GiNaC/ginac/pseries.cpp

/** @file pseries.cpp
 *
 *  Implementation of class for extended truncated power series and
 *  methods for series expansion. */

/*
 *  GiNaC Copyright (C) 1999-2002 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  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
 */

#include <iostream>
#include <stdexcept>

#include "pseries.h"
#include "add.h"
#include "inifcns.h" // for Order function
#include "lst.h"
#include "mul.h"
#include "power.h"
#include "relational.h"
#include "symbol.h"
#include "print.h"
#include "archive.h"
#include "utils.h"

namespace GiNaC {

GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)


/*
 *  Default ctor, dtor, copy ctor, assignment operator and helpers
 */

pseries::pseries() : inherited(TINFO_pseries) { }

void pseries::copy(const pseries &other)
{
        inherited::copy(other);
        seq = other.seq;
        var = other.var;
        point = other.point;
}

DEFAULT_DESTROY(pseries)


/*
 *  Other ctors
 */

/** 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,
 *  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_)
{
        GINAC_ASSERT(is_exactly_a<relational>(rel_));
        GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
        point = rel_.rhs();
        var = rel_.lhs();
}


/*
 *  Archiving
 */

pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
{
        for (unsigned int i=0; true; ++i) {
                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("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;
        }
        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
{
        if (is_a<print_tree>(c)) {

                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;
                unsigned num = seq.size();
                for (unsigned 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);

        } else if (is_a<print_python_repr>(c)) {
                c.s << class_name() << "(relational(";
                var.print(c);
                c.s << ',';
                point.print(c);
                c.s << "),[";
                unsigned num = seq.size();
                for (unsigned 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 {

                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;
                                        } 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);
                                                }
                                        }
                                }
                        } else
                                Order(power(var-point,i->coeff)).print(c);
                        ++i;
                }

                if (precedence() <= level)
                        c.s << ")";
        }
}

int pseries::compare_same_type(const basic & other) const
{
        GINAC_ASSERT(is_a<pseries>(other));
        const pseries &o = static_cast<const pseries &>(other);
        
        // first compare the lengths of the series...
        if (seq.size()>o.seq.size())
                return 1;
        if (seq.size()<o.seq.size())
                return -1;
        
        // ...then the expansion point...
        int cmpval = var.compare(o.var);
        if (cmpval)
                return cmpval;
        cmpval = point.compare(o.point);
        if (cmpval)
                return cmpval;
        
        // ...and if that failed the individual elements
        epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
        while (it!=seq.end() && o_it!=o.seq.end()) {
                cmpval = it->compare(*o_it);
                if (cmpval)
                        return cmpval;
                ++it;
                ++o_it;
        }

        // so they are equal.
        return 0;
}

/** Return the number of operands including a possible order term. */
unsigned pseries::nops(void) const
{
        return seq.size();
}

/** Return the ith term in the series when represented as a sum. */
ex pseries::op(int i) const
{
        if (i < 0 || unsigned(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"));
}

/** 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 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;
        }
}

/** Return degree of lowest power of the series.  This is usually the exponent
 *  of the leading term.  If s is not the expansion variable of the series, the
 *  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 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;
        }
}

/** Return coefficient of degree n in power series if s is the expansion
 *  variable.  If the expansion point is nonzero, by definition the n=1
 *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 *  the expansion took place in the s in the first place).
 *  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 ex &s, int n) const
{
        if (var.is_equal(s)) {
                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_exactly_a<numeric>(seq[mid].coeff));
                        int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
                        switch (cmp) {
                                case -1:
                                        lo = mid + 1;
                                        break;
                                case 0:
                                        return seq[mid].rest;
                                case 1:
                                        hi = mid - 1;
                                        break;
                                default:
                                        throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
                        }
                }
                return _ex0;
        } else
                return convert_to_poly().coeff(s, n);
}

/** Does nothing. */
ex pseries::collect(const ex &s, bool distributed) const
{
        return *this;
}

