* methods for series expansion. */
/*
- * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2019 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 <iostream>
-#include <stdexcept>
-
#include "pseries.h"
#include "add.h"
#include "inifcns.h" // for Order function
#include "relational.h"
#include "operators.h"
#include "symbol.h"
-#include "print.h"
+#include "integral.h"
#include "archive.h"
#include "utils.h"
+#include <limits>
+#include <numeric>
+#include <stdexcept>
+
namespace GiNaC {
-GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
+ print_func<print_context>(&pseries::do_print).
+ print_func<print_latex>(&pseries::do_print_latex).
+ print_func<print_tree>(&pseries::do_print_tree).
+ print_func<print_python>(&pseries::do_print_python).
+ print_func<print_python_repr>(&pseries::do_print_python_repr))
/*
* Default constructor
*/
-pseries::pseries() : inherited(TINFO_pseries) { }
+pseries::pseries() { }
/*
* @param rel_ expansion variable and point (must hold a relational)
* @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
* @return newly constructed pseries */
-pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
+pseries::pseries(const ex &rel_, const epvector &ops_)
+ : seq(ops_)
{
+#ifdef DO_GINAC_ASSERT
+ auto i = seq.begin();
+ while (i != seq.end()) {
+ auto ip1 = i+1;
+ if (ip1 != seq.end())
+ GINAC_ASSERT(!is_order_function(i->rest));
+ else
+ break;
+ GINAC_ASSERT(is_a<numeric>(i->coeff));
+ GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
+ ++i;
+ }
+#endif // def DO_GINAC_ASSERT
+ GINAC_ASSERT(is_a<relational>(rel_));
+ GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
+ point = rel_.rhs();
+ var = rel_.lhs();
+}
+pseries::pseries(const ex &rel_, epvector &&ops_)
+ : seq(std::move(ops_))
+{
+#ifdef DO_GINAC_ASSERT
+ auto i = seq.begin();
+ while (i != seq.end()) {
+ auto ip1 = i+1;
+ if (ip1 != seq.end())
+ GINAC_ASSERT(!is_order_function(i->rest));
+ else
+ break;
+ GINAC_ASSERT(is_a<numeric>(i->coeff));
+ GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
+ ++i;
+ }
+#endif // def DO_GINAC_ASSERT
GINAC_ASSERT(is_a<relational>(rel_));
GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
point = rel_.rhs();
* Archiving
*/
-pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
+void pseries::read_archive(const archive_node &n, lst &sym_lst)
{
- for (unsigned int i=0; true; ++i) {
+ inherited::read_archive(n, sym_lst);
+ auto range = n.find_property_range("coeff", "power");
+ seq.reserve((range.end-range.begin)/2);
+
+ for (auto loc = range.begin; loc < range.end;) {
ex rest;
ex coeff;
- if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
- seq.push_back(expair(rest, coeff));
- else
- break;
+ n.find_ex_by_loc(loc++, rest, sym_lst);
+ n.find_ex_by_loc(loc++, coeff, sym_lst);
+ seq.emplace_back(expair(rest, coeff));
}
+
n.find_ex("var", var, sym_lst);
n.find_ex("point", point, sym_lst);
}
void pseries::archive(archive_node &n) const
{
inherited::archive(n);
- epvector::const_iterator i = seq.begin(), iend = seq.end();
- while (i != iend) {
- n.add_ex("coeff", i->rest);
- n.add_ex("power", i->coeff);
- ++i;
+ for (auto & it : seq) {
+ n.add_ex("coeff", it.rest);
+ n.add_ex("power", it.coeff);
}
n.add_ex("var", var);
n.add_ex("point", point);
}
-DEFAULT_UNARCHIVE(pseries)
//////////
// functions overriding virtual functions from base classes
//////////
-void pseries::print(const print_context & c, unsigned level) const
+void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
{
- if (is_a<print_tree>(c)) {
+ if (precedence() <= level)
+ c.s << '(';
+
+ // objects of type pseries must not have any zero entries, so the
+ // trivial (zero) pseries needs a special treatment here:
+ if (seq.empty())
+ c.s << '0';
- c.s << std::string(level, ' ') << class_name()
- << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
- << std::endl;
- unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
- size_t num = seq.size();
- for (size_t i=0; i<num; ++i) {
- seq[i].rest.print(c, level + delta_indent);
- seq[i].coeff.print(c, level + delta_indent);
- c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
- }
- var.print(c, level + delta_indent);
- point.print(c, level + delta_indent);
+ auto i = seq.begin(), end = seq.end();
+ while (i != end) {
- } else if (is_a<print_python_repr>(c)) {
- c.s << class_name() << "(relational(";
- var.print(c);
- c.s << ',';
- point.print(c);
- c.s << "),[";
- size_t num = seq.size();
- for (size_t i=0; i<num; ++i) {
- if (i)
- c.s << ',';
- c.s << '(';
- seq[i].rest.print(c);
- c.s << ',';
