Gt_helpers.cpp

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/*
 *  GiNaC Copyright (C) 1999-2026 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 "add.h"
#include "mul.h"
#include "power.h"
#include "lst.h"
#include "operators.h"
#include "relational.h"
#include "inifcns.h"
#include "function.h"
#include "Gt.h"
#include "Gt_helpers.h"
 
using namespace GiNaC;
 
namespace GiNaC {
namespace Gt_detail {

//TransformExpressionWithCache//
template<typename Type>
template<typename T>
typename std::enable_if<std::is_base_of<basic, T>::value, bool>::type
TransformExpressionWithCache<Type>::check_type(const ex& expr)
{
    return is_exactly_a<Type>(expr);
}
template<typename Type>
template<typename T>
typename std::enable_if<!std::is_base_of<basic, T>::value, bool>::type
TransformExpressionWithCache<Type>::check_type(const ex& expr)
{
    return is_the_function<Type>(expr);
}
 
template<typename Type>
void TransformExpressionWithCache<Type>::print_cache() const
{
    std::cout << "Cache" << std::endl;
    for (const auto& n : cache)
        std::cout << "  " << n.first << " -> " << n.second << std::endl;
}
 
template<typename Type>
ex TransformExpressionWithCache<Type>::impl(const ex& expr)
{
    // If the input is of the desired type, apply the transformation, update cache
    if (check_type(expr)) {
        const auto pos = cache.find(expr);
        if (pos != cache.end())
            return pos->second;
        return cache.emplace(expr, func_obj(expr)).first->second;
    }
 
    // If the input is an addition, multiplication, exponentiation or list, run recurisvely over elements
    if (is_exactly_a<add>(expr) || is_exactly_a<mul>(expr)) {
        std::vector<ex> terms;
        terms.reserve(expr.nops());
        for (size_t i = 0; i < expr.nops(); i++)
            terms.emplace_back(impl(expr.op(i)));
        if (is_exactly_a<add>(expr))
            return add(terms);
        else
            return mul(terms);
    }
    if (is_exactly_a<lst>(expr)) {
        lst result;
        for (size_t i = 0; i < expr.nops(); i++)
            result.append(impl(expr.op(i)));
        return result;
    }
    if (is_exactly_a<power>(expr))
        return power(impl(expr.op(0)), expr.op(1));
 
    // Otherwise, use default transformation function (e.g. for number, symbols, unrelated functions)
    return func_default(expr);
}
 
// Instantiate template for elliptic and regular MPLs
template class TransformExpressionWithCache<Gt>;
template class TransformExpressionWithCache<G2_SERIAL>;
 
 

// Path deconcatenation tools //
std::vector<std::vector<int>> integer_partition(const int n, const int m)
{
    std::vector<std::vector<int>> result;
    std::vector<int> hlp(m, 0);
    int k = 1;
    hlp[1] = n;
    while (k != 0) {
        int x = hlp[k-1] + 1;
        int y = hlp[k] - 1;
        k -= 1;
        while (x <= y && k < m-1) {
            hlp[k] = x;
            y -= x;
            k += 1;
        }
        hlp[k] = x + y;
 
        std::vector<int> decomposition(m, 0);
        for (size_t i = 0; i < k+1; i++)
            decomposition[i] = hlp[i];
        std::sort(decomposition.begin(), decomposition.end());
        do {
            result.emplace_back(decomposition);
        } while (next_permutation(decomposition.begin(), decomposition.end()));
    }
 