/** Perform coefficient-wise automatic term rewriting rules in this class. */
ex pseries::eval(int level) const
{
        if (level == 1)
                return this->hold();
        
        if (level == -max_recursion_level)
                throw (std::runtime_error("pseries::eval(): 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.eval(level-1), it->coeff));
                ++it;
        }
        return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
}

/** Evaluate coefficients numerically. */
ex pseries::evalf(int level) 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;
        }
        return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
}

ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) 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, no_pattern);
        
        // Otherwise construct a new series with substituted coefficients and
        // 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(ls, lr, no_pattern), it->coeff));
                ++it;
        }
        return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), 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;
        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 | (options == 0 ? status_flags::expanded : 0));
}

/** Implementation of ex::diff() for a power series.  It treats the series as a
 *  polynomial.
 *  @see ex::diff */
ex pseries::derivative(const symbol & s) const
{
        if (s == var) {
                epvector new_seq;
                epvector::const_iterator it = seq.begin(), itend = seq.end();
                
                // 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));
                        } else {
                                ex c = it->rest * it->coeff;
                                if (!c.is_zero())
                                        new_seq.push_back(expair(c, it->coeff - 1));
                        }
                        ++it;
                }
                return pseries(relational(var,point), new_seq);
        } else {
                return *this;
        }
}

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)) {
                        if (!no_order)
                                e += Order(power(var - point, it->coeff));
                } else
                        e += it->rest * power(var - point, it->coeff);
                ++it;
        }
        return e;
}

bool pseries::is_terminating(void) const
{
        return seq.empty() || !is_order_function((seq.end()-1)->rest);
}


/*
 *  Implementations of series expansion
 */

/** Default implementation of ex::series(). This performs Taylor expansion.
 *  @see ex::series */
ex basic::series(const relational & r, int order, unsigned options) const
{
        epvector seq;
        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));
        
        int n;
        for (n=1; n<order; ++n) {
                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);
                if (!coeff.is_zero())
                        seq.push_back(expair(fac.inverse() * 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);
}


/** Implementation of ex::series() for symbols.
 *  @see ex::series */
ex symbol::series(const relational & r, int order, unsigned options) const
{
        epvector seq;
        const ex point = r.rhs();
        GINAC_ASSERT(is_exactly_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));
                if (order > 1)
                        seq.push_back(expair(_ex1, _ex1));
                else
                        seq.push_back(expair(Order(_ex1), numeric(order)));
        } else
                seq.push_back(expair(*this, _ex0));
        return pseries(r, seq);
}


/** Add one series object to another, producing a pseries object that
 *  represents the sum.
 *
 *  @param other  pseries object to add with
 *  @return the sum as a pseries */
ex pseries::add_series(const pseries &other) const
{
        // 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);
        }
        
        // 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;
        for (;;) {
                // If a is empty, fill up with elements from b and stop
                if (a == a_end) {
                        while (b != b_end) {
                                new_seq.push_back(*b);
                                ++b;
                        }
                        break;
                } else
                        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) {
                        while (a != a_end) {
                                new_seq.push_back(*a);
                                ++a;
                        }
                        break;
                } else
                        pow_b = ex_to<numeric>((*b).coeff).to_int();
                
                // a and b are non-empty, compare powers
                if (pow_a < pow_b) {
                        // a has lesser power, get coefficient from a
                        new_seq.push_back(*a);
                        if (is_order_function((*a).rest))
                                break;
                        ++a;
                } else if (pow_b < pow_a) {
                        // b has lesser power, get coefficient from b
                        new_seq.push_back(*b);
                        if (is_order_function((*b).rest))
                                break;
                        ++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));
                                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)));
                                ++a;
                                ++b;
                        }
                }
        }
        return pseries(relational(var,point), new_seq);
}


/** Implementation of ex::series() for sums. This performs series addition when
 *  adding pseries objects.
 *  @see ex::series */
ex add::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);
        
        // Add remaining terms
        epvector::const_iterator it = seq.begin();
        epvector::const_iterator itend = seq.end();
        for (; it!=itend; ++it) {
                ex op;
                if (is_ex_exactly_of_type(it->rest, pseries))
                        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));
                
                // Series addition
                acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
        }
        return acc;
}