- seq[i].coeff.print(c);
- c.s << ')';
- }
- c.s << "])";
- } else {
+ // print a sign, if needed
+ if (i != seq.begin())
+ c.s << '+';
- if (precedence() <= level)
- c.s << "(";
-
- std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
- std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
-
- // objects of type pseries must not have any zero entries, so the
- // trivial (zero) pseries needs a special treatment here:
- if (seq.empty())
- c.s << '0';
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- // print a sign, if needed
- if (i != seq.begin())
- c.s << '+';
- if (!is_order_function(i->rest)) {
- // print 'rest', i.e. the expansion coefficient
- if (i->rest.info(info_flags::numeric) &&
- i->rest.info(info_flags::positive)) {
- i->rest.print(c);
- } else {
- c.s << par_open;
- i->rest.print(c);
- c.s << par_close;
- }
- // print 'coeff', something like (x-1)^42
- if (!i->coeff.is_zero()) {
- if (is_a<print_latex>(c))
- c.s << ' ';
- else
- c.s << '*';
- if (!point.is_zero()) {
- c.s << par_open;
- (var-point).print(c);
- c.s << par_close;
+ if (!is_order_function(i->rest)) {
+
+ // print 'rest', i.e. the expansion coefficient
+ if (i->rest.info(info_flags::numeric) &&
+ i->rest.info(info_flags::positive)) {
+ i->rest.print(c);
+ } else {
+ c.s << openbrace << '(';
+ i->rest.print(c);
+ c.s << ')' << closebrace;
+ }
+
+ // print 'coeff', something like (x-1)^42
+ if (!i->coeff.is_zero()) {
+ c.s << mul_sym;
+ if (!point.is_zero()) {
+ c.s << openbrace << '(';
+ (var-point).print(c);
+ c.s << ')' << closebrace;
+ } else
+ var.print(c);
+ if (i->coeff.compare(_ex1)) {
+ c.s << pow_sym;
+ c.s << openbrace;
+ if (i->coeff.info(info_flags::negative)) {
+ c.s << '(';
+ i->coeff.print(c);
+ c.s << ')';
} else
- var.print(c);
- if (i->coeff.compare(_ex1)) {
- if (is_a<print_python>(c))
- c.s << "**";
- else
- c.s << '^';
- if (i->coeff.info(info_flags::negative)) {
- c.s << par_open;
- i->coeff.print(c);
- c.s << par_close;
- } else {
- if (is_a<print_latex>(c)) {
- c.s << '{';
- i->coeff.print(c);
- c.s << '}';
- } else
- i->coeff.print(c);
- }
- }
+ i->coeff.print(c);
+ c.s << closebrace;
}
- } else
- Order(power(var-point,i->coeff)).print(c);
- ++i;
- }
+ }
+ } else
+ Order(pow(var - point, i->coeff)).print(c);
+ ++i;
+ }
+
+ if (precedence() <= level)
+ c.s << ')';
+}
+
+void pseries::do_print(const print_context & c, unsigned level) const
+{
+ print_series(c, "", "", "*", "^", level);
+}
+
+void pseries::do_print_latex(const print_latex & c, unsigned level) const
+{
+ print_series(c, "{", "}", " ", "^", level);
+}
+
+void pseries::do_print_python(const print_python & c, unsigned level) const
+{
+ print_series(c, "", "", "*", "**", level);
+}
- if (precedence() <= level)
- c.s << ")";
+void pseries::do_print_tree(const print_tree & c, unsigned level) const
+{
+ c.s << std::string(level, ' ') << class_name() << " @" << this
+ << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
+ << std::endl;
+ size_t num = seq.size();
+ for (size_t i=0; i<num; ++i) {
+ seq[i].rest.print(c, level + c.delta_indent);
+ seq[i].coeff.print(c, level + c.delta_indent);
+ c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
+ }
+ var.print(c, level + c.delta_indent);
+ point.print(c, level + c.delta_indent);
+}
+
+void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
+{
+ c.s << class_name() << "(relational(";
+ var.print(c);
+ c.s << ',';
+ point.print(c);
+ c.s << "),[";
+ size_t num = seq.size();
+ for (size_t i=0; i<num; ++i) {
+ if (i)
+ c.s << ',';
+ c.s << '(';
+ seq[i].rest.print(c);
+ c.s << ',';
+ seq[i].coeff.print(c);
+ c.s << ')';
}
+ c.s << "])";
}
int pseries::compare_same_type(const basic & other) const
return cmpval;
// ...and if that failed the individual elements
- epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
+ auto it = seq.begin(), o_it = o.seq.begin();
while (it!=seq.end() && o_it!=o.seq.end()) {
cmpval = it->compare(*o_it);
if (cmpval)
if (i >= seq.size())
throw (std::out_of_range("op() out of range"));
- return seq[i].rest * power(var - point, seq[i].coeff);
+ if (is_order_function(seq[i].rest))
+ return Order(pow(var-point, seq[i].coeff));
+ return seq[i].rest * pow(var - point, seq[i].coeff);
}
/** Return degree of highest power of the series. This is usually the exponent
* series is examined termwise. */
int pseries::degree(const ex &s) const
{
- if (var.is_equal(s)) {
- // Return last exponent
- if (seq.size())
- return ex_to<numeric>((seq.end()-1)->coeff).to_int();
- else
- return 0;
- } else {
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- if (it == itend)
