// <copyright file="GaussKronrodRule.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2019 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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// This file uses code from the Boost Project.
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// Copyright John Maddock 2017.
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// Copyright Nick Thompson 2017.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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// https://github.com/boostorg/math/blob/develop/include/boost/math/quadrature/gauss_kronrod.hpp
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using IStation.Numerics.Integration.GaussRule;
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using System;
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using System.Numerics;
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namespace IStation.Numerics.Integration
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{
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public class GaussKronrodRule
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{
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private readonly GaussPointPair gaussKronrodPoint;
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/// <summary>
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/// Getter for the order.
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/// </summary>
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public int Order => gaussKronrodPoint.Order;
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/// <summary>
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/// Getter that returns a clone of the array containing the Kronrod abscissas.
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/// </summary>
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public double[] KronrodAbscissas => gaussKronrodPoint.Abscissas.Clone() as double[];
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/// <summary>
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/// Getter that returns a clone of the array containing the Kronrod weights.
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/// </summary>
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public double[] KronrodWeights => gaussKronrodPoint.Weights.Clone() as double[];
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/// <summary>
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/// Getter that returns a clone of the array containing the Gauss weights.
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/// </summary>
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public double[] GaussWeights => gaussKronrodPoint.SecondWeights.Clone() as double[];
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public GaussKronrodRule(int order)
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{
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gaussKronrodPoint = GaussKronrodPointFactory.GetGaussPoint(order);
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}
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/// <summary>
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/// Performs adaptive Gauss-Kronrod quadrature on function f over the range (a,b)
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/// </summary>
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/// <param name="f">The analytic smooth function to integrate</param>
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/// <param name="intervalBegin">Where the interval starts</param>
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/// <param name="intervalEnd">Where the interval stops</param>
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/// <param name="error">The difference between the (N-1)/2 point Gauss approximation and the N-point Gauss-Kronrod approximation</param>
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/// <param name="L1Norm">The L1 norm of the result, if there is a significant difference between this and the returned value, then the result is likely to be ill-conditioned.</param>
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/// <param name="targetRelativeError">The maximum relative error in the result</param>
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/// <param name="maximumDepth">The maximum number of interval splittings permitted before stopping</param>
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/// <param name="order">The number of Gauss-Kronrod points. Pre-computed for 15, 21, 31, 41, 51 and 61 points</param>
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public static double Integrate(Func<double, double> f, double intervalBegin, double intervalEnd, out double error, out double L1Norm, double targetRelativeError = 1E-10, int maximumDepth = 15, int order = 15)
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{
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// Formula used for variable subsitution from
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// 1. Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131-140.
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// 2. quadgk.m, GNU Octave
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if (f == null)
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{
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throw new ArgumentNullException(nameof(f));
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}
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if (intervalBegin > intervalEnd)
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{
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return -Integrate(f, intervalEnd, intervalBegin, out error, out L1Norm, targetRelativeError, maximumDepth, order);
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}
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GaussPointPair gaussKronrodPoint = GaussKronrodPointFactory.GetGaussPoint(order);
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// (-oo, oo) => [-1, 1]
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//
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// integral_(-oo)^(oo) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
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// g(t) = t / (1 - t^2)
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// g'(t) = (1 + t^2) / (1 - t^2)^2
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if ((intervalBegin < double.MinValue) && (intervalEnd > double.MaxValue))
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{
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Func<double, double> u = (t) =>
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{
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return f(t / (1 - t * t)) * (1 + t * t) / ((1 - t * t) * (1 - t * t));
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};
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return recursive_adaptive_integrate(u, -1, 1, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint);
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}
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// [a, oo) => [0, 1]
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//
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// integral_(a)^(oo) f(x) dx = integral_(0)^(oo) f(a + t^2) 2 t dt
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// = integral_(0)^(1) f(a + g(s)^2) 2 g(s) g'(s) ds
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// g(s) = s / (1 - s)
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// g'(s) = 1 / (1 - s)^2
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else if (intervalEnd > double.MaxValue)
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{
