nh-rep/code/pre_processing/ParametricSample/Mathematics/Integration.h

// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2021
// Distributed under the Boost Software License, Version 1.0.
// https://www.boost.org/LICENSE_1_0.txt
// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
// Version: 4.0.2019.08.13

#pragma once

#include <Mathematics/RootsPolynomial.h>
#include <array>
#include <functional>

namespace gte
{
    template <typename Real>
    class Integration
    {
    public:
        // A simple algorithm, but slow to converge as the number of samples
        // is increased.  The 'numSamples' needs to be two or larger.
        static Real TrapezoidRule(int numSamples, Real a, Real b,
            std::function<Real(Real)> const& integrand)
        {
            Real h = (b - a) / (Real)(numSamples - 1);
            Real result = (Real)0.5 * (integrand(a) + integrand(b));
            for (int i = 1; i <= numSamples - 2; ++i)
            {
                result += integrand(a + i * h);
            }
            result *= h;
            return result;
        }

        // The trapezoid rule is used to generate initial estimates, but then
        // Richardson extrapolation is used to improve the estimates.  This is
        // preferred over TrapezoidRule.  The 'order' must be positive.
        static Real Romberg(int order, Real a, Real b,
            std::function<Real(Real)> const& integrand)
        {
            Real const half = (Real)0.5;
            std::vector<std::array<Real, 2>> rom(order);
            Real h = b - a;
            rom[0][0] = half * h * (integrand(a) + integrand(b));
            for (int i0 = 2, p0 = 1; i0 <= order; ++i0, p0 *= 2, h *= half)
            {
                // Approximations via the trapezoid rule.
                Real sum = (Real)0;
                int i1;
                for (i1 = 1; i1 <= p0; ++i1)
                {
                    sum += integrand(a + h * (i1 - half));
                }

                // Richardson extrapolation.
                rom[0][1] = half * (rom[0][0] + h * sum);
                for (int i2 = 1, p2 = 4; i2 < i0; ++i2, p2 *= 4)
                {
                    rom[i2][1] = (p2 * rom[i2 - 1][1] - rom[i2 - 1][0]) / (p2 - 1);
                }

                for (i1 = 0; i1 < i0; ++i1)
                {
                    rom[i1][0] = rom[i1][1];
                }
            }

            Real result = rom[order - 1][0];
            return result;
        }

        // Gaussian quadrature estimates the integral of a function f(x)
        // defined on [-1,1] using
        //   integral_{-1}^{1} f(t) dt = sum_{i=0}^{n-1} c[i]*f(r[i])
        // where r[i] are the roots to the Legendre polynomial p(t) of degree
        // n and
        //   c[i] = integral_{-1}^{1} prod_{j=0,j!=i} (t-r[j]/(r[i]-r[j]) dt
        // To integrate over [a,b], a transformation to [-1,1] is applied
        // internally:  x - ((b-a)*t + (b+a))/2.  The Legendre polynomials are
        // generated by
        //   P[0](x) = 1, P[1](x) = x,
        //   P[k](x) = ((2*k-1)*x*P[k-1](x) - (k-1)*P[k-2](x))/k, k >= 2
        // Implementing the polynomial generation is simple, and computing the
        // roots requires a numerical method for finding polynomial roots.
        // The challenging task is to develop an efficient algorithm for
        // computing the coefficients c[i] for a specified degree.  The
        // 'degree' must be two or larger.

