 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 (int 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;
+ int 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(), den_itend = den.end();
+ for (int 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);
+ ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup, modifier);
+ ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup, modifier);
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(int 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(int 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(int 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(int i=0; i < modifier.nops(); ++i)
+ e = e.subs(modifier.op(i), subs_options::no_pattern);
+
return e.subs(repl, subs_options::no_pattern);
+ }
}
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