// <copyright file="CubicSpline.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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/// Cubic Spline Interpolation.
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/// </summary>
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/// <remarks>Supports both differentiation and integration.</remarks>
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public class CubicSpline : IInterpolation
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{
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readonly double[] _x;
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readonly double[] _c0;
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readonly double[] _c1;
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readonly double[] _c2;
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readonly double[] _c3;
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readonly Lazy<double[]> _indefiniteIntegral;
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/// <param name="x">sample points (N+1), sorted ascending</param>
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/// <param name="c0">Zero order spline coefficients (N)</param>
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/// <param name="c1">First order spline coefficients (N)</param>
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/// <param name="c2">second order spline coefficients (N)</param>
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/// <param name="c3">third order spline coefficients (N)</param>
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public CubicSpline(double[] x, double[] c0, double[] c1, double[] c2, double[] c3)
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{
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if (x.Length != c0.Length + 1 || x.Length != c1.Length + 1 || x.Length != c2.Length + 1 || x.Length != c3.Length + 1)
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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 < 2)
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{
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throw new ArgumentException("The given array is too small. It must be at least 2 long.", nameof(x));
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}
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_x = x;
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_c0 = c0;
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_c1 = c1;
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_c2 = c2;
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_c3 = c3;
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_indefiniteIntegral = new Lazy<double[]>(ComputeIndefiniteIntegral);
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}
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/// <summary>
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/// Create a Hermite cubic spline interpolation from a set of (x,y) value pairs and their slope (first derivative), sorted ascendingly by x.
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/// </summary>
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public static CubicSpline InterpolateHermiteSorted(double[] x, double[] y, double[] firstDerivatives)
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{
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if (x.Length != y.Length || x.Length != firstDerivatives.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 < 2)
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{
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throw new ArgumentException("The given array is too small. It must be at least 2 long.", nameof(x));
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}
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var c0 = new double[x.Length - 1];
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var c1 = new double[x.Length - 1];
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var c2 = new double[x.Length - 1];
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var c3 = new double[x.Length - 1];
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for (int i = 0; i < c1.Length; i++)
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{
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double w = x[i + 1] - x[i];
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double w2 = w*w;
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c0[i] = y[i];
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c1[i] = firstDerivatives[i];
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c2[i] = (3*(y[i + 1] - y[i])/w - 2*firstDerivatives[i] - firstDerivatives[i + 1])/w;
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c3[i] = (2*(y[i] - y[i + 1])/w + firstDerivatives[i] + firstDerivatives[i + 1])/w2;
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}
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return new CubicSpline(x, c0, c1, c2, c3);
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}
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/// <summary>
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/// Create a Hermite cubic spline interpolation from an unsorted set of (x,y) value pairs and their slope (first derivative).
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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 CubicSpline InterpolateHermiteInplace(double[] x, double[] y, double[] firstDerivatives)
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{
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if (x.Length != y.Length || x.Length != firstDerivatives.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 < 2)
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{
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throw new ArgumentException("The given array is too small. It must be at least 2 long.", nameof(x));
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}
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Sorting.Sort(x, y, firstDerivatives);
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return InterpolateHermiteSorted(x, y, firstDerivatives);
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}
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/// <summary>
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/// Create a Hermite cubic spline interpolation from an unsorted set of (x,y) value pairs and their slope (first derivative).
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/// </summary>
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public static CubicSpline InterpolateHermite(IEnumerable<double> x, IEnumerable<double> y, IEnumerable<double> firstDerivatives)
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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 InterpolateHermiteInplace(x.ToArray(), y.ToArray(), firstDerivatives.ToArray());
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}
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/// <summary>
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/// Create an Akima cubic spline interpolation from a set of (x,y) value pairs, sorted ascendingly by x.
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/// Akima splines are robust to outliers.
