pseries.cpp

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/*
 *  GiNaC Copyright (C) 1999-2023 Johannes Gutenberg University Mainz, Germany
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */
 
#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 "operators.h"
#include "symbol.h"
#include "integral.h"
#include "archive.h"
#include "utils.h"
 
#include <limits>
#include <numeric>
#include <stdexcept>
 
namespace GiNaC {
 
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() { }
 
 
/*
 *  Other ctors
 */
 
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();
    var = rel_.lhs();
}
 
 
/*
 *  Archiving
 */
 
void pseries::read_archive(const archive_node &n, lst &sym_lst) 
{
    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;
        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);
    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);
}
 
 
// functions overriding virtual functions from base classes
 
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 (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';
 
    auto 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 << 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
                        i->coeff.print(c);
                    c.s << closebrace;
                }
            }
        } 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);
}
 
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
{
    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
    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)
            return cmpval;
        ++it;
        ++o_it;
    }
 
    // so they are equal.
    return 0;
}
 
size_t pseries::nops() const
{
    return seq.size();
}
 
ex pseries::op(size_t i) const
{
    if (i >= seq.size())
        throw (std::out_of_range("op() out of range"));
 
    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);
}
 
int pseries::degree(const ex &s) const
{
    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;
}
 
int pseries::ldegree(const ex &s) const
{
    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;
}
 
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);
}
 
ex pseries::collect(const ex &s, bool distributed) const
{
    return *this;
}
 
ex pseries::eval() const
{
    return hold();
}
 
ex pseries::evalf() const
{
    // Construct a new series with evaluated coefficients
    epvector new_seq;
    new_seq.reserve(seq.size());
    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));
            }
        }
    }
    if (something_changed)
        return dynallocate<pseries>(var==point, std::move(newseq));
    else
        return *this;
}
 
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 (m.find(var) != m.end())
        return convert_to_poly(true).subs(m, options);
    
    // Otherwise construct a new series with substituted coefficients and
    // expansion point
    epvector newseq;
    newseq.reserve(seq.size());
    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));
}
 
ex pseries::expand(unsigned options) const
{
    epvector newseq;
    for (auto & it : seq) {
        ex restexp = it.rest.expand();
        if (!restexp.is_zero())
            newseq.emplace_back(expair(restexp, it.coeff));
    }
    return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
}
 
ex pseries::derivative(const symbol & s) const
{
    epvector new_seq;
 
    if (s == var) {
        
        // FIXME: coeff might depend on var
        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;
                if (!c.is_zero())
                    new_seq.emplace_back(expair(c, it.coeff - 1));
            }
        }
 
    } else {
 
        for (auto & it : seq) {
            if (is_order_function(it.rest)) {
                new_seq.push_back(it);
            } else {
                ex c = it.rest.diff(s);
                if (!c.is_zero())
                    new_seq.emplace_back(expair(c, it.coeff));
            }
        }
    }
 
    return pseries(relational(var,point), std::move(new_seq));
}
 
ex pseries::convert_to_poly(bool no_order) const
{
    ex e;
    for (auto & it : seq) {
        if (is_order_function(it.rest)) {
            if (!no_order)
                e += Order(pow(var - point, it.coeff));
        } else
            e += it.rest * pow(var - point, it.coeff);
    }
    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;
}
 
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, subs_options::no_pattern);
 
    if (!coeff.is_zero()) {
        seq.emplace_back(expair(coeff, _ex0));
    }
 
    int n;
    for (n=1; n<order; ++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, std::move(seq));
 
        coeff = deriv.subs(r, subs_options::no_pattern);
        if (!coeff.is_zero())
            seq.emplace_back(expair(fac * coeff, n));
    }
    
    // Higher-order terms, if present
    deriv = deriv.diff(s);
    if (!deriv.expand().is_zero())
        seq.emplace_back(expair(Order(_ex1), n));
    return pseries(r, std::move(seq));
}
 
 
ex symbol::series(const relational & r, int order, unsigned options) const
{
    epvector seq;
    const ex point = r.rhs();
    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.emplace_back(expair(point, _ex0));
        if (order > 1)
            seq.emplace_back(expair(_ex1, _ex1));
        else
            seq.emplace_back(expair(Order(_ex1), numeric(order)));
    } else
        seq.emplace_back(expair(*this, _ex0));
    return pseries(r, std::move(seq));
}
 
 
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 { expair(Order(_ex1), _ex0) };
        return pseries(relational(var,point), std::move(nul));
    }
    
    // Series addition
    epvector new_seq;
    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) {
            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.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.emplace_back(expair(sum, numeric(pow_a)));
                ++a;
                ++b;
            }
        }
    }
    return pseries(relational(var,point), std::move(new_seq));
}
 
 
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
    for (auto & it : seq) {
        ex op;
        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));
        
        // Series addition
        acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
    }
    return acc;
}
 
 
ex pseries::mul_const(const numeric &other) const
{
    epvector new_seq;
    new_seq.reserve(seq.size());
    
    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);
    }
    return pseries(relational(var,point), std::move(new_seq));
}
 
 
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 { 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;
    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 = 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;
    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) {
            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.emplace_back(expair(co, numeric(cdeg)));
    }
    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));
}
 
 
ex mul::series(const relational & r, int order, unsigned options) const
{
    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
    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 == 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));
}
 
 
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)^p is
    // then of course a_0^p*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 base_ldeg = ldegree(var);
    if (!(p*base_ldeg).is_integer())
        throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
    int new_ldeg = (p*base_ldeg).to_int();
 
    const int base_deg = degree(var);
    int new_deg = deg;
    if (p.is_pos_integer()) {
        // No need to compute beyond p*base_deg.
        new_deg = std::min((p*base_deg).to_int(), deg);
    }
 
    // adjust number of coefficients
    int numcoeff = new_deg - new_ldeg;
    if (new_deg < deg)
        numcoeff += 1;
 
    if (numcoeff <= 0) {
        return dynallocate<pseries>(relational(var, point),
                                    epvector{{Order(_ex1), deg}});
    }
    
    // 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(numcoeff);
    co.push_back(pow(coeff(var, base_ldeg), p));
    for (int i=1; i<numcoeff; ++i) {
        ex sum = _ex0;
        for (int j=1; j<=i; ++j) {
            ex c = coeff(var, j + base_ldeg);
            if (is_order_function(c)) {
                co.push_back(Order(_ex1));
                break;
            } else
                sum += (p * j - (i - j)) * co[i - j] * c;
        }
        co.push_back(sum / coeff(var, base_ldeg) / i);
    }
    
    // Construct new series (of non-zero coefficients)
    epvector new_seq;
    bool higher_order = false;
    for (int i=0; i<numcoeff; ++i) {
        if (!co[i].is_zero()) {
            new_seq.emplace_back(expair(co[i], new_ldeg + i));
        }
        if (is_order_function(co[i])) {
            higher_order = true;
            break;
        }
    }
    if (!higher_order && new_deg == deg) {
        new_seq.emplace_back(expair{Order(_ex1), new_deg});
    }
 
    return pseries(relational(var,point), std::move(new_seq));
}
 
 
pseries pseries::shift_exponents(int deg) const
{
    epvector newseq = seq;
    for (auto & it : newseq)
        it.coeff += deg;
    return pseries(relational(var, point), std::move(newseq));
}
 
 
ex power::series(const relational & r, int order, unsigned options) const
{
    // 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.emplace_back(expair(_ex1, exponent));
        else
            new_seq.emplace_back(expair(Order(_ex1), exponent));
        return pseries(r, std::move(new_seq));
    }
 
    // No, expand basis into series
 
    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");
    int extra_terms = (real_ldegree*(1-numexp)).to_int();
    ex e = basis.series(r, order + std::max(0, extra_terms), 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;
}
 
 
ex pseries::series(const relational & r, int order, unsigned options) const
{
    const ex p = r.rhs();
    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))
            return *this;
        else {
            epvector new_seq;
            for (auto & it : seq) {
                int o = ex_to<numeric>(it.coeff).to_int();
                if (o >= order) {
                    new_seq.emplace_back(expair(Order(_ex1), o));
                    break;
                }
                new_seq.push_back(it);
            }
            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;
}
 
 
ex ex::series(const ex & r, int order, unsigned options) const
{
    ex e;
    relational rel_;
    
    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"));
    
    e = bp->series(rel_, order, options);
    return e;
}
 
GINAC_BIND_UNARCHIVER(pseries);
 
} // namespace GiNaC