// <copyright file="QuadraticSpline.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) 2009-2013 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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namespace IStation.Numerics.Interpolation
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{
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/// <summary>
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/// Quadratic Spline Interpolation.
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/// </summary>
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/// <remarks>Supports both differentiation and integration.</remarks>
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public class QuadraticSpline : 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 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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public QuadraticSpline(double[] x, double[] c0, double[] c1, double[] c2)
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{
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if (x.Length != c0.Length + 1 || x.Length != c1.Length + 1 || x.Length != c2.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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_indefiniteIntegral = new Lazy<double[]>(ComputeIndefiniteIntegral);
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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 => true;
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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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int k = LeftSegmentIndex(t);
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var x = t - _x[k];
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return _c0[k] + x*(_c1[k] + x*_c2[k]);
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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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int k = LeftSegmentIndex(t);
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return _c1[k] + (t - _x[k])*2*_c2[k];
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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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int k = LeftSegmentIndex(t);
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return 2*_c2[k];
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}
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/// <summary>
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/// Indefinite integral at point t.
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/// </summary>
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/// <param name="t">Point t to integrate at.</param>
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public double Integrate(double t)
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{
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int k = LeftSegmentIndex(t);
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var x = t - _x[k];
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return _indefiniteIntegral.Value[k] + x*(_c0[k] + x*(_c1[k]/2 + x*_c2[k]/3));
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}
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/// <summary>
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/// Definite integral between points a and b.
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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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public double Integrate(double a, double b)
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{
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return Integrate(b) - Integrate(a);
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}
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double[] ComputeIndefiniteIntegral()
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{
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var integral = new double[_c1.Length];
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for (int i = 0; i < integral.Length - 1; i++)
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{
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double w = _x[i + 1] - _x[i];
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integral[i + 1] = integral[i] + w*(_c0[i] + w*(_c1[i]/2 + w*_c2[i]/3));
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}
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return integral;
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}
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/// <summary>
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/// Find the index of the greatest sample point smaller than t,
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/// or the left index of the closest segment for extrapolation.
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/// </summary>
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int LeftSegmentIndex(double t)
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{
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int index = Array.BinarySearch(_x, t);
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if (index < 0)
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{
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index = ~index - 1;
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}
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return Math.Min(Math.Max(index, 0), _x.Length - 2);
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}
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}
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}
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