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/// </summary>
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public static CubicSpline InterpolateAkimaSorted(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 < 5)
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{
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throw new ArgumentException("The given array is too small. It must be at least 5 long.", nameof(x));
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}
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/* Prepare divided differences (diff) and weights (w) */
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var diff = new double[x.Length - 1];
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var weights = new double[x.Length - 1];
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for (int i = 0; i < diff.Length; i++)
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{
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diff[i] = (y[i + 1] - y[i])/(x[i + 1] - x[i]);
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}
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for (int i = 1; i < weights.Length; i++)
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{
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weights[i] = Math.Abs(diff[i] - diff[i - 1]);
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}
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/* Prepare Hermite interpolation scheme */
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var dd = new double[x.Length];
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for (int i = 2; i < dd.Length - 2; i++)
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{
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dd[i] = weights[i - 1].AlmostEqual(0.0) && weights[i + 1].AlmostEqual(0.0)
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? (((x[i + 1] - x[i])*diff[i - 1]) + ((x[i] - x[i - 1])*diff[i]))/(x[i + 1] - x[i - 1])
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: ((weights[i + 1]*diff[i - 1]) + (weights[i - 1]*diff[i]))/(weights[i + 1] + weights[i - 1]);
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}
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dd[0] = DifferentiateThreePoint(x, y, 0, 0, 1, 2);
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dd[1] = DifferentiateThreePoint(x, y, 1, 0, 1, 2);
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dd[x.Length - 2] = DifferentiateThreePoint(x, y, x.Length - 2, x.Length - 3, x.Length - 2, x.Length - 1);
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dd[x.Length - 1] = DifferentiateThreePoint(x, y, x.Length - 1, x.Length - 3, x.Length - 2, x.Length - 1);
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/* Build Akima spline using Hermite interpolation scheme */
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return InterpolateHermiteSorted(x, y, dd);
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}
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/// <summary>
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/// Create an Akima cubic spline interpolation from an unsorted set of (x,y) value pairs.
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/// Akima splines are robust to outliers.
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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 CubicSpline InterpolateAkimaInplace(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 InterpolateAkimaSorted(x, y);
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}
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/// <summary>
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/// Create an Akima cubic spline interpolation from an unsorted set of (x,y) value pairs.
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/// Akima splines are robust to outliers.
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/// </summary>
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public static CubicSpline InterpolateAkima(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 InterpolateAkimaInplace(x.ToArray(), y.ToArray());
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}
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/// <summary>
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/// Create a piecewise cubic Hermite interpolating polynomial from an unsorted set of (x,y) value pairs.
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/// Monotone-preserving interpolation with continuous first derivative.
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/// </summary>
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public static CubicSpline InterpolatePchipSorted(double[] x, double[] y)
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{
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// Implementation based on "Numerical Computing with Matlab" (Moler, 2004).
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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 < 3)
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{
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throw new ArgumentException("The given array is too small. It must be at least 3 long.", nameof(x));
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}
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var m = new double[x.Length - 1];
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for (int i = 0; i < m.Length; i++)
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{
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m[i] = (y[i + 1] - y[i])/(x[i + 1] - x[i]);
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}
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var dd = new double[x.Length];
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var hPrev = x[1] - x[0];
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// This check is quite costly as it usually involves a Math.Pow().
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var mPrevIs0 = m[0].AlmostEqual(0.0);
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for (var i = 1; i < x.Length - 1; ++i)
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{
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var h = x[i + 1] - x[i];
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var mIs0 = m[i].AlmostEqual(0.0);
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if (mIs0 || mPrevIs0 || Math.Sign(m[i]) != Math.Sign(m[i - 1]))
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{
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dd[i] = 0;
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}
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else
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{
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// Weighted harmonic mean of each slope.
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var w1 = 2 * h + hPrev;
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var w2 = h + 2 * hPrev;
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dd[i] = (w1 + w2) / (w1 / m[i - 1] + w2 / m[i]);
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}
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hPrev = h;
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mPrevIs0 = mIs0;
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}
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// Special case end-points.