- return 0;
- int max_pow = INT_MIN;
- while (it != itend) {
- int pow = it->rest.degree(s);
- if (pow > max_pow)
- max_pow = pow;
- ++it;
- }
- return max_pow;
- }
+ if (seq.empty())
+ return 0;
+
+ if (var.is_equal(s))
+ // Return last/greatest exponent
+ return ex_to<numeric>((seq.end()-1)->coeff).to_int();
+
+ int max_pow = std::numeric_limits<int>::min();
+ for (auto & it : seq)
+ max_pow = std::max(max_pow, it.rest.degree(s));
+ return max_pow;
}
/** Return degree of lowest power of the series. This is usually the exponent
* I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
int pseries::ldegree(const ex &s) const
{
- if (var.is_equal(s)) {
- // Return first exponent
- if (seq.size())
- return ex_to<numeric>((seq.begin())->coeff).to_int();
- else
- return 0;
- } else {
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- if (it == itend)
- return 0;
- int min_pow = INT_MAX;
- while (it != itend) {
- int pow = it->rest.ldegree(s);
- if (pow < min_pow)
- min_pow = pow;
- ++it;
- }
- return min_pow;
- }
+ if (seq.empty())
+ return 0;
+
+ if (var.is_equal(s))
+ // Return first/smallest exponent
+ return ex_to<numeric>((seq.begin())->coeff).to_int();
+
+ int min_pow = std::numeric_limits<int>::max();
+ for (auto & it : seq)
+ min_pow = std::min(min_pow, it.rest.degree(s));
+ return min_pow;
}
/** Return coefficient of degree n in power series if s is the expansion
}
/** Perform coefficient-wise automatic term rewriting rules in this class. */
-ex pseries::eval(int level) const
+ex pseries::eval() const
{
- if (level == 1)
- return this->hold();
-
- if (level == -max_recursion_level)
- throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
-
+ if (flags & status_flags::evaluated) {
+ return *this;
+ }
+
// Construct a new series with evaluated coefficients
epvector new_seq;
new_seq.reserve(seq.size());
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
- ++it;
- }
- return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ for (auto & it : seq)
+ new_seq.emplace_back(expair(it.rest, it.coeff));
+
+ return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
}
/** Evaluate coefficients numerically. */
-ex pseries::evalf(int level) const
+ex pseries::evalf() const
{
- if (level == 1)
- return *this;
-
- if (level == -max_recursion_level)
- throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
-
// Construct a new series with evaluated coefficients
epvector new_seq;
new_seq.reserve(seq.size());
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
- ++it;
+ for (auto & it : seq)
+ new_seq.emplace_back(expair(it.rest.evalf(), it.coeff));
+
+ return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
+}
+
+ex pseries::conjugate() const
+{
+ if(!var.info(info_flags::real))
+ return conjugate_function(*this).hold();
+
+ std::unique_ptr<epvector> newseq(conjugateepvector(seq));
+ ex newpoint = point.conjugate();
+
+ if (!newseq && are_ex_trivially_equal(point, newpoint)) {
+ return *this;
+ }
+
+ return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
+}
+
+ex pseries::real_part() const
+{
+ if(!var.info(info_flags::real))
+ return real_part_function(*this).hold();
+ ex newpoint = point.real_part();
+ if(newpoint != point)
+ return real_part_function(*this).hold();
+
+ epvector v;
+ v.reserve(seq.size());
+ for (auto & it : seq)
+ v.emplace_back(expair(it.rest.real_part(), it.coeff));
+ return dynallocate<pseries>(var==point, std::move(v));
+}
+
+ex pseries::imag_part() const
+{
+ if(!var.info(info_flags::real))
+ return imag_part_function(*this).hold();
+ ex newpoint = point.real_part();
+ if(newpoint != point)
+ return imag_part_function(*this).hold();
+
+ epvector v;
+ v.reserve(seq.size());
+ for (auto & it : seq)
+ v.emplace_back(expair(it.rest.imag_part(), it.coeff));
+ return dynallocate<pseries>(var==point, std::move(v));
+}
+
+ex pseries::eval_integ() const
+{
+ std::unique_ptr<epvector> newseq(nullptr);
+ for (auto i=seq.begin(); i!=seq.end(); ++i) {
+ if (newseq) {
+ newseq->emplace_back(expair(i->rest.eval_integ(), i->coeff));
+ continue;
+ }
+ ex newterm = i->rest.eval_integ();
+ if (!are_ex_trivially_equal(newterm, i->rest)) {
+ newseq.reset(new epvector);
+ newseq->reserve(seq.size());
+ for (auto j=seq.begin(); j!=i; ++j)
+ newseq->push_back(*j);
+ newseq->emplace_back(expair(newterm, i->coeff));
+ }
+ }
+
+ ex newpoint = point.eval_integ();
+ if (newseq || !are_ex_trivially_equal(newpoint, point))
+ return dynallocate<pseries>(var==newpoint, std::move(*newseq));
+ return *this;
+}
+
+ex pseries::evalm() const
+{