/** Multiply a pseries object with a numeric constant, producing a pseries
 *  object that represents the product.
 *
 *  @param other  constant to multiply with
 *  @return the product as a pseries */
ex pseries::mul_const(const numeric &other) const
{
        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));
                else
                        new_seq.push_back(*it);
                ++it;
        }
        return pseries(relational(var,point), new_seq);
}


/** Multiply one pseries object to another, producing a pseries object that
 *  represents the product.
 *
 *  @param other  pseries object to multiply with
 *  @return the product as a pseries */
ex pseries::mul_series(const pseries &other) const
{
        // 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);
        }
        
        // 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;
        int cdeg_max = a_max + b_max;
        
        int higher_order_a = INT_MAX;
        int higher_order_b = 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);
        if (cdeg_max >= higher_order_c)
                cdeg_max = higher_order_c - 1;
        
        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;
                }
                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);
}


/** Implementation of ex::series() for product. This performs series
 *  multiplication when multiplying series.
 *  @see ex::series */
ex mul::series(const relational & r, int order, unsigned options) const
{
        pseries acc; // Series accumulator

        // 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);

                // Series multiplication
                if (it==itbeg)
                        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));
}


/** Compute the p-th power of a series.
 *
 *  @param p  power to compute
 *  @param deg  truncation order of series calculation */
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 + ...
        // We want to compute
        //     C(x) = A(x)^p
        //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
        // Taking the derivative on both sides and multiplying with A(x) one
        // immediately arrives at
        //     C'(x)*A(x) = p*C(x)*A'(x)
        // Multiplying this out and comparing coefficients we get the recurrence
        // formula
        //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
        //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
        // which can easily be solved given the starting value c_0 = (a_0)^p.
        // For the more general case where the leading coefficient of A(x) is not
        // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
        // 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.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");
                else if (p.real().is_negative())
                        throw pole_error("pseries::power_const(): division by zero",1);
                else
                        return *this;
        }
        
        const int ldeg = ldegree(var);
        if (!(p*ldeg).is_integer())
                throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");

        // 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) {
                ex sum = _ex0;
                for (int j=1; j<=i; ++j) {
                        ex c = coeff(var, j + ldeg);
                        if (is_order_function(c)) {
                                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(var, ldeg) / i);
        }
        
        // Construct new series (of non-zero coefficients)
        epvector new_seq;
        bool higher_order = false;
        for (int i=0; i<deg; ++i) {
                if (!co[i].is_zero())
                        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), p * ldeg + deg));
        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;
        epvector::iterator i = newseq.begin(), end  = newseq.end();
        while (i != end) {
                i->coeff += deg;
                ++i;
        }
        return pseries(relational(var, point), newseq);
}


/** Implementation of ex::series() for powers. This performs Laurent expansion
 *  of reciprocals of series at singularities.
 *  @see ex::series */
ex power::series(const relational & r, int order, unsigned options) const
{
        // If basis is already a series, just power it
        if (is_ex_exactly_of_type(basis, pseries))
                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);
        } 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, 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
        ex e = basis.series(r, order, options);
        return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
}


/** Re-expansion of a pseries object. */
ex pseries::series(const relational & r, int order, unsigned options) const
{
        const ex p = r.rhs();
        GINAC_ASSERT(is_exactly_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))
                        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();
                                if (o >= order) {
                                        new_seq.push_back(expair(Order(_ex1), o));
                                        break;
                                }
                                new_seq.push_back(*it);
                                ++it;
                        }
                        return pseries(r, new_seq);
                }
        } else
                return convert_to_poly().series(r, order, options);
}


/** Compute the truncated series expansion of an expression.
 *  This function returns an expression containing an object of class pseries 
 *  to represent the series. If the series does not terminate within the given
 *  truncation order, the last term of the series will be an order term.
 *
 *  @param r  expansion relation, lhs holds variable and rhs holds point
 *  @param order  truncation order of series calculations
 *  @param options  of class series_options
 *  @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);
        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() + ")"));
        }
        return e;
}

} // namespace GiNaC
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