* methods for series expansion. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2011 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 <stdexcept>
-
#include "pseries.h"
#include "add.h"
-#include "inifcns.h"
+#include "inifcns.h" // for Order function
#include "lst.h"
#include "mul.h"
#include "power.h"
#include "relational.h"
+#include "operators.h"
#include "symbol.h"
+#include "integral.h"
#include "archive.h"
#include "utils.h"
-#include "debugmsg.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 ctor, dtor, copy ctor, assignment operator and helpers
+ * Default constructor
*/
-pseries::pseries() : basic(TINFO_pseries)
-{
- debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
-}
-
-void pseries::copy(const pseries &other)
-{
- inherited::copy(other);
- seq = other.seq;
- var = other.var;
- point = other.point;
-}
-
-void pseries::destroy(bool call_parent)
-{
- if (call_parent)
- inherited::destroy(call_parent);
-}
+pseries::pseries() { }
/*
/** Construct pseries from a vector of coefficients and powers.
* expair.rest holds the coefficient, expair.coeff holds the power.
* The powers must be integers (positive or negative) and in ascending order;
- * the last coefficient can be Order(_ex1()) to represent a truncated,
+ * the last coefficient can be Order(_ex1) to represent a truncated,
* non-terminating series.
*
* @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_)
{
- debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
- GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
- GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
+ GINAC_ASSERT(is_a<relational>(rel_));
+ GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
point = rel_.rhs();
- var = *static_cast<symbol *>(rel_.lhs().bp);
+ var = rel_.lhs();
}
* Archiving
*/
-/** Construct object from archive_node. */
-pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+void pseries::read_archive(const archive_node &n, lst &sym_lst)
{
- debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
- for (unsigned int i=0; true; ++i) {
+ inherited::read_archive(n, sym_lst);
+ archive_node::archive_node_cit first = n.find_first("coeff");
+ archive_node::archive_node_cit last = n.find_last("power");
+ ++last;
+ seq.reserve((last-first)/2);
+
+ for (archive_node::archive_node_cit loc = first; loc < last;) {
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.push_back(expair(rest, coeff));
}
+
n.find_ex("var", var, sym_lst);
n.find_ex("point", point, sym_lst);
}
-/** Unarchive the object. */
-ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
-{
- return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
-}
-
-/** Archive the object. */
void pseries::archive(archive_node &n) const
{
inherited::archive(n);
n.add_ex("point", point);
}
+
//////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
//////////
-void pseries::print(std::ostream &os, unsigned upper_precedence) 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
{
- debugmsg("pseries print", LOGLEVEL_PRINT);
- if (precedence<=upper_precedence) os << "(";
+ 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.size()==0)
- os << '0';
- for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+ 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())
- os << '+';
+ 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)) {
- os << i->rest;
- } else
- os << "(" << i->rest << ')';
+ 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()) {
- os << '*';
- if (!point.is_zero())
- os << '(' << var-point << ')';
- else
- os << var;
- if (i->coeff.compare(_ex1())) {
- os << '^';
- if (i->coeff.info(info_flags::negative))
- os << '(' << i->coeff << ')';
- else
- os << i->coeff;
+ 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
+ i->coeff.print(c);
+ c.s << closebrace;
}
}
- } else {
- os << Order(power(var-point,i->coeff));
- }
+ } else
+ Order(power(var-point,i->coeff)).print(c);
+ ++i;
}
- if (precedence<=upper_precedence) os << ")";
+
+ if (precedence() <= level)
+ c.s << ')';
}
+void pseries::do_print(const print_context & c, unsigned level) const
+{
+ print_series(c, "", "", "*", "^", level);
+}
-void pseries::printraw(std::ostream &os) const
+void pseries::do_print_latex(const print_latex & c, unsigned level) const
{
- debugmsg("pseries printraw", LOGLEVEL_PRINT);
- os << class_name() << "(" << var << ";" << point << ";";
- for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
- os << "(" << (*i).rest << "," << (*i).coeff << "),";
- os << ")";
+ print_series(c, "{", "}", " ", "^", level);
}
+void pseries::do_print_python(const print_python & c, unsigned level) const
+{
+ print_series(c, "", "", "*", "**", level);
+}
-void pseries::printtree(std::ostream & os, unsigned indent) const
+void pseries::do_print_tree(const print_tree & c, unsigned level) const
{
- debugmsg("pseries printtree",LOGLEVEL_PRINT);
- os << std::string(indent,' ') << class_name()
- << ", hash=" << hashvalue
- << " (0x" << std::hex << hashvalue << std::dec << ")"
- << ", flags=" << flags << std::endl;
- for (unsigned i=0; i<seq.size(); ++i) {
- seq[i].rest.printtree(os,indent+delta_indent);
- seq[i].coeff.printtree(os,indent+delta_indent);
- if (i!=seq.size()-1)
- os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
+ 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.printtree(os, indent+delta_indent);
- point.printtree(os, indent+delta_indent);
+ 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
{
- GINAC_ASSERT(is_of_type(other, pseries));
+ GINAC_ASSERT(is_a<pseries>(other));
const pseries &o = static_cast<const pseries &>(other);
// first compare the lengths of the series...