+ // evalm each coefficient
+ epvector newseq;
+ bool something_changed = false;
+ for (auto i=seq.begin(); i!=seq.end(); ++i) {
+ if (something_changed) {
+ ex newcoeff = i->rest.evalm();
+ if (!newcoeff.is_zero())
+ newseq.emplace_back(expair(newcoeff, i->coeff));
+ } else {
+ ex newcoeff = i->rest.evalm();
+ if (!are_ex_trivially_equal(newcoeff, i->rest)) {
+ something_changed = true;
+ newseq.reserve(seq.size());
+ std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
+ if (!newcoeff.is_zero())
+ newseq.emplace_back(expair(newcoeff, i->coeff));
+ }
+ }
}
- return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ if (something_changed)
+ return dynallocate<pseries>(var==point, std::move(newseq));
+ else
+ return *this;
}
ex pseries::subs(const exmap & m, unsigned options) const
// expansion point
epvector newseq;
newseq.reserve(seq.size());
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
- ++it;
- }
- return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
+ for (auto & it : seq)
+ newseq.emplace_back(expair(it.rest.subs(m, options), it.coeff));
+ return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
}
/** Implementation of ex::expand() for a power series. It expands all the
ex pseries::expand(unsigned options) const
{
epvector newseq;
- epvector::const_iterator i = seq.begin(), end = seq.end();
- while (i != end) {
- ex restexp = i->rest.expand();
+ for (auto & it : seq) {
+ ex restexp = it.rest.expand();
if (!restexp.is_zero())
- newseq.push_back(expair(restexp, i->coeff));
- ++i;
+ newseq.emplace_back(expair(restexp, it.coeff));
}
- return (new pseries(relational(var,point), newseq))
- ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
+ return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
}
/** Implementation of ex::diff() for a power series.
ex pseries::derivative(const symbol & s) const
{
epvector new_seq;
- epvector::const_iterator it = seq.begin(), itend = seq.end();
if (s == var) {
// FIXME: coeff might depend on var
- while (it != itend) {
- if (is_order_function(it->rest)) {
- new_seq.push_back(expair(it->rest, it->coeff - 1));
+ for (auto & it : seq) {
+ if (is_order_function(it.rest)) {
+ new_seq.emplace_back(expair(it.rest, it.coeff - 1));
} else {
- ex c = it->rest * it->coeff;
+ ex c = it.rest * it.coeff;
if (!c.is_zero())
- new_seq.push_back(expair(c, it->coeff - 1));
+ new_seq.emplace_back(expair(c, it.coeff - 1));
}
- ++it;
}
} else {
- while (it != itend) {
- if (is_order_function(it->rest)) {
- new_seq.push_back(*it);
+ for (auto & it : seq) {
+ if (is_order_function(it.rest)) {
+ new_seq.push_back(it);
} else {
- ex c = it->rest.diff(s);
+ ex c = it.rest.diff(s);
if (!c.is_zero())
- new_seq.push_back(expair(c, it->coeff));
+ new_seq.emplace_back(expair(c, it.coeff));
}
- ++it;
}
}
- return pseries(relational(var,point), new_seq);
+ return pseries(relational(var,point), std::move(new_seq));
}
ex pseries::convert_to_poly(bool no_order) const
{
ex e;
- epvector::const_iterator it = seq.begin(), itend = seq.end();
-
- while (it != itend) {
- if (is_order_function(it->rest)) {
+ for (auto & it : seq) {
+ if (is_order_function(it.rest)) {
if (!no_order)
- e += Order(power(var - point, it->coeff));
+ e += Order(pow(var - point, it.coeff));
} else
- e += it->rest * power(var - point, it->coeff);
- ++it;
+ e += it.rest * pow(var - point, it.coeff);
}
return e;
}
return seq.empty() || !is_order_function((seq.end()-1)->rest);
}
+ex pseries::coeffop(size_t i) const
+{
+ if (i >= nops())
+ throw (std::out_of_range("coeffop() out of range"));
+ return seq[i].rest;
+}
+
+ex pseries::exponop(size_t i) const
+{
+ if (i >= nops())
+ throw (std::out_of_range("exponop() out of range"));
+ return seq[i].coeff;
+}
+
/*
* Implementations of series expansion
ex basic::series(const relational & r, int order, unsigned options) const
{
epvector seq;
+ const symbol &s = ex_to<symbol>(r.lhs());
+
+ // default for order-values that make no sense for Taylor expansion
+ if ((order <= 0) && this->has(s)) {
+ seq.emplace_back(expair(Order(_ex1), order));
+ return pseries(r, std::move(seq));
+ }
+
+ // do Taylor expansion
numeric fac = 1;
ex deriv = *this;
- ex coeff = deriv.subs(r);
- const symbol &s = ex_to<symbol>(r.lhs());
-
- if (!coeff.is_zero())
- seq.push_back(expair(coeff, _ex0));
-
+ ex coeff = deriv.subs(r, subs_options::no_pattern);
+
+ if (!coeff.is_zero()) {
+ seq.emplace_back(expair(coeff, _ex0));
+ }
+
int n;
for (n=1; n<order; ++n) {
- fac = fac.mul(n);
+ fac = fac.div(n);
// We need to test for zero in order to see if the series terminates.
