// <copyright file="GaussLegendreRule.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) 2009-2016 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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// Software is furnished to do so, subject to the following
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// conditions:
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// </copyright>
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using System;
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using System.Numerics;
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using IStation.Numerics.Integration.GaussRule;
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namespace IStation.Numerics.Integration
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{
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/// <summary>
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/// Approximates a definite integral using an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2-20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calculated on the fly.
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/// </summary>
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public class GaussLegendreRule
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{
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private readonly GaussPoint _gaussLegendrePoint;
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/// <summary>
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/// Initializes a new instance of the <see cref="GaussLegendreRule"/> class.
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/// </summary>
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/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
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/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
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/// <param name="order">Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2-20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calculated on the fly.</param>
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public GaussLegendreRule(double intervalBegin, double intervalEnd, int order)
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{
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_gaussLegendrePoint = GaussLegendrePointFactory.GetGaussPoint(intervalBegin, intervalEnd, order);
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}
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/// <summary>
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/// Gettter for the ith abscissa.
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/// </summary>
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/// <param name="index">Index of the ith abscissa.</param>
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/// <returns>The ith abscissa.</returns>
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public double GetAbscissa(int index)
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{
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return _gaussLegendrePoint.Abscissas[index];
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}
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/// <summary>
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/// Getter that returns a clone of the array containing the abscissas.
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/// </summary>
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public double[] Abscissas => _gaussLegendrePoint.Abscissas.Clone() as double[];
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/// <summary>
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/// Getter for the ith weight.
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/// </summary>
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/// <param name="index">Index of the ith weight.</param>
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/// <returns>The ith weight.</returns>
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public double GetWeight(int index)
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{
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return _gaussLegendrePoint.Weights[index];
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}
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/// <summary>
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/// Getter that returns a clone of the array containing the weights.
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/// </summary>
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public double[] Weights => _gaussLegendrePoint.Weights.Clone() as double[];
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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 => _gaussLegendrePoint.Order;
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/// <summary>
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/// Getter for the InvervalBegin.
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/// </summary>
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public double IntervalBegin => _gaussLegendrePoint.IntervalBegin;
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/// <summary>
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/// Getter for the InvervalEnd.
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/// </summary>
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public double IntervalEnd => _gaussLegendrePoint.IntervalEnd;
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/// <summary>
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/// Approximates a definite integral using an Nth order Gauss-Legendre rule.
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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="invervalBegin">Where the interval starts, exclusive and finite.</param>
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/// <param name="invervalEnd">Where the interval ends, exclusive and finite.</param>
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/// <param name="order">Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2-20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calculated on the fly.</param>
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/// <returns>Approximation of the finite integral in the given interval.</returns>
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public static double Integrate(Func<double, double> f, double invervalBegin, double invervalEnd, int order)
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{
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GaussPoint gaussLegendrePoint = GaussLegendrePointFactory.GetGaussPoint(order);
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double sum, ax;
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int i;
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int m = (order + 1) >> 1;
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double a = 0.5*(invervalEnd - invervalBegin);
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double b = 0.5*(invervalEnd + invervalBegin);
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if (order.IsOdd())
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{
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sum = gaussLegendrePoint.Weights[0]*f(b);
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for (i = 1; i < m; i++)
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{
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ax = a*gaussLegendrePoint.Abscissas[i];
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sum += gaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax));
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}
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}
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else
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{
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sum = 0.0;
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for (i = 0; i < m; i++)
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{
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ax = a*gaussLegendrePoint.Abscissas[i];
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sum += gaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax));
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}
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}
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return a*sum;
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}
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/// <summary>
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/// Approximates a definite integral using an Nth order Gauss-Legendre rule.