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dd[0] = PchipEndPoints(x[1] - x[0], x[2] - x[1], m[0], m[1]);
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dd[dd.Length - 1] = PchipEndPoints(
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x[x.Length - 1] - x[x.Length - 2], x[x.Length - 2] - x[x.Length - 3],
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m[m.Length - 1], m[m.Length - 2]);
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return InterpolateHermiteSorted(x, y, dd);
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}
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static double PchipEndPoints(double h0, double h1, double m0, double m1)
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{
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// One-sided, shape-preserving, three-point estimate for the derivative.
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var d = ((2 * h0 + h1) * m0 - h0 * m1) / (h0 + h1);
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if (Math.Sign(d) != Math.Sign(m0))
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{
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return 0.0;
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}
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if (Math.Sign(m0) != Math.Sign(m1) && (Math.Abs(d) > 3 * Math.Abs(m0)))
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{
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return 3 * m0;
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}
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return d;
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}
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/// <summary>
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/// Create a piecewise cubic Hermite interpolating polynomial from an unsorted set of (x,y) value pairs.
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/// Monotone-preserving interpolation with continuous first derivative.
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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 CubicSpline InterpolatePchipInplace(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 InterpolatePchipSorted(x, y);
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}
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/// <summary>
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/// Create a piecewise cubic Hermite interpolating polynomial from an unsorted set of (x,y) value pairs.
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/// Monotone-preserving interpolation with continuous first derivative.
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/// </summary>
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public static CubicSpline InterpolatePchip(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 InterpolatePchipInplace(x.ToArray(), y.ToArray());
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}
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/// <summary>
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/// Create a cubic spline interpolation from a set of (x,y) value pairs, sorted ascendingly by x,
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/// and custom boundary/termination conditions.
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/// </summary>
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public static CubicSpline InterpolateBoundariesSorted(double[] x, double[] y,
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SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
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SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
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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 < 2)
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{
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throw new ArgumentException("The given array is too small. It must be at least 2 long.", nameof(x));
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}
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int n = x.Length;
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// normalize special cases
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if ((n == 2)
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&& (leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated)
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&& (rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
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{
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leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
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leftBoundary = 0d;
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rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
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rightBoundary = 0d;
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}
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if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
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{
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leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
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leftBoundary = 0d;
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}
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if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
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{
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rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
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rightBoundary = 0d;
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}
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var a1 = new double[n];
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var a2 = new double[n];
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var a3 = new double[n];
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var b = new double[n];
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// Left Boundary
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switch (leftBoundaryCondition)
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{
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case SplineBoundaryCondition.ParabolicallyTerminated:
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a1[0] = 0;
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a2[0] = 1;
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a3[0] = 1;
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b[0] = 2*(y[1] - y[0])/(x[1] - x[0]);
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break;
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case SplineBoundaryCondition.FirstDerivative:
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a1[0] = 0;
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a2[0] = 1;
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a3[0] = 0;
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b[0] = leftBoundary;
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break;
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case SplineBoundaryCondition.SecondDerivative:
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a1[0] = 0;
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a2[0] = 2;
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a3[0] = 1;
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b[0] = (3*((y[1] - y[0])/(x[1] - x[0]))) - (0.5*leftBoundary*(x[1] - x[0]));
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break;
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default:
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throw new NotSupportedException("Invalid Left Boundary Condition.");
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}
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// Central Conditions
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for (int i = 1; i < x.Length - 1; i++)
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{
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a1[i] = x[i + 1] - x[i];
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a2[i] = 2*(x[i + 1] - x[i - 1]);
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a3[i] = x[i] - x[i - 1];
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b[i] = (3*(y[i] - y[i - 1])/(x[i] - x[i - 1])*(x[i + 1] - x[i])) + (3*(y[i + 1] - y[i])/(x[i + 1] - x[i])*(x[i] - x[i - 1]));
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}