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Func<double, double> u = (s) =>
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{
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return 2 * s * f(intervalBegin + (s / (1 - s)) * (s / (1 - s))) / ((1 - s) * (1 - s) * (1 - s));
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};
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return recursive_adaptive_integrate(u, 0, 1, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint);
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}
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// (-oo, b] => [-1, 0]
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//
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// integral_(-oo)^(b) f(x) dx = -integral_(-oo)^(0) f(b - t^2) 2 t dt
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// = -integral_(-1)^(0) f(b - g(s)^2) 2 g(s) g'(s) ds
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// g(s) = s / (1 + s)
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// g'(s) = 1 / (1 + s)^2
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else if (intervalBegin < double.MinValue)
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{
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Func<double, double> u = (s) =>
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{
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return -2 * s * f(intervalEnd - s / (1 + s) * (s / (1 + s))) / ((1 + s) * (1 + s) * (1 + s));
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};
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return recursive_adaptive_integrate(u, -1, 0, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint);
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}
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// [a, b] => [-1, 1]
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//
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// integral_(a)^(b) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
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// g(t) = (b - a) * t * (3 - t^2) / 4 + (b + a) / 2
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// g'(t) = 3 / 4 * (b - a) * (1 - t^2)
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else
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{
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Func<double, double> u = (t) =>
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{
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return f((intervalEnd - intervalBegin) / 4 * t * (3 - t * t) + (intervalEnd + intervalBegin) / 2) * 3 * (intervalEnd - intervalBegin) / 4 * (1 - t * t);
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};
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return recursive_adaptive_integrate(u, -1, 1, maximumDepth, targetRelativeError, 0d, out error, out L1Norm, gaussKronrodPoint);
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}
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}
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/// <summary>
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/// Performs adaptive Gauss-Kronrod quadrature on function f over the range (a,b)
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/// </summary>
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/// <param name="f">The analytic smooth complex function to integrate, defined on the real axis.</param>
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/// <param name="intervalBegin">Where the interval starts</param>
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/// <param name="intervalEnd">Where the interval stops</param>
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/// <param name="error">The difference between the (N-1)/2 point Gauss approximation and the N-point Gauss-Kronrod approximation</param>
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/// <param name="L1Norm">The L1 norm of the result, if there is a significant difference between this and the returned value, then the result is likely to be ill-conditioned.</param>
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/// <param name="targetRelativeError">The maximum relative error in the result</param>
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/// <param name="maximumDepth">The maximum number of interval splittings permitted before stopping</param>
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/// <param name="order">The number of Gauss-Kronrod points. Pre-computed for 15, 21, 31, 41, 51 and 61 points</param>
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/// <returns></returns>
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public static Complex ContourIntegrate(Func<double, Complex> f, double intervalBegin, double intervalEnd, out double error, out double L1Norm, double targetRelativeError = 1E-10, int maximumDepth = 15, int order = 15)
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{
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// Formula used for variable subsitution from
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// 1. Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131-140.
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// 2. quadgk.m, GNU Octave
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if (f == null)
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{
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throw new ArgumentNullException(nameof(f));
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}
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if (intervalBegin > intervalEnd)
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{
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return -ContourIntegrate(f, intervalEnd, intervalBegin, out error, out L1Norm, targetRelativeError, maximumDepth, order);
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}
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GaussPointPair gaussKronrodPoint = GaussKronrodPointFactory.GetGaussPoint(order);
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// (-oo, oo) => [-1, 1]
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//
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// integral_(-oo)^(oo) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
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// g(t) = t / (1 - t^2)
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// g'(t) = (1 + t^2) / (1 - t^2)^2
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if ((intervalBegin < double.MinValue) && (intervalEnd > double.MaxValue))
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{
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Func<double, Complex> u = (t) =>
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{
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return f(t / (1 - t * t)) * (1 + t * t) / ((1 - t * t) * (1 - t * t));
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};
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return contour_recursive_adaptive_integrate(u, -1, 1, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint);
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}
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// [a, oo) => [0, 1]
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//
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// integral_(a)^(oo) f(x) dx = integral_(0)^(oo) f(a + t^2) 2 t dt
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// = integral_(0)^(1) f(a + g(s)^2) 2 g(s) g'(s) ds
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// g(s) = s / (1 - s)
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// g'(s) = 1 / (1 - s)^2