}
/** Return the number of operands including a possible order term. */
-unsigned pseries::nops(void) const
+size_t pseries::nops() const
{
return seq.size();
}
-
/** Return the ith term in the series when represented as a sum. */
-ex pseries::op(int i) const
+ex pseries::op(size_t i) const
{
- if (i < 0 || unsigned(i) >= seq.size())
+ if (i >= seq.size())
throw (std::out_of_range("op() out of range"));
- return seq[i].rest * power(var - point, seq[i].coeff);
-}
-
-ex &pseries::let_op(int i)
-{
- throw (std::logic_error("let_op not defined for pseries"));
+ if (is_order_function(seq[i].rest))
+ return Order(power(var-point, seq[i].coeff));
+ return seq[i].rest * power(var - point, seq[i].coeff);
}
-
/** Return degree of highest power of the series. This is usually the exponent
* of the Order term. If s is not the expansion variable of the series, the
* series is examined termwise. */
-int pseries::degree(const symbol &s) const
+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();
+ 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;
+ int max_pow = std::numeric_limits<int>::min();
while (it != itend) {
int pow = it->rest.degree(s);
if (pow > max_pow)
* series is examined termwise. If s is the expansion variable but the
* expansion point is not zero the series is not expanded to find the degree.
* I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
-int pseries::ldegree(const symbol &s) const
+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();
+ 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;
+ int min_pow = std::numeric_limits<int>::max();
while (it != itend) {
int pow = it->rest.ldegree(s);
if (pow < min_pow)
* If s is not the expansion variable, an attempt is made to convert the
* series to a polynomial and return the corresponding coefficient from
* there. */
-ex pseries::coeff(const symbol &s, int n) const
+ex pseries::coeff(const ex &s, int n) const
{
if (var.is_equal(s)) {
- if (seq.size() == 0)
- return _ex0();
+ if (seq.empty())
+ return _ex0;
// Binary search in sequence for given power
numeric looking_for = numeric(n);
int lo = 0, hi = seq.size() - 1;
while (lo <= hi) {
int mid = (lo + hi) / 2;
- GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
- int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
+ GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
+ int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
switch (cmp) {
case -1:
lo = mid + 1;
throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
}
}
- return _ex0();
+ return _ex0;
} else
return convert_to_poly().coeff(s, n);
}
/** Does nothing. */
-ex pseries::collect(const symbol &s) const
+ex pseries::collect(const ex &s, bool distributed) const
{
return *this;
}
-
-/** Evaluate coefficients. */
+/** Perform coefficient-wise automatic term rewriting rules in this class. */
ex pseries::eval(int level) const
{
if (level == 1)
return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
}
-
/** Evaluate coefficients numerically. */
ex pseries::evalf(int level) const
{
return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
}
+ex pseries::conjugate() const
+{
+ if(!var.info(info_flags::real))
+ return conjugate_function(*this).hold();
+
+ epvector * newseq = conjugateepvector(seq);
+ ex newpoint = point.conjugate();
+
+ if (!newseq && are_ex_trivially_equal(point, newpoint)) {
+ return *this;
+ }
+
+ ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
+ delete newseq;
+ return result;
+}
+
+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(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
+ v.push_back(expair((i->rest).real_part(), i->coeff));
+ return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
+}
+
+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(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