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// Right Boundary
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switch (rightBoundaryCondition)
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{
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case SplineBoundaryCondition.ParabolicallyTerminated:
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a1[n - 1] = 1;
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a2[n - 1] = 1;
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a3[n - 1] = 0;
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b[n - 1] = 2*(y[n - 1] - y[n - 2])/(x[n - 1] - x[n - 2]);
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break;
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case SplineBoundaryCondition.FirstDerivative:
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a1[n - 1] = 0;
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a2[n - 1] = 1;
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a3[n - 1] = 0;
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b[n - 1] = rightBoundary;
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break;
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case SplineBoundaryCondition.SecondDerivative:
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a1[n - 1] = 1;
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a2[n - 1] = 2;
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a3[n - 1] = 0;
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b[n - 1] = (3*(y[n - 1] - y[n - 2])/(x[n - 1] - x[n - 2])) + (0.5*rightBoundary*(x[n - 1] - x[n - 2]));
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break;
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default:
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throw new NotSupportedException("Invalid Right Boundary Condition.");
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}
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// Build Spline
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double[] dd = SolveTridiagonal(a1, a2, a3, b);
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return InterpolateHermiteSorted(x, y, dd);
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}
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/// <summary>
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/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
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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 CubicSpline InterpolateBoundariesInplace(double[] x, double[] y,
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SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
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SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
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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 InterpolateBoundariesSorted(x, y, leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary);
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}
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/// <summary>
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/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
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/// </summary>
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public static CubicSpline InterpolateBoundaries(IEnumerable<double> x, IEnumerable<double> y,
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SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
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SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
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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 InterpolateBoundariesInplace(x.ToArray(), y.ToArray(), leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary);
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}
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/// <summary>
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/// Create a natural cubic spline interpolation from a set of (x,y) value pairs
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/// and zero second derivatives at the two boundaries, sorted ascendingly by x.
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/// </summary>
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public static CubicSpline InterpolateNaturalSorted(double[] x, double[] y)
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{
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return InterpolateBoundariesSorted(x, y, SplineBoundaryCondition.SecondDerivative, 0.0, SplineBoundaryCondition.SecondDerivative, 0.0);
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}
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/// <summary>
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/// Create a natural cubic spline interpolation from an unsorted set of (x,y) value pairs
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/// and zero second derivatives at the two boundaries.
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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 CubicSpline InterpolateNaturalInplace(double[] x, double[] y)
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{
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return InterpolateBoundariesInplace(x, y, SplineBoundaryCondition.SecondDerivative, 0.0, SplineBoundaryCondition.SecondDerivative, 0.0);
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}
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/// <summary>
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/// Create a natural cubic spline interpolation from an unsorted set of (x,y) value pairs
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/// and zero second derivatives at the two boundaries.
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/// </summary>
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public static CubicSpline InterpolateNatural(IEnumerable<double> x, IEnumerable<double> y)
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{
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return InterpolateBoundaries(x, y, SplineBoundaryCondition.SecondDerivative, 0.0, SplineBoundaryCondition.SecondDerivative, 0.0);
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}
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/// <summary>
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/// Three-Point Differentiation Helper.
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/// </summary>
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/// <param name="xx">Sample Points t.</param>
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/// <param name="yy">Sample Values x(t).</param>
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/// <param name="indexT">Index of the point of the differentiation.</param>
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/// <param name="index0">Index of the first sample.</param>
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/// <param name="index1">Index of the second sample.</param>
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/// <param name="index2">Index of the third sample.</param>
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/// <returns>The derivative approximation.</returns>
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static double DifferentiateThreePoint(double[] xx, double[] yy, int indexT, int index0, int index1, int index2)
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{
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double x0 = yy[index0];
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double x1 = yy[index1];
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double x2 = yy[index2];
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double t = xx[indexT] - xx[index0];
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double t1 = xx[index1] - xx[index0];
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double t2 = xx[index2] - xx[index0];
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double a = (x2 - x0 - (t2/t1*(x1 - x0)))/(t2*(t2 - t1));
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double b = (x1 - x0 - a*t1*t1)/t1;
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return (2*a*t) + b;
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}
|
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/// <summary>
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/// Tridiagonal Solve Helper.