        static void ComputeQuadratureInfo(int degree, std::vector<Real>& roots,
            std::vector<Real>& coefficients)
        {
            Real const zero = (Real)0;
            Real const one = (Real)1;
            Real const half = (Real)0.5;

            std::vector<std::vector<Real>> poly(degree + 1);

            poly[0].resize(1);
            poly[0][0] = one;

            poly[1].resize(2);
            poly[1][0] = zero;
            poly[1][1] = one;

            for (int n = 2; n <= degree; ++n)
            {
                Real mult0 = (Real)(n - 1) / (Real)n;
                Real mult1 = (Real)(2 * n - 1) / (Real)n;

                poly[n].resize(n + 1);
                poly[n][0] = -mult0 * poly[n - 2][0];
                for (int i = 1; i <= n - 2; ++i)
                {
                    poly[n][i] = mult1 * poly[n - 1][i - 1] - mult0 * poly[n - 2][i];
                }
                poly[n][n - 1] = mult1 * poly[n - 1][n - 2];
                poly[n][n] = mult1 * poly[n - 1][n - 1];
            }

            roots.resize(degree);
            RootsPolynomial<Real>::Find(degree, &poly[degree][0], 2048, &roots[0]);

            coefficients.resize(roots.size());
            size_t n = roots.size() - 1;
            std::vector<Real> subroots(n);
            for (size_t i = 0; i < roots.size(); ++i)
            {
                Real denominator = (Real)1;
                for (size_t j = 0, k = 0; j < roots.size(); ++j)
                {
                    if (j != i)
                    {
                        subroots[k++] = roots[j];
                        denominator *= roots[i] - roots[j];
                    }
                }

                std::array<Real, 2> delta =
                {
                    -one - subroots.back(),
                    +one - subroots.back()
                };

                std::vector<std::array<Real, 2>> weights(n);
                weights[0][0] = half * delta[0] * delta[0];
                weights[0][1] = half * delta[1] * delta[1];
                for (size_t k = 1; k < n; ++k)
                {
                    Real dk = (Real)k;
                    Real mult = -dk / (dk + (Real)2);
                    weights[k][0] = mult * delta[0] * weights[k - 1][0];
                    weights[k][1] = mult * delta[1] * weights[k - 1][1];
                }

                struct Info
                {
                    int numBits;
                    std::array<Real, 2> product;
                };

                int numElements = (1 << static_cast<unsigned int>(n - 1));
                std::vector<Info> info(numElements);
                info[0].numBits = 0;
                info[0].product[0] = one;
                info[0].product[1] = one;
                for (int ipow = 1, r = 0; ipow < numElements; ipow <<= 1, ++r)
                {
                    info[ipow].numBits = 1;
                    info[ipow].product[0] = -one - subroots[r];
                    info[ipow].product[1] = +one - subroots[r];
                    for (int m = 1, j = ipow + 1; m < ipow; ++m, ++j)
                    {
                        info[j].numBits = info[m].numBits + 1;
                        info[j].product[0] =
                            info[ipow].product[0] * info[m].product[0];
                        info[j].product[1] =
                            info[ipow].product[1] * info[m].product[1];
                    }
                }

                std::vector<std::array<Real, 2>> sum(n);
                std::array<Real, 2> zero2 = { zero, zero };
                std::fill(sum.begin(), sum.end(), zero2);
                for (size_t k = 0; k < info.size(); ++k)
                {
                    sum[info[k].numBits][0] += info[k].product[0];
                    sum[info[k].numBits][1] += info[k].product[1];
                }

                std::array<Real, 2> total = zero2;
                for (size_t k = 0; k < n; ++k)
                {
                    total[0] += weights[n - 1 - k][0] * sum[k][0];
                    total[1] += weights[n - 1 - k][1] * sum[k][1];
                }

                coefficients[i] = (total[1] - total[0]) / denominator;
            }
        }

        static Real GaussianQuadrature(std::vector<Real> const& roots,
            std::vector<Real>const& coefficients, Real a, Real b,
            std::function<Real(Real)> const& integrand)
        {
            Real const half = (Real)0.5;
            Real radius = half * (b - a);
            Real center = half * (b + a);
            Real result = (Real)0;
            for (size_t i = 0; i < roots.size(); ++i)
            {
                result += coefficients[i] * integrand(radius * roots[i] + center);
            }
            result *= radius;
            return result;
        }
    };
}