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else if (intervalEnd > double.MaxValue)
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{
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Func<double, Complex> u = (s) =>
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{
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return 2 * s * f(intervalBegin + (s / (1 - s)) * (s / (1 - s))) / ((1 - s) * (1 - s) * (1 - s));
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};
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return contour_recursive_adaptive_integrate(u, 0, 1, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint);
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}
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// (-oo, b] => [-1, 0]
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//
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// integral_(-oo)^(b) f(x) dx = -integral_(-oo)^(0) f(b - t^2) 2 t dt
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// = -integral_(-1)^(0) f(b - g(s)^2) 2 g(s) g'(s) ds
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// g(s) = s / (1 + s)
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// g'(s) = 1 / (1 + s)^2
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else if (intervalBegin < double.MinValue)
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{
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Func<double, Complex> u = (s) =>
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{
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return -2 * s * f(intervalEnd - s / (1 + s) * (s / (1 + s))) / ((1 + s) * (1 + s) * (1 + s));
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};
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return contour_recursive_adaptive_integrate(u, -1, 0, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint);
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}
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// [a, b] => [-1, 1]
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//
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// integral_(a)^(b) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
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// g(t) = (b - a) * t * (3 - t^2) / 4 + (b + a) / 2
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// g'(t) = 3 / 4 * (b - a) * (1 - t^2)
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else
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{
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Func<double, Complex> u = (t) =>
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{
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return f((intervalEnd - intervalBegin) / 4 * t * (3 - t * t) + (intervalEnd + intervalBegin) / 2) * 3 * (intervalEnd - intervalBegin) / 4 * (1 - t * t);
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};
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return contour_recursive_adaptive_integrate(u, -1, 1, maximumDepth, targetRelativeError, 0d, out error, out L1Norm, gaussKronrodPoint);
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}
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}
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private static double integrate_non_adaptive_m1_1(Func<double, double> f, out double error, out double pL1, GaussPointPair gaussKronrodPoint)
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{
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int gauss_start = 2;
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int kronrod_start = 1;
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int gauss_order = (gaussKronrodPoint.Order - 1) / 2;
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double kronrod_result = 0d;
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double gauss_result = 0d;
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double fp, fm;
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var KAbscissa = gaussKronrodPoint.Abscissas;
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var KWeights = gaussKronrodPoint.Weights;
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var GWeights = gaussKronrodPoint.SecondWeights;
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if ((gauss_order & 1) == 1)
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{
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fp = f(0);
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kronrod_result = fp * KWeights[0];
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gauss_result += fp * GWeights[0];
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}
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else
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{
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fp = f(0);
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kronrod_result = fp * KWeights[0];
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gauss_start = 1;
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kronrod_start = 2;
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}
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double L1 = Math.Abs(kronrod_result);
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for (int i = gauss_start; i < KAbscissa.Length; i += 2)
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{
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fp = f(KAbscissa[i]);
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fm = f(-KAbscissa[i]);
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kronrod_result += (fp + fm) * KWeights[i];
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L1 += (Math.Abs(fp) + Math.Abs(fm)) * KWeights[i];
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gauss_result += (fp + fm) * GWeights[i / 2];
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}
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for (int i = kronrod_start; i < KAbscissa.Length; i += 2)
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{
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fp = f(KAbscissa[i]);
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fm = f(-KAbscissa[i]);
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kronrod_result += (fp + fm) * KWeights[i];
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L1 += (Math.Abs(fp) + Math.Abs(fm)) * KWeights[i];
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}
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pL1 = L1;
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error = Math.Max(Math.Abs(kronrod_result - gauss_result), Math.Abs(kronrod_result * Precision.MachineEpsilon * 2d));
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return kronrod_result;
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}
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private static Complex contour_integrate_non_adaptive_m1_1(Func<double, Complex> f, out double error, out double pL1, GaussPointPair gaussKronrodPoint)
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{
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int gauss_start = 2;
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int kronrod_start = 1;
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int gauss_order = (gaussKronrodPoint.Order - 1) / 2;
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Complex kronrod_result = new Complex();
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Complex gauss_result = new Complex();
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Complex fp, fm;
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var KAbscissa = gaussKronrodPoint.Abscissas;
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var KWeights = gaussKronrodPoint.Weights;
