// <copyright file="NevillePolynomialInterpolation.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) 2009-2014 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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using System;
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using System.Collections.Generic;
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using System.Linq;
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namespace IStation.Numerics.Interpolation
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{
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/// <summary>
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/// Lagrange Polynomial Interpolation using Neville's Algorithm.
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/// </summary>
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/// <remarks>
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/// <para>
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/// This algorithm supports differentiation, but doesn't support integration.
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/// </para>
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/// <para>
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/// When working with equidistant or Chebyshev sample points it is
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/// recommended to use the barycentric algorithms specialized for
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/// these cases instead of this arbitrary Neville algorithm.
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/// </para>
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/// </remarks>
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public class NevillePolynomialInterpolation : IInterpolation
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{
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readonly double[] _x;
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readonly double[] _y;
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/// <param name="x">Sample Points t, sorted ascendingly.</param>
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/// <param name="y">Sample Values x(t), sorted ascendingly by x.</param>
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public NevillePolynomialInterpolation(double[] x, double[] y)
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{
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if (x.Length != y.Length)
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{
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throw new ArgumentException("All vectors must have the same dimensionality.");
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}
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if (x.Length < 1)
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{
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throw new ArgumentException("The given array is too small. It must be at least 1 long.", nameof(x));
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}
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for (var i = 1; i < x.Length; ++i)
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{
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if (x[i] == x[i - 1])
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{
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throw new ArgumentException("All sample points should be unique.", nameof(x));
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}
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}
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_x = x;
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_y = y;
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}
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/// <summary>
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/// Create a Neville polynomial interpolation from a set of (x,y) value pairs, sorted ascendingly by x.
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/// </summary>
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public static NevillePolynomialInterpolation InterpolateSorted(double[] x, double[] y)
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{
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return new NevillePolynomialInterpolation(x, y);
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}
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/// <summary>
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/// Create a Neville polynomial interpolation from an unsorted set of (x,y) value pairs.
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/// WARNING: Works in-place and can thus causes the data array to be reordered.
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/// </summary>
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public static NevillePolynomialInterpolation InterpolateInplace(double[] x, double[] y)
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{
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if (x.Length != y.Length)
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{
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throw new ArgumentException("All vectors must have the same dimensionality.");
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}
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Sorting.Sort(x, y);
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return InterpolateSorted(x, y);
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}
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/// <summary>
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/// Create a Neville polynomial interpolation from an unsorted set of (x,y) value pairs.
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/// </summary>
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public static NevillePolynomialInterpolation Interpolate(IEnumerable<double> x, IEnumerable<double> y)
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{
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// note: we must make a copy, even if the input was arrays already
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return InterpolateInplace(x.ToArray(), y.ToArray());
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}
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/// <summary>
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/// Gets a value indicating whether the algorithm supports differentiation (interpolated derivative).
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/// </summary>
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bool IInterpolation.SupportsDifferentiation => true;
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/// <summary>
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/// Gets a value indicating whether the algorithm supports integration (interpolated quadrature).
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/// </summary>
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bool IInterpolation.SupportsIntegration => false;
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/// <summary>
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/// Interpolate at point t.
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/// </summary>
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/// <param name="t">Point t to interpolate at.</param>
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/// <returns>Interpolated value x(t).</returns>
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public double Interpolate(double t)
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{
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var x = new double[_y.Length];
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_y.CopyTo(x, 0);
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for (int level = 1; level < x.Length; level++)
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{
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for (int i = 0; i < x.Length - level; i++)
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{
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double hp = t - _x[i + level];
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double ho = _x[i] - t;
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double den = _x[i] - _x[i + level];
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x[i] = ((hp*x[i]) + (ho*x[i + 1]))/den;
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}
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}
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return x[0];
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}
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/// <summary>
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/// Differentiate at point t.
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/// </summary>
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/// <param name="t">Point t to interpolate at.</param>
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/// <returns>Interpolated first derivative at point t.</returns>
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public double Differentiate(double t)
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{
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var x = new double[_y.Length];
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var dx = new double[_y.Length];
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_y.CopyTo(x, 0);
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for (int level = 1; level < x.Length; level++)
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{
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for (int i = 0; i < x.Length - level; i++)
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{
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double hp = t - _x[i + level];
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double ho = _x[i] - t;
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double den = _x[i] - _x[i + level];
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dx[i] = ((hp*dx[i]) + x[i] + (ho*dx[i + 1]) - x[i + 1])/den;
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x[i] = ((hp*x[i]) + (ho*x[i + 1]))/den;
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}
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}
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return dx[0];
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}
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/// <summary>
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/// Differentiate twice at point t.
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/// </summary>
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/// <param name="t">Point t to interpolate at.</param>
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/// <returns>Interpolated second derivative at point t.</returns>
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public double Differentiate2(double t)
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{
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var x = new double[_y.Length];
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var dx = new double[_y.Length];
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var ddx = new double[_y.Length];
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_y.CopyTo(x, 0);
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for (int level = 1; level < x.Length; level++)
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{
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for (int i = 0; i < x.Length - level; i++)
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{
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double hp = t - _x[i + level];
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double ho = _x[i] - t;
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double den = _x[i] - _x[i + level];
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ddx[i] = ((hp*ddx[i]) + (ho*ddx[i + 1]) + (2*dx[i]) - (2*dx[i + 1]))/den;
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dx[i] = ((hp*dx[i]) + x[i] + (ho*dx[i + 1]) - x[i + 1])/den;
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x[i] = ((hp*x[i]) + (ho*x[i + 1]))/den;
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}
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}
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return ddx[0];
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}
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/// <summary>
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/// Indefinite integral at point t. NOT SUPPORTED.
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/// </summary>
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/// <param name="t">Point t to integrate at.</param>
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double IInterpolation.Integrate(double t)
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{
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throw new NotSupportedException();
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}
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/// <summary>
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/// Definite integral between points a and b. NOT SUPPORTED.
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/// </summary>
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/// <param name="a">Left bound of the integration interval [a,b].</param>
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/// <param name="b">Right bound of the integration interval [a,b].</param>
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double IInterpolation.Integrate(double a, double b)
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{
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throw new NotSupportedException();
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}
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}
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}
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