|
/// </summary>
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/// <param name="a">The a-vector[n].</param>
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/// <param name="b">The b-vector[n], will be modified by this function.</param>
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/// <param name="c">The c-vector[n].</param>
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/// <param name="d">The d-vector[n], will be modified by this function.</param>
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/// <returns>The x-vector[n]</returns>
|
static double[] SolveTridiagonal(double[] a, double[] b, double[] c, double[] d)
|
{
|
for (int k = 1; k < a.Length; k++)
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{
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double t = a[k]/b[k - 1];
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b[k] = b[k] - (t*c[k - 1]);
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d[k] = d[k] - (t*d[k - 1]);
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}
|
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var x = new double[a.Length];
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x[x.Length - 1] = d[d.Length - 1]/b[b.Length - 1];
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for (int k = x.Length - 2; k >= 0; k--)
|
{
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x[k] = (d[k] - (c[k]*x[k + 1]))/b[k];
|
}
|
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return x;
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}
|
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/// <summary>
|
/// Gets a value indicating whether the algorithm supports differentiation (interpolated derivative).
|
/// </summary>
|
bool IInterpolation.SupportsDifferentiation => true;
|
|
/// <summary>
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/// Gets a value indicating whether the algorithm supports integration (interpolated quadrature).
|
/// </summary>
|
bool IInterpolation.SupportsIntegration => true;
|
|
/// <summary>
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/// Interpolate at point t.
|
/// </summary>
|
/// <param name="t">Point t to interpolate at.</param>
|
/// <returns>Interpolated value x(t).</returns>
|
public double Interpolate(double t)
|
{
|
int k = LeftSegmentIndex(t);
|
var x = t - _x[k];
|
return _c0[k] + x*(_c1[k] + x*(_c2[k] + x*_c3[k]));
|
}
|
|
/// <summary>
|
/// Differentiate at point t.
|
/// </summary>
|
/// <param name="t">Point t to interpolate at.</param>
|
/// <returns>Interpolated first derivative at point t.</returns>
|
public double Differentiate(double t)
|
{
|
int k = LeftSegmentIndex(t);
|
var x = t - _x[k];
|
return _c1[k] + x*(2*_c2[k] + x*3*_c3[k]);
|
}
|
|
/// <summary>
|
/// Differentiate twice at point t.
|
/// </summary>
|
/// <param name="t">Point t to interpolate at.</param>
|
/// <returns>Interpolated second derivative at point t.</returns>
|
public double Differentiate2(double t)
|
{
|
int k = LeftSegmentIndex(t);
|
var x = t - _x[k];
|
return 2*_c2[k] + x*6*_c3[k];
|
}
|
|
/// <summary>
|
/// Indefinite integral at point t.
|
/// </summary>
|
/// <param name="t">Point t to integrate at.</param>
|
public double Integrate(double t)
|
{
|
int k = LeftSegmentIndex(t);
|
var x = t - _x[k];
|
return _indefiniteIntegral.Value[k] + x*(_c0[k] + x*(_c1[k]/2 + x*(_c2[k]/3 + x*_c3[k]/4)));
|
}
|
|
/// <summary>
|
/// Definite integral between points a and b.
|
/// </summary>
|
/// <param name="a">Left bound of the integration interval [a,b].</param>
|
/// <param name="b">Right bound of the integration interval [a,b].</param>
|
public double Integrate(double a, double b)
|
{
|
return Integrate(b) - Integrate(a);
|
}
|
|
double[] ComputeIndefiniteIntegral()
|
{
|
var integral = new double[_c1.Length];
|
for (int i = 0; i < integral.Length - 1; i++)
|
{
|
double w = _x[i + 1] - _x[i];
|
integral[i + 1] = integral[i] + w*(_c0[i] + w*(_c1[i]/2 + w*(_c2[i]/3 + w*_c3[i]/4)));
|
}
|
|
return integral;
|
}
|
|
/// <summary>
|
/// Find the index of the greatest sample point smaller than t,
|
/// or the left index of the closest segment for extrapolation.
|
/// </summary>
|
int LeftSegmentIndex(double t)
|
{
|
int index = Array.BinarySearch(_x, t);
|
if (index < 0)
|
{
|
index = ~index - 1;
|
}
|
|
return Math.Min(Math.Max(index, 0), _x.Length - 2);
|
}
|
}
|
}
|