    return result;
}
 
template<typename Kernel>
ex deconcatenate_path(
    const std::vector<Kernel>& args,
    const std::vector<ex>& endpoints,
    const std::function<ex(std::vector<Kernel>& new_args, const ex& start, const ex& end)>& construct
)
{
    // Assemble partition information into iterated integrals
    const size_t n_segments = endpoints.size()-1;
    const std::vector<std::vector<int>> partitions = integer_partition(args.size(), n_segments);
    std::vector<ex> terms;
    terms.reserve(partitions.size());
    for (const std::vector<int>& partition : partitions) { // loop over all partitions, each gives one term
        size_t kernel_count = 0;
        std::vector<ex> factors;
        factors.reserve(n_segments);
        for (size_t i = 0; i < n_segments; i++) { // loop over paths, each gives zero or one integral
            if (partition[i] == 0)
                continue;
            // copy partition[i] consecutive kernels
            std::vector<Kernel> new_args(args.begin()+kernel_count, args.begin()+kernel_count+partition[i]);
            kernel_count += partition[i];
            // Construct the shifted iterated integral, shift kernel arguments if necessary
            factors.emplace_back(construct(new_args, endpoints[n_segments-1-i], endpoints[n_segments-i]));
        }
        terms.emplace_back(mul(factors));
    }
    return add(terms);
}
// Instantiate for regular and elliptic MPL kernels
template ex deconcatenate_path<ex>(
    const std::vector<ex>& args,
    const std::vector<ex>& endpoints,
    const std::function<ex(std::vector<ex>& new_args, const ex& start, const ex& end)>& construct
);
template ex deconcatenate_path<Gt::kernel>(
    const std::vector<Gt::kernel>& args,
    const std::vector<ex>& endpoints,
    const std::function<ex(std::vector<Gt::kernel>& new_args, const ex& start, const ex& end)>& construct
);
 
 

//     pathintegral_term      //
 
pathintegral_term::pathintegral_term(const pathintegral_term& a, const pathintegral_term& b, int power_offset) // construct as product
    : prefactor(a.prefactor * b.prefactor),
      power(a.power + b.power + power_offset),
      polylog(a.polylog.empty() ? b.polylog : a.polylog),
      denom(a.denom.is_zero() ? b.denom : a.denom)
{
    if ((!a.polylog.empty() && !b.polylog.empty()) || (!a.denom.is_zero() && !b.denom.is_zero()))
        throw std::logic_error("Gt: Cannot multiply these terms");
}
 
void pathintegral_term::integrate(const ex& start, pathintegral_term_list& integrand, pathintegral_term_list& result) const
{
    if (power != 0 && !denom.is_zero()) { // Power and denominator -> partial fractioning
        for (int i = std::min(0, power); i < std::max(0, power); i++)
            integrand.add((power > 0 ? 1 : -1) * prefactor * pow(denom, power-i-1), i, polylog);
        integrand.add(prefactor * pow(denom, power), 0, polylog, denom);
        return;
    }
 
    if (polylog.empty() && denom.is_zero()) { // only power of integration variable
        if (power == -1)
            result.add(prefactor, 0, std::vector<ex>(1,ex{0}));
        else {
            result.add( prefactor / (power + 1), power+1);
            result.add(-prefactor / (power + 1)*pow(start,power+1), 0);
        }
        return;
    }
 
    if (!polylog.empty() && denom.is_zero()) { // (power and) polylog
        std::vector<ex> args = polylog;
        const ex& first = polylog[0];
        if (power == -1) {
            args.insert(args.begin(), ex{0});
            result.add(prefactor, 0, args);
        }
        else {
            result.add(prefactor / (power + 1), power+1, args);
            args.erase(args.begin());
            // If first==0 the denominator 1/([int.var.] - first) turns into [int.var]^(-1)
            integrand.add(-prefactor / (power + 1), power+1 - first.is_zero(), args, first);
        }
        return;
    }
 
    if (polylog.empty() and power == 0) { // only denominator: 1/([int.var.] - C)
        result.add(prefactor, 0, std::vector<ex>(1, denom));
        return;
    }
 
    if (!polylog.empty() && power==0 && !denom.is_zero()) { // polylog and denominator
        std::vector<ex> args = polylog;
        args.insert(args.begin(), denom);
        result.add(prefactor, 0, args);
        return;
    }
 