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var GWeights = gaussKronrodPoint.SecondWeights;
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if (gauss_order.IsOdd())
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{
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fp = f(0);
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kronrod_result = fp * KWeights[0];
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gauss_result += fp * GWeights[0];
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}
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else
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{
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fp = f(0);
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kronrod_result = fp * KWeights[0];
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gauss_start = 1;
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kronrod_start = 2;
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}
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double L1 = Complex.Abs(kronrod_result);
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for (int i = gauss_start; i < KAbscissa.Length; i += 2)
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{
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fp = f(KAbscissa[i]);
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fm = f(-KAbscissa[i]);
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kronrod_result += (fp + fm) * KWeights[i];
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L1 += (Complex.Abs(fp) + Complex.Abs(fm)) * KWeights[i];
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gauss_result += (fp + fm) * GWeights[i / 2];
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}
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for (int i = kronrod_start; i < KAbscissa.Length; i += 2)
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{
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fp = f(KAbscissa[i]);
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fm = f(-KAbscissa[i]);
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kronrod_result += (fp + fm) * KWeights[i];
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L1 += (Complex.Abs(fp) + Complex.Abs(fm)) * KWeights[i];
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}
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pL1 = L1;
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error = Math.Max(Complex.Abs(kronrod_result - gauss_result), Complex.Abs(kronrod_result * Precision.MachineEpsilon * 2d));
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return kronrod_result;
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}
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private static double recursive_adaptive_integrate(Func<double, double> f, double a, double b, int max_levels, double rel_tol, double abs_tol, out double error, out double L1, GaussPointPair gaussKronrodPoint)
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{
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double error_local;
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double mean = (b + a) / 2;
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double scale = (b - a) / 2;
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var r1 = integrate_non_adaptive_m1_1((x) => f(scale * x + mean), out error_local, out L1, gaussKronrodPoint);
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var estimate = scale * r1;
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var tmp = estimate * rel_tol;
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var abs_tol1 = Math.Abs(tmp);
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if (abs_tol == 0)
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{
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abs_tol = abs_tol1;
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}
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if (max_levels > 0 && (abs_tol1 < error_local) && (abs_tol < error_local))
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{
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double mid = (a + b) / 2d;
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double L1_local;
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estimate = recursive_adaptive_integrate(f, a, mid, max_levels - 1, rel_tol, abs_tol / 2, out error, out L1, gaussKronrodPoint);
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estimate += recursive_adaptive_integrate(f, mid, b, max_levels - 1, rel_tol, abs_tol / 2, out error_local, out L1_local, gaussKronrodPoint);
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error += error_local;
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L1 += L1_local;
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return estimate;
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}
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L1 *= scale;
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error = error_local;
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return estimate;
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}
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private static Complex contour_recursive_adaptive_integrate(Func<double, Complex> f, double a, double b, int max_levels, double rel_tol, double abs_tol, out double error, out double L1, GaussPointPair gaussKronrodPoint)
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{
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double error_local;
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double mean = (b + a) / 2;
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double scale = (b - a) / 2;
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var r1 = contour_integrate_non_adaptive_m1_1((x) => f(scale * x + mean), out error_local, out L1, gaussKronrodPoint);
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var estimate = scale * r1;
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var tmp = estimate * rel_tol;
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var abs_tol1 = Complex.Abs(tmp);
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if (abs_tol == 0)
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{
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abs_tol = abs_tol1;
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}
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if (max_levels > 0 && (abs_tol1 < error_local) && (abs_tol < error_local))
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{
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double mid = (a + b) / 2d;
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double L1_local;
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estimate = contour_recursive_adaptive_integrate(f, a, mid, max_levels - 1, rel_tol, abs_tol / 2, out error, out L1, gaussKronrodPoint);
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estimate += contour_recursive_adaptive_integrate(f, mid, b, max_levels - 1, rel_tol, abs_tol / 2, out error_local, out L1_local, gaussKronrodPoint);
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error += error_local;
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L1 += L1_local;
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return estimate;
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}
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L1 *= scale;
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error = error_local;
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return estimate;
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}
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}
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}
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