Christoph Ruegg
17 years ago
5 changed files with 465 additions and 0 deletions
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// <copyright file="CubicSplineInterpolation.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://mathnet.opensourcedotnet.info
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//
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// Copyright (c) 2009 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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namespace MathNet.Numerics.Interpolation.Algorithms |
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{ |
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using System; |
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using System.Collections.Generic; |
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using Properties; |
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/// <summary>
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/// Cubic Spline Interpolation Algorithm with continuous first and second derivatives.
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/// </summary>
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/// <remarks>
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/// This algorithm supports both differentiation and integration.
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/// </remarks>
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public class CubicSplineInterpolation : IInterpolation |
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{ |
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/// <summary>
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/// Internal Spline Interpolation
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/// </summary>
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private readonly CubicHermiteSplineInterpolation _spline; |
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/// <summary>
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/// Initializes a new instance of the CubicSplineInterpolation class.
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/// </summary>
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public CubicSplineInterpolation() |
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{ |
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_spline = new CubicHermiteSplineInterpolation(); |
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} |
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/// <summary>
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/// Initializes a new instance of the CubicSplineInterpolation class.
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/// </summary>
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/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
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/// <param name="sampleValues">Sample Values x(t)</param>
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public CubicSplineInterpolation( |
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IList<double> samplePoints, |
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IList<double> sampleValues) |
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{ |
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_spline = new CubicHermiteSplineInterpolation(); |
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Initialize(samplePoints, sampleValues); |
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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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/// <seealso cref="Differentiate(double)"/>
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/// <seealso cref="Differentiate(double, out double, out double)"/>
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bool IInterpolation.SupportsDifferentiation |
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{ |
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get { return true; } |
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} |
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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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/// <seealso cref="Integrate"/>
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bool IInterpolation.SupportsIntegration |
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{ |
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get { return true; } |
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} |
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/// <summary>
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/// Initialize the interpolation method with the given spline coefficients (sorted by the sample points t).
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/// </summary>
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/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
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/// <param name="sampleValues">Sample Values x(t)</param>
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public void Initialize( |
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IList<double> samplePoints, |
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IList<double> sampleValues) |
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{ |
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double[] derivatives = EvaluateSplineDerivatives( |
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samplePoints, |
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sampleValues, |
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SplineBoundaryCondition.SecondDerivative, |
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0.0, |
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SplineBoundaryCondition.SecondDerivative, |
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0.0); |
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_spline.Initialize(samplePoints, sampleValues, derivatives); |
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} |
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/// <summary>
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/// Initialize the interpolation method with the given spline coefficients (sorted by the sample points t).
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/// </summary>
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/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
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/// <param name="sampleValues">Sample Values x(t)</param>
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/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
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/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
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/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
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/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
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public void Initialize( |
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IList<double> samplePoints, |
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IList<double> sampleValues, |
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SplineBoundaryCondition leftBoundaryCondition, |
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double leftBoundary, |
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SplineBoundaryCondition rightBoundaryCondition, |
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double rightBoundary) |
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{ |
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double[] derivatives = EvaluateSplineDerivatives( |
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samplePoints, |
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sampleValues, |
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leftBoundaryCondition, |
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leftBoundary, |
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rightBoundaryCondition, |
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rightBoundary); |
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_spline.Initialize(samplePoints, sampleValues, derivatives); |
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} |
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/// <summary>
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/// Evaluate the spline derivatives as used
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/// internally by this interpolation algorithm.
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/// </summary>
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/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
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/// <param name="sampleValues">Sample Values x(t)</param>
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/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
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/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
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/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
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/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
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/// <returns>Spline Derivative Vector</returns>
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public static double[] EvaluateSplineDerivatives( |
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IList<double> samplePoints, |
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IList<double> sampleValues, |
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SplineBoundaryCondition leftBoundaryCondition, |
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double leftBoundary, |
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SplineBoundaryCondition rightBoundaryCondition, |
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double rightBoundary) |
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{ |
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if (null == samplePoints) |
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{ |
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throw new ArgumentNullException("samplePoints"); |
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} |
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if (null == sampleValues) |
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{ |
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throw new ArgumentNullException("sampleValues"); |
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} |
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if (samplePoints.Count < 2) |
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{ |
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throw new ArgumentOutOfRangeException("samplePoints"); |
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} |
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if (samplePoints.Count != sampleValues.Count) |
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{ |
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throw new ArgumentException(Resources.ArgumentVectorsSameLengths); |
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} |
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int n = samplePoints.Count; |
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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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double[] a1 = new double[n]; |
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double[] a2 = new double[n]; |
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double[] a3 = new double[n]; |
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double[] 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 * (sampleValues[1] - sampleValues[0]) / (samplePoints[1] - samplePoints[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 * ((sampleValues[1] - sampleValues[0]) / (samplePoints[1] - samplePoints[0]))) - (0.5 * leftBoundary * (samplePoints[1] - samplePoints[0])); |
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break; |
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default: |
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throw new NotSupportedException(Resources.InvalidLeftBoundaryCondition); |
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} |
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// Central Conditions
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for (int i = 1; i < samplePoints.Count - 1; i++) |
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{ |
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a1[i] = samplePoints[i + 1] - samplePoints[i]; |
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a2[i] = 2 * (samplePoints[i + 1] - samplePoints[i - 1]); |
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a3[i] = samplePoints[i] - samplePoints[i - 1]; |
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b[i] = (3 * (sampleValues[i] - sampleValues[i - 1]) / (samplePoints[i] - samplePoints[i - 1]) * (samplePoints[i + 1] - samplePoints[i])) + (3 * (sampleValues[i + 1] - sampleValues[i]) / (samplePoints[i + 1] - samplePoints[i]) * (samplePoints[i] - samplePoints[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 * (sampleValues[n - 1] - sampleValues[n - 2]) / (samplePoints[n - 1] - samplePoints[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 * (sampleValues[n - 1] - sampleValues[n - 2]) / (samplePoints[n - 1] - samplePoints[n - 2])) + (0.5 * rightBoundary * (samplePoints[n - 1] - samplePoints[n - 2])); |
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break; |
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default: |
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throw new NotSupportedException(Resources.InvalidRightBoundaryCondition); |
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} |
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// Build Spline
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return SolveTridiagonal(a1, a2, a