// The problem is that there is no such thing as a perfect test for
// zero. Expanding the term occasionally helps a little...
deriv = deriv.diff(s).expand();
if (deriv.is_zero()) // Series terminates
- return pseries(r, seq);
+ return pseries(r, std::move(seq));
- coeff = deriv.subs(r);
+ coeff = deriv.subs(r, subs_options::no_pattern);
if (!coeff.is_zero())
- seq.push_back(expair(fac.inverse() * coeff, n));
+ seq.emplace_back(expair(fac * coeff, n));
}
// Higher-order terms, if present
deriv = deriv.diff(s);
if (!deriv.expand().is_zero())
- seq.push_back(expair(Order(_ex1), n));
- return pseries(r, seq);
+ seq.emplace_back(expair(Order(_ex1), n));
+ return pseries(r, std::move(seq));
}
if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
if (order > 0 && !point.is_zero())
- seq.push_back(expair(point, _ex0));
+ seq.emplace_back(expair(point, _ex0));
if (order > 1)
- seq.push_back(expair(_ex1, _ex1));
+ seq.emplace_back(expair(_ex1, _ex1));
else
- seq.push_back(expair(Order(_ex1), numeric(order)));
+ seq.emplace_back(expair(Order(_ex1), numeric(order)));
} else
- seq.push_back(expair(*this, _ex0));
- return pseries(r, seq);
+ seq.emplace_back(expair(*this, _ex0));
+ return pseries(r, std::move(seq));
}
// Adding two series with different variables or expansion points
// results in an empty (constant) series
if (!is_compatible_to(other)) {
- epvector nul;
- nul.push_back(expair(Order(_ex1), _ex0));
- return pseries(relational(var,point), nul);
+ epvector nul { expair(Order(_ex1), _ex0) };
+ return pseries(relational(var,point), std::move(nul));
}
// Series addition
epvector new_seq;
- epvector::const_iterator a = seq.begin();
- epvector::const_iterator b = other.seq.begin();
- epvector::const_iterator a_end = seq.end();
- epvector::const_iterator b_end = other.seq.end();
- int pow_a = INT_MAX, pow_b = INT_MAX;
+ auto a = seq.begin(), a_end = seq.end();
+ auto b = other.seq.begin(), b_end = other.seq.end();
+ int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
for (;;) {
// If a is empty, fill up with elements from b and stop
if (a == a_end) {
} else {
// Add coefficient of a and b
if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
- new_seq.push_back(expair(Order(_ex1), (*a).coeff));
+ new_seq.emplace_back(expair(Order(_ex1), (*a).coeff));
break; // Order term ends the sequence
} else {
ex sum = (*a).rest + (*b).rest;
if (!(sum.is_zero()))
- new_seq.push_back(expair(sum, numeric(pow_a)));
+ new_seq.emplace_back(expair(sum, numeric(pow_a)));
++a;
++b;
}
}
}
- return pseries(relational(var,point), new_seq);
+ return pseries(relational(var,point), std::move(new_seq));
}
acc = overall_coeff.series(r, order, options);
// Add remaining terms
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- for (; it!=itend; ++it) {
+ for (auto & it : seq) {
ex op;
- if (is_exactly_a<pseries>(it->rest))
- op = it->rest;
+ if (is_exactly_a<pseries>(it.rest))
+ op = it.rest;
else
- op = it->rest.series(r, order, options);
- if (!it->coeff.is_equal(_ex1))
- op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
+ op = it.rest.series(r, order, options);
+ if (!it.coeff.is_equal(_ex1))
+ op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
// Series addition
acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
epvector new_seq;
new_seq.reserve(seq.size());
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- if (!is_order_function(it->rest))
- new_seq.push_back(expair(it->rest * other, it->coeff));
+ for (auto & it : seq) {
+ if (!is_order_function(it.rest))
+ new_seq.emplace_back(expair(it.rest * other, it.coeff));
else
- new_seq.push_back(*it);
- ++it;
+ new_seq.push_back(it);
}
- return pseries(relational(var,point), new_seq);
+ return pseries(relational(var,point), std::move(new_seq));
}
// Multiplying two series with different variables or expansion points
// results in an empty (constant) series
if (!is_compatible_to(other)) {
- epvector nul;
- nul.push_back(expair(Order(_ex1), _ex0));