+ v.push_back(expair((i->rest).imag_part(), i->coeff));
+ return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
+}
+
+ex pseries::eval_integ() const
+{
+ epvector *newseq = NULL;
+ for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+ if (newseq) {
+ newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
+ continue;
+ }
+ ex newterm = i->rest.eval_integ();
+ if (!are_ex_trivially_equal(newterm, i->rest)) {
+ newseq = new epvector;
+ newseq->reserve(seq.size());
+ for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
+ newseq->push_back(*j);
+ newseq->push_back(expair(newterm, i->coeff));
+ }
+ }
+
+ ex newpoint = point.eval_integ();
+ if (newseq || !are_ex_trivially_equal(newpoint, point))
+ return (new pseries(var==newpoint, *newseq))
+ ->setflag(status_flags::dynallocated);
+ return *this;
+}
+
+ex pseries::evalm() const
+{
+ // evalm each coefficient
+ epvector newseq;
+ bool something_changed = false;
+ for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+ if (something_changed) {
+ ex newcoeff = i->rest.evalm();
+ if (!newcoeff.is_zero())
+ newseq.push_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.push_back(expair(newcoeff, i->coeff));
+ }
+ }
+ }
+ if (something_changed)
+ return (new pseries(var==point, newseq))->setflag(status_flags::dynallocated);
+ else
+ return *this;
+}
-ex pseries::subs(const lst & ls, const lst & lr) const
+ex pseries::subs(const exmap & m, unsigned options) const
{
// If expansion variable is being substituted, convert the series to a
// polynomial and do the substitution there because the result might
// no longer be a power series
- if (ls.has(var))
- return convert_to_poly(true).subs(ls, lr);
+ if (m.find(var) != m.end())
+ return convert_to_poly(true).subs(m, options);
// Otherwise construct a new series with substituted coefficients and
// expansion point
newseq.reserve(seq.size());
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
- newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
+ newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
++it;
}
- return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
+ return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
}
-
/** Implementation of ex::expand() for a power series. It expands all the
* terms individually and returns the resulting series as a new pseries. */
ex pseries::expand(unsigned options) const
{
epvector newseq;
- for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+ epvector::const_iterator i = seq.begin(), end = seq.end();
+ while (i != end) {
ex restexp = i->rest.expand();
if (!restexp.is_zero())
newseq.push_back(expair(restexp, i->coeff));
+ ++i;
}
return (new pseries(relational(var,point), newseq))
- ->setflag(status_flags::dynallocated | status_flags::expanded);
+ ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
}
-
-/** Implementation of ex::diff() for a power series. It treats the series as a
- * polynomial.
+/** Implementation of ex::diff() for a power series.
* @see ex::diff */
ex pseries::derivative(const symbol & s) const
{
+ epvector new_seq;
+ epvector::const_iterator it = seq.begin(), itend = seq.end();
+
if (s == var) {
- epvector new_seq;
- epvector::const_iterator it = seq.begin(), itend = seq.end();
// FIXME: coeff might depend on var
while (it != itend) {
}
++it;
}
- return pseries(relational(var,point), new_seq);
+
} else {
- return *this;
+
+ while (it != itend) {
+ if (is_order_function(it->rest)) {
+ new_seq.push_back(*it);
+ } else {
+ ex c = it->rest.diff(s);
+ if (!c.is_zero())
+ new_seq.push_back(expair(c, it->coeff));
+ }
+ ++it;
+ }
}
-}
+ return pseries(relational(var,point), new_seq);
+}
-/** Convert a pseries object to an ordinary polynomial.
- *
- * @param no_order flag: discard higher order terms */
ex pseries::convert_to_poly(bool no_order) const
{
ex e;
return e;
}
+bool pseries::is_terminating() const
+{
+ 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;
+}
-/** Returns true if there is no order term, i.e. the series terminates and
- * false otherwise. */
-bool pseries::is_terminating(void) const
+ex pseries::exponop(size_t i) const
{
- return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
+ if (i >= nops())
+ throw (std::out_of_range("exponop() out of range"));
+ return seq[i].coeff;
}
ex basic::series(const relational & r, int order, unsigned options) const
{
epvector seq;
- numeric fac(1);
+ 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.push_back(expair(Order(_ex1), order));
+ return pseries(r, seq);
+ }
+
+ // do Taylor expansion
+ numeric fac = 1;
ex deriv = *this;
- ex coeff = deriv.subs(r);
- const symbol *s = static_cast<symbol *>(r.lhs().bp);
-
- if (!coeff.is_zero())
- seq.push_back(expair(coeff, numeric(0)));
-
+ ex coeff = deriv.subs(r, subs_options::no_pattern);
+
+ if (!coeff.is_zero()) {
+ seq.push_back(expair(coeff, _ex0));
+ }
+
int n;
for (n=1; n<order; ++n) {
- fac = fac.mul(numeric(n));
- deriv = deriv.diff(*s).expand();
- if (deriv.is_zero()) {
- // Series terminates
+ fac = fac.mul(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);
- }
- coeff = deriv.subs(r);
+
+ coeff = deriv.subs(r, subs_options::no_pattern);
if (!coeff.is_zero())
- seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
+ seq.push_back(expair(fac.inverse() * coeff, n));
}
// Higher-order terms, if present
- deriv = deriv.diff(*s);
+ deriv = deriv.diff(s);
if (!deriv.expand().is_zero())
- seq.push_back(expair(Order(_ex1()), numeric(n)));
+ seq.push_back(expair(Order(_ex1), n));
return pseries(r, seq);
}
{
epvector seq;
const ex point = r.rhs();
- GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
- const symbol *s = static_cast<symbol *>(r.lhs().bp);
-
- if (this->is_equal(*s)) {
+ GINAC_ASSERT(is_a<symbol>(r.lhs()));
+
+ if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
if (order > 0 && !point.is_zero())
- seq.push_back(expair(point, _ex0()));
+ seq.push_back(expair(point, _ex0));
if (order > 1)
- seq.push_back(expair(_ex1(), _ex1()));
+ seq.push_back(expair(_ex1, _ex1));
else
- seq.push_back(expair(Order(_ex1()), numeric(order)));
+ seq.push_back(expair(Order(_ex1), numeric(order)));
} else
- seq.push_back(expair(*this, _ex0()));
+ seq.push_back(expair(*this, _ex0));
return pseries(r, seq);
}
// results in an empty (constant) series
if (!is_compatible_to(other)) {
epvector nul;
- nul.push_back(expair(Order(_ex1()), _ex0()));
+ nul.push_back(expair(Order(_ex1), _ex0));
return pseries(relational(var,point), nul);
}
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;
+ 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) {
}
break;
} else
- pow_a = ex_to_numeric((*a).coeff).to_int();
+ pow_a = ex_to<numeric>((*a).coeff).to_int();
// If b is empty, fill up with elements from a and stop
if (b == b_end) {
}
break;
} else
- pow_b = ex_to_numeric((*b).coeff).to_int();
+ pow_b = ex_to<numeric>((*b).coeff).to_int();
// a and b are non-empty, compare powers
if (pow_a < pow_b) {
} 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.push_back(expair(Order(_ex1), (*a).coeff));
break; // Order term ends the sequence
} else {
ex sum = (*a).rest + (*b).rest;
epvector::const_iterator itend = seq.end();
for (; it!=itend; ++it) {
ex op;
- if (is_ex_exactly_of_type(it->rest, pseries))
+ 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));
+ 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));
+ acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
}
return acc;
}
// results in an empty (constant) series
if (!is_compatible_to(other)) {
epvector nul;
- nul.push_back(expair(Order(_ex1()), _ex0()));
+ nul.push_back(expair(Order(_ex1), _ex0));
return pseries(relational(var,point), nul);
}
+
+ if (seq.empty() || other.seq.empty()) {
+ return (new pseries(var==point, epvector()))
+ ->setflag(status_flags::dynallocated);
+ }
// Series multiplication
epvector new_seq;
-
- const symbol *s = static_cast<symbol *>(var.bp);
- int a_max = degree(*s);
- int b_max = other.degree(*s);
- int a_min = ldegree(*s);
- int b_min = other.ldegree(*s);
+ 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;
int cdeg_max = a_max + b_max;
- int higher_order_a = INT_MAX;
- int higher_order_b = INT_MAX;
- if (is_order_function(coeff(*s, a_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(*s, b_max)))
+ 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);
if (cdeg_max >= higher_order_c)
cdeg_max = higher_order_c - 1;
for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
- ex co = _ex0();
+ 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(*s, i);
- ex b_coeff = other.coeff(*s, cdeg-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;
}
if (!co.is_zero())
new_seq.push_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.push_back(expair(Order(_ex1), numeric(higher_order_c)));
+ return pseries(relational(var, point), new_seq);
}
* @see ex::series */
ex mul::series(const relational & r, int order, unsigned options) const
{
- ex acc; // Series accumulator
-
- // Get first term from overall_coeff
- acc = overall_coeff.series(r, order, options);
-
+ 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
+ const epvector::const_iterator itbeg = seq.begin();
+ const epvector::const_iterator itend = seq.end();
+ // first round: obtain a bound up to which minimal degrees have to be
+ // considered
+ for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
+
+ 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 (epvector::const_iterator it=itbeg; it!=itend; ++it) {
+ 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) {
+ epvector epv;
+ epv.push_back(expair(Order(_ex1), order));
+ return (new pseries(r, epv))->setflag(status_flags::dynallocated);
+ }
+
// Multiply with remaining terms
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- for (; it!=itend; ++it) {
- ex op = it->rest;
- if (op.info(info_flags::numeric)) {
- // series * const (special case, faster)
- ex f = power(op, it->coeff);
- acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
- continue;
- } else if (!is_ex_exactly_of_type(op, pseries))
- op = op.series(r, order, options);
- if (!it->coeff.is_equal(_ex1()))
- op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
+ std::vector<int>::const_iterator itd = ldegrees.begin();
+ for (epvector::const_iterator it=itbeg; 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
- acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
+ if (it == itbeg)
+ acc = ex_to<pseries>(op);
+ else
+ acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
}
- return acc;
+
+ return acc.mul_const(ex_to<numeric>(overall_coeff));
}
ex pseries::power_const(const numeric &p, int deg) const
{
// method:
+ // (due to Leonhard Euler)
// let A(x) be this series and for the time being let it start with a
// constant (later we'll generalize):
// A(x) = a_0 + a_1*x + a_2*x^2 + ...
// repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
// then of course x^(p*m) but the recurrence formula still holds.
- if (seq.size()==0) {
- // as a spacial case, handle the empty (zero) series honoring the
+ if (seq.empty()) {
+ // as a special case, handle the empty (zero) series honoring the
// usual power laws such as implemented in power::eval()
if (p.real().is_zero())
- throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
+ throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
else if (p.real().is_negative())
- throw (pole_error("pseries::power_const(): division by zero",1));
+ throw pole_error("pseries::power_const(): division by zero",1);
else
return *this;
}
- const symbol *s = static_cast<symbol *>(var.bp);
- int ldeg = ldegree(*s);
+ const int ldeg = ldegree(var);
+ 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;
+ epv.reserve(1);
+ epv.push_back(expair(Order(_ex1), deg));
+ return (new pseries(relational(var,point), epv))
+ ->setflag(status_flags::dynallocated);
+ }
+
+ // 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(*s, ldeg), p));
- bool all_sums_zero = true;
- for (int i=1; i<deg; ++i) {
- ex sum = _ex0();
+ co.reserve(numcoeff);
+ co.push_back(power(coeff(var, ldeg), p));
+ for (int i=1; i<numcoeff; ++i) {
+ ex sum = _ex0;
for (int j=1; j<=i; ++j) {
- ex c = coeff(*s, j + ldeg);
+ ex c = coeff(var, j + ldeg);
if (is_order_function(c)) {
- co.push_back(Order(_ex1()));
+ co.push_back(Order(_ex1));
break;
} else
sum += (p * j - (i - j)) * co[i - j] * c;
}
- if (!sum.is_zero())
- all_sums_zero = false;
- co.push_back(sum / coeff(*s, ldeg) / numeric(i));
+ 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], numeric(i) + p * ldeg));
+ new_seq.push_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()), numeric(deg) + p * ldeg));
+ if (!higher_order)
+ new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
+
return pseries(relational(var,point), new_seq);
}
/** Return a new pseries object with the powers shifted by deg. */
pseries pseries::shift_exponents(int deg) const
{
- epvector newseq(seq);
- for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
- i->coeff = i->coeff + deg;
+ epvector newseq = seq;
+ epvector::iterator i = newseq.begin(), end = newseq.end();
+ while (i != end) {
+ i->coeff += deg;
+ ++i;
+ }
return pseries(relational(var, point), newseq);
}
* @see ex::series */
ex power::series(const relational & r, int order, unsigned options) const
{
- ex e;
- if (!is_ex_exactly_of_type(basis, pseries)) {
- // Basis is not a series, may there be a singularity?
- bool must_expand_basis = false;
- try {
- basis.subs(r);
- } catch (pole_error) {
- must_expand_basis = true;
- }
-
- // Is the expression of type something^(-int)?
- if (!must_expand_basis && !exponent.info(info_flags::negint))
- return basic::series(r, order, options);
-
- // Is the expression of type 0^something?
- if (!must_expand_basis && !basis.subs(r).is_zero())
- return basic::series(r, order, options);
-
- // Singularity encountered, expand basis into series
- e = basis.series(r, order, options);
+ // If basis is already a series, just power it
+ if (is_exactly_a<pseries>(basis))
+ return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
+
+ // Basis is not a series, may there be a singularity?
+ bool must_expand_basis = false;
+ try {
+ 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)
+ && (!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, 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));
+ else
+ new_seq.push_back(expair(Order(_ex1), exponent));
+ return pseries(r, new_seq);
+ }
+
+ // No, expand basis into series
+
+ numeric numexp;
+ if (is_a<numeric>(exponent)) {
+ numexp = ex_to<numeric>(exponent);
} else {
- // Basis is a series
- e = basis;
+ 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);
- // Power e
- return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
+ ex result;
+ try {
+ result = ex_to<pseries>(e).power_const(numexp, order);
+ } catch (pole_error) {
+ epvector ser;
+ ser.push_back(expair(Order(_ex1), order));
+ result = pseries(r, ser);
+ }
+
+ return result;
}
ex pseries::series(const relational & r, int order, unsigned options) const
{
const ex p = r.rhs();
- GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
- const symbol *s = static_cast<symbol *>(r.lhs().bp);
+ GINAC_ASSERT(is_a<symbol>(r.lhs()));
+ const symbol &s = ex_to<symbol>(r.lhs());
- if (var.is_equal(*s) && point.is_equal(p)) {
- if (order > degree(*s))
+ if (var.is_equal(s) && point.is_equal(p)) {
+ if (order > degree(s))
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();
+ int o = ex_to<numeric>(it->coeff).to_int();
if (o >= order) {
- new_seq.push_back(expair(Order(_ex1()), o));
+ new_seq.push_back(expair(Order(_ex1), o));
break;
}
new_seq.push_back(*it);
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 integrant 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.push_back(
+ expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
+ }
+
+ // Expanding lower boundary
+ ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
+ 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
* @return an expression holding a pseries object */
ex ex::series(const ex & r, int order, unsigned options) const
{
- GINAC_ASSERT(bp!=0);
ex e;
relational rel_;
- if (is_ex_exactly_of_type(r,relational))
- rel_ = ex_to_relational(r);
- else if (is_ex_exactly_of_type(r,symbol))
- rel_ = relational(r,_ex0());
+ if (is_a<relational>(r))
+ rel_ = ex_to<relational>(r);
+ else if (is_a<symbol>(r))
+ rel_ = relational(r,_ex0);
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;
}
-//////////
-// static member variables
-//////////
-
-// protected
-
-unsigned pseries::precedence = 38; // for clarity just below add::precedence
+GINAC_BIND_UNARCHIVER(pseries);
} // namespace GiNaC
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