    throw std::logic_error("Gt: Invalid integration");
}
 
ex pathintegral_term::evaluate(const ex& upper_bound, const std::vector<ex>& path) const
{
    ex result = prefactor * pow(upper_bound, power);
    if (!denom.is_zero())
        result /= denom - upper_bound;
    if (!polylog.empty())
        result *= G_path(polylog, path);
    return result;
}
 
ex pathintegral_term::G_path(const std::vector<ex>& args, const std::vector<ex>& path)
{
    // Multiple segments -> deconcatenate path
    if (path.size() > 2) {
        return deconcatenate_path<ex>(args, path, [](std::vector<ex>& new_args, const ex& start, const ex& end) -> ex {
            return G_path(new_args,start,end);
        });
    }
    return G_path(args, path[0], path[1]);
}
ex pathintegral_term::G_path(const std::vector<ex>& args, const ex& start, const ex& end)
{
    // Optimisation in case all arguments are the same
    bool all_args_same = args.size() >= 2;
    for (size_t i = 1; i < args.size() && all_args_same; i++)
        all_args_same = (args[i] == args[0]);
    if (all_args_same)
        return (pow(G((args[0] - start)/(end - start),1).hold(), args.size()) / factorial(args.size()));
 
    // Shift arguments, such that integration path is 0 -> 1
    lst new_args;
    for (const ex& arg : args)
        new_args.append((arg - start)/(end - start));
    return G(new_args,1).hold();
}
 
std::ostream & operator<<(std::ostream & os, const pathintegral_term & term)
{
    os << "(" << term.prefactor << ")";
    if (term.power)
        os << "*[int.var.]^" << term.power;
    if (!term.polylog.empty()) {
        os << "*G({";
        for (const ex& kern : term.polylog)
            os << kern << ",";
        os << "},[int.var.],[path])";
    }
    if (!term.denom.is_zero())
        os << "/([int.var.]-(" << term.denom << "))";
    return os;
}
 
// Terms are ordered such that more complicated terms (for integration) come first
bool operator<(const pathintegral_term & a, const pathintegral_term & b)
{
    // 1) Sort decending by polylog order (integration-by-parts yields shorter polylogs)
    if (a.polylog.size() > b.polylog.size()) return true;
    if (a.polylog.size() < b.polylog.size()) return false;
    // 2) Terms with denominator in beginning (partial fractioning yields terms without denominators)
    if (!a.denom.is_zero() && b.denom.is_zero()) return true;
    if (a.denom.is_zero() && !b.denom.is_zero()) return false;
    // 3) Sort ascending by powers (integration-by-parts yields terms with higher power (if polylog present))
    if (a.power < b.power) return true;
    if (a.power > b.power) return false;
 
    // Arbitrarily decide order based on remaining info
    // Sort by denominator
    int cmpval = a.denom.compare(b.denom);
    if (cmpval < 0) return true;
    if (cmpval > 0) return false;
    // Sort by Polylog
    for (size_t i = 0; i < a.polylog.size(); i++) {
        cmpval = a.polylog[i].compare(b.polylog[i]);
        if (cmpval < 0) return true;
        if (cmpval > 0) return false;
    }
    // Terms which differ only in their prefactor are considered identical
    // Calling find() on a container thus yields a term which is equivalent up to prefactor
    // This means we can add a term to a container by calling find() and summing the prefactor
    return false;
}
 
void pathintegral_term_list::add(pathintegral_term&& term)
{
    std::set<pathintegral_term>::iterator pos = terms.find(term);
    if (pos == end())
        terms.insert(std::move(term));
    else {
        pos->prefactor += term.prefactor;
        if (pos->prefactor.is_zero())
            terms.erase(pos);
    }
}
 
void pathintegral_term_list::add(const pathintegral_term_list& other)
{
    // Cannot use terms.insert here, as we need to properly add prefacttors
    for (const pathintegral_term& term : other)
        add(pathintegral_term(term));
}
 
pathintegral_term pathintegral_term_list::pop()
{
    pathintegral_term term = *begin();
    terms.erase(begin());
    return term;
}
 
} // namespace Gt_detail
} // namespace GiNaC