- return pseries(relational(var,point), nul);
+ epvector nul { expair(Order(_ex1), _ex0) };
+ return pseries(relational(var,point), std::move(nul));
+ }
+
+ if (seq.empty() || other.seq.empty()) {
+ return dynallocate<pseries>(var==point, epvector());
}
// Series multiplication
epvector new_seq;
- int a_max = degree(var);
- int b_max = other.degree(var);
- int a_min = ldegree(var);
- int b_min = other.ldegree(var);
- int cdeg_min = a_min + b_min;
+ const int a_max = degree(var);
+ const int b_max = other.degree(var);
+ const int a_min = ldegree(var);
+ const int b_min = other.ldegree(var);
+ const int cdeg_min = a_min + b_min;
int cdeg_max = a_max + b_max;
- int higher_order_a = INT_MAX;
- int higher_order_b = INT_MAX;
+ int higher_order_a = std::numeric_limits<int>::max();
+ int higher_order_b = std::numeric_limits<int>::max();
if (is_order_function(coeff(var, a_max)))
higher_order_a = a_max + b_min;
if (is_order_function(other.coeff(var, b_max)))
higher_order_b = b_max + a_min;
- int higher_order_c = std::min(higher_order_a, higher_order_b);
+ const int higher_order_c = std::min(higher_order_a, higher_order_b);
if (cdeg_max >= higher_order_c)
cdeg_max = higher_order_c - 1;
-
+
+ std::map<int, ex> rest_map_a, rest_map_b;
+ for (const auto& it : seq)
+ rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
+
+ if (other.var.is_equal(var))
+ for (const auto& it : other.seq)
+ rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
+
for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
ex co = _ex0;
// c(i)=a(0)b(i)+...+a(i)b(0)
for (int i=a_min; cdeg-i>=b_min; ++i) {
- ex a_coeff = coeff(var, i);
- ex b_coeff = other.coeff(var, cdeg-i);
- if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
- co += a_coeff * b_coeff;
+ const auto& ita = rest_map_a.find(i);
+ if (ita == rest_map_a.end())
+ continue;
+ const auto& itb = rest_map_b.find(cdeg-i);
+ if (itb == rest_map_b.end())
+ continue;
+ if (!is_order_function(ita->second) && !is_order_function(itb->second))
+ co += ita->second * itb->second;
}
if (!co.is_zero())
- new_seq.push_back(expair(co, numeric(cdeg)));
+ new_seq.emplace_back(expair(co, numeric(cdeg)));
}
- if (higher_order_c < INT_MAX)
- new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
- return pseries(relational(var, point), new_seq);
+ if (higher_order_c < std::numeric_limits<int>::max())
+ new_seq.emplace_back(expair(Order(_ex1), numeric(higher_order_c)));
+ return pseries(relational(var, point), std::move(new_seq));
}
{
pseries acc; // Series accumulator
+ GINAC_ASSERT(is_a<symbol>(r.lhs()));
+ const ex& sym = r.lhs();
+
+ // holds ldegrees of the series of individual factors
+ std::vector<int> ldegrees;
+ std::vector<bool> ldegree_redo;
+
+ // find minimal degrees
+ // first round: obtain a bound up to which minimal degrees have to be
+ // considered
+ for (auto & it : seq) {
+
+ ex expon = it.coeff;
+ int factor = 1;
+ ex buf;
+ if (expon.info(info_flags::integer)) {
+ buf = it.rest;
+ factor = ex_to<numeric>(expon).to_int();
+ } else {
+ buf = recombine_pair_to_ex(it);
+ }
+
+ int real_ldegree = 0;
+ bool flag_redo = false;
+ try {
+ real_ldegree = buf.expand().ldegree(sym-r.rhs());
+ } catch (std::runtime_error) {}
+
+ if (real_ldegree == 0) {
+ if ( factor < 0 ) {
+ // This case must terminate, otherwise we would have division by
+ // zero.
+ int orderloop = 0;
+ do {
+ orderloop++;
+ real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
+ } while (real_ldegree == orderloop);
+ } else {
+ // Here it is possible that buf does not have a ldegree, therefore
+ // check only if ldegree is negative, otherwise reconsider the case
+ // in the second round.
+ real_ldegree = buf.series(r, 0, options).ldegree(sym);
+ if (real_ldegree == 0)
+ flag_redo = true;
+ }
+ }
+
+ ldegrees.push_back(factor * real_ldegree);
+ ldegree_redo.push_back(flag_redo);
+ }
+
+ int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
+ // Second round: determine the remaining positive ldegrees by the series
+ // method.