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/// </summary>
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/// <param name="f">The analytic smooth complex function to integrate, defined on the real domain.</param>
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/// <param name="invervalBegin">Where the interval starts, exclusive and finite.</param>
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/// <param name="invervalEnd">Where the interval ends, exclusive and finite.</param>
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/// <param name="order">Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2-20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calculated on the fly.</param>
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/// <returns>Approximation of the finite integral in the given interval.</returns>
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public static Complex ContourIntegrate(Func<double, Complex> f, double invervalBegin, double invervalEnd, int order)
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{
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GaussPoint gaussLegendrePoint = GaussLegendrePointFactory.GetGaussPoint(order);
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Complex sum;
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double ax;
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int i;
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int m = (order + 1) >> 1;
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double a = 0.5 * (invervalEnd - invervalBegin);
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double b = 0.5 * (invervalEnd + invervalBegin);
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if (order.IsOdd())
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{
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sum = gaussLegendrePoint.Weights[0] * f(b);
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for (i = 1; i < m; i++)
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{
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ax = a * gaussLegendrePoint.Abscissas[i];
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sum += gaussLegendrePoint.Weights[i] * (f(b + ax) + f(b - ax));
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}
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}
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else
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{
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sum = 0.0;
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for (i = 0; i < m; i++)
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{
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ax = a * gaussLegendrePoint.Abscissas[i];
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sum += gaussLegendrePoint.Weights[i] * (f(b + ax) + f(b - ax));
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}
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}
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return a * sum;
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}
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/// <summary>
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/// Approximates a 2-dimensional definite integral using an Nth order Gauss-Legendre rule over the rectangle [a,b] x [c,d].
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/// </summary>
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/// <param name="f">The 2-dimensional analytic smooth function to integrate.</param>
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/// <param name="invervalBeginA">Where the interval starts for the first (inside) integral, exclusive and finite.</param>
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/// <param name="invervalEndA">Where the interval ends for the first (inside) integral, exclusive and finite.</param>
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/// <param name="invervalBeginB">Where the interval starts for the second (outside) integral, exclusive and finite.</param>
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/// /// <param name="invervalEndB">Where the interval ends for the second (outside) integral, exclusive and finite.</param>
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/// <param name="order">Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2-20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calculated on the fly.</param>
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/// <returns>Approximation of the finite integral in the given interval.</returns>
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public static double Integrate(Func<double, double, double> f, double invervalBeginA, double invervalEndA, double invervalBeginB, double invervalEndB, int order)
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{
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GaussPoint gaussLegendrePoint = GaussLegendrePointFactory.GetGaussPoint(order);
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double ax, cy, sum;
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int i, j;
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int m = (order + 1) >> 1;
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double a = 0.5*(invervalEndA - invervalBeginA);
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double b = 0.5*(invervalEndA + invervalBeginA);
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double c = 0.5*(invervalEndB - invervalBeginB);
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double d = 0.5*(invervalEndB + invervalBeginB);
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if (order.IsOdd())
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{
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sum = gaussLegendrePoint.Weights[0]*gaussLegendrePoint.Weights[0]*f(b, d);
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double t;
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for (j = 1, t = 0.0; j < m; j++)
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{
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cy = c*gaussLegendrePoint.Abscissas[j];
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t += gaussLegendrePoint.Weights[j]*(f(b, d + cy) + f(b, d - cy));
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}
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sum += gaussLegendrePoint.Weights[0]*t;
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for (i = 1, t = 0.0; i < m; i++)
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{
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ax = a*gaussLegendrePoint.Abscissas[i];
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t += gaussLegendrePoint.Weights[i]*(f(b + ax, d) + f(b - ax, d));
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}
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sum += gaussLegendrePoint.Weights[0]*t;
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for (i = 1; i < m; i++)
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{
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ax = a*gaussLegendrePoint.Abscissas[i];
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for (j = 1; j < m; j++)
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{
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cy = c*gaussLegendrePoint.Abscissas[j];
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sum += gaussLegendrePoint.Weights[i]*gaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy));
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}
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}
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}
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else
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{
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sum = 0.0;
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for (i = 0; i < m; i++)
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{
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ax = a*gaussLegendrePoint.Abscissas[i];
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for (j = 0; j < m; j++)
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{
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cy = c*gaussLegendrePoint.Abscissas[j];
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sum += gaussLegendrePoint.Weights[i]*gaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy));
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}
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}
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}
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return c*a*sum;
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}
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}
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}
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