+ // here we can ignore ldegrees larger than degbound
+ size_t j = 0;
+ for (auto & it : seq) {
+ if ( ldegree_redo[j] ) {
+ ex expon = it.coeff;
+ int factor = 1;
+ ex buf;
+ if (expon.info(info_flags::integer)) {
+ buf = it.rest;
+ factor = ex_to<numeric>(expon).to_int();
+ } else {
+ buf = recombine_pair_to_ex(it);
+ }
+ int real_ldegree = 0;
+ int orderloop = 0;
+ do {
+ orderloop++;
+ real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
+ } while ((real_ldegree == orderloop)
+ && (factor*real_ldegree < degbound));
+ ldegrees[j] = factor * real_ldegree;
+ degbound -= factor * real_ldegree;
+ }
+ j++;
+ }
+
+ int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
+
+ if (degsum > order) {
+ return dynallocate<pseries>(r, epvector{{Order(_ex1), order}});
+ }
+
// Multiply with remaining terms
- const epvector::const_iterator itbeg = seq.begin();
- const epvector::const_iterator itend = seq.end();
- for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
- ex op = recombine_pair_to_ex(*it).series(r, order, options);
+ auto itd = ldegrees.begin();
+ for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
+
+ // do series expansion with adjusted order
+ ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
// Series multiplication
- if (it==itbeg)
+ if (it == seq.begin())
acc = ex_to<pseries>(op);
else
acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
}
+
return acc.mul_const(ex_to<numeric>(overall_coeff));
}
if (!(p*ldeg).is_integer())
throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
+ // adjust number of coefficients
+ int numcoeff = deg - (p*ldeg).to_int();
+ if (numcoeff <= 0) {
+ epvector epv { expair(Order(_ex1), deg) };
+ return dynallocate<pseries>(relational(var,point), std::move(epv));
+ }
+
// O(x^n)^(-m) is undefined
if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
throw pole_error("pseries::power_const(): division by zero",1);
// Compute coefficients of the powered series
exvector co;
- co.reserve(deg);
- co.push_back(power(coeff(var, ldeg), p));
- bool all_sums_zero = true;
- for (int i=1; i<deg; ++i) {
+ co.reserve(numcoeff);
+ co.push_back(pow(coeff(var, ldeg), p));
+ for (int i=1; i<numcoeff; ++i) {
ex sum = _ex0;
for (int j=1; j<=i; ++j) {
ex c = coeff(var, j + ldeg);
} else
sum += (p * j - (i - j)) * co[i - j] * c;
}
- if (!sum.is_zero())
- all_sums_zero = false;
co.push_back(sum / coeff(var, ldeg) / i);
}
// Construct new series (of non-zero coefficients)
epvector new_seq;
bool higher_order = false;
- for (int i=0; i<deg; ++i) {
+ for (int i=0; i<numcoeff; ++i) {
if (!co[i].is_zero())
- new_seq.push_back(expair(co[i], p * ldeg + i));
+ new_seq.emplace_back(expair(co[i], p * ldeg + i));
if (is_order_function(co[i])) {
higher_order = true;
break;
}
}
- if (!higher_order && !all_sums_zero)
- new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
- return pseries(relational(var,point), new_seq);
+ if (!higher_order)
+ new_seq.emplace_back(expair(Order(_ex1), p * ldeg + numcoeff));
+
+ return pseries(relational(var,point), std::move(new_seq));
}
pseries pseries::shift_exponents(int deg) const
{
epvector newseq = seq;
- epvector::iterator i = newseq.begin(), end = newseq.end();
- while (i != end) {
- i->coeff += deg;
- ++i;
- }
- return pseries(relational(var, point), newseq);
+ for (auto & it : newseq)
+ it.coeff += deg;
+ return pseries(relational(var, point), std::move(newseq));
}
// Basis is not a series, may there be a singularity?
bool must_expand_basis = false;
try {
- basis.subs(r);
+ basis.subs(r, subs_options::no_pattern);
} catch (pole_error) {
must_expand_basis = true;
}
-
+
+ bool exponent_is_regular = true;
+ try {
+ exponent.subs(r, subs_options::no_pattern);
+ } catch (pole_error) {
+ exponent_is_regular = false;
+ }
+
+ if (!exponent_is_regular) {
+ ex l = exponent*log(basis);
+ // this == exp(l);
+ ex le = l.series(r, order, options);
+ // Note: expanding exp(l) won't help, since that will attempt
+ // Taylor expansion, and fail (because exponent is "singular")
+ // Still l itself might be expanded in Taylor series.
+ // Examples:
+ // sin(x)/x*log(cos(x))
+ // 1/x*log(1 + x)
+ return exp(le).series(r, order, options);
+ // Note: if l happens to have a Laurent expansion (with
+ // negative powers of (var - point)), expanding exp(le)
+ // will barf (which is The Right Thing).
+ }
+
// Is the expression of type something^(-int)?
- if (!must_expand_basis && !exponent.info(info_flags::negint))
+ if (!must_expand_basis && !exponent.info(info_flags::negint)
+ && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
return basic::series(r, order, options);
-
+
// Is the expression of type 0^something?
- if (!must_expand_basis && !basis.subs(r).is_zero())
+ if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
+ && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
return basic::series(r, order, options);
// Singularity encountered, is the basis equal to (var - point)?
if (basis.is_equal(r.lhs() - r.rhs())) {
epvector new_seq;
if (ex_to<numeric>(exponent).to_int() < order)
- new_seq.push_back(expair(_ex1, exponent));
+ new_seq.emplace_back(expair(_ex1, exponent));
else
- new_seq.push_back(expair(Order(_ex1), exponent));
- return pseries(r, new_seq);
+ new_seq.emplace_back(expair(Order(_ex1), exponent));
+ return pseries(r, std::move(new_seq));
}
// No, expand basis into series
- ex e = basis.series(r, order, options);
- return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
+
+ numeric numexp;
+ if (is_a<numeric>(exponent)) {
+ numexp = ex_to<numeric>(exponent);
+ } else {
+ numexp = 0;
+ }
+ const ex& sym = r.lhs();
+ // find existing minimal degree
+ ex eb = basis.expand();
+ int real_ldegree = 0;
+ if (eb.info(info_flags::rational_function))
+ real_ldegree = eb.ldegree(sym-r.rhs());
+ if (real_ldegree == 0) {
+ int orderloop = 0;
+ do {
+ orderloop++;
+ real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
+ } while (real_ldegree == orderloop);
+ }
+
+ if (!(real_ldegree*numexp).is_integer())
+ throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
+ ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
+
+ ex result;
+ try {
+ result = ex_to<pseries>(e).power_const(numexp, order);
+ } catch (pole_error) {
+ epvector ser { expair(Order(_ex1), order) };
+ result = pseries(r, std::move(ser));
+ }
+
+ return result;
}
return *this;
else {
epvector new_seq;
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- int o = ex_to<numeric>(it->coeff).to_int();
+ for (auto & it : seq) {
+ int o = ex_to<numeric>(it.coeff).to_int();
if (o >= order) {
- new_seq.push_back(expair(Order(_ex1), o));
+ new_seq.emplace_back(expair(Order(_ex1), o));
break;
}
- new_seq.push_back(*it);
- ++it;
+ new_seq.push_back(it);
}
- return pseries(r, new_seq);
+ return pseries(r, std::move(new_seq));
}
} else
return convert_to_poly().series(r, order, options);
}
+ex integral::series(const relational & r, int order, unsigned options) const
+{
+ if (x.subs(r) != x)
+ throw std::logic_error("Cannot series expand wrt dummy variable");
+
+ // Expanding integrand with r substituted taken in boundaries.
+ ex fseries = f.series(r, order, options);
+ epvector fexpansion;
+ fexpansion.reserve(fseries.nops());
+ for (size_t i=0; i<fseries.nops(); ++i) {
+ ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
+ currcoeff = (currcoeff == Order(_ex1))
+ ? currcoeff
+ : integral(x, a.subs(r), b.subs(r), currcoeff);
+ if (currcoeff != 0)
+ fexpansion.emplace_back(
+ expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
+ }
+
+ // Expanding lower boundary
+ ex result = dynallocate<pseries>(r, std::move(fexpansion));
+ ex aseries = (a-a.subs(r)).series(r, order, options);
+ fseries = f.series(x == (a.subs(r)), order, options);
+ for (size_t i=0; i<fseries.nops(); ++i) {
+ ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
+ if (is_order_function(currcoeff))
+ break;
+ ex currexpon = ex_to<pseries>(fseries).exponop(i);
+ int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
+ currcoeff = currcoeff.series(r, orderforf);
+ ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
+ term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
+ term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
+ result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
+ }
+
+ // Expanding upper boundary
+ ex bseries = (b-b.subs(r)).series(r, order, options);
+ fseries = f.series(x == (b.subs(r)), order, options);
+ for (size_t i=0; i<fseries.nops(); ++i) {
+ ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
+ if (is_order_function(currcoeff))
+ break;
+ ex currexpon = ex_to<pseries>(fseries).exponop(i);
+ int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
+ currcoeff = currcoeff.series(r, orderforf);
+ ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
+ term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
+ term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
+ result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
+ }
+
+ return result;
+}
+
/** Compute the truncated series expansion of an expression.
* This function returns an expression containing an object of class pseries
else
throw (std::logic_error("ex::series(): expansion point has unknown type"));
- try {
- e = bp->series(rel_, order, options);
- } catch (std::exception &x) {
- throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
- }
+ e = bp->series(rel_, order, options);
return e;
}
+GINAC_BIND_UNARCHIVER(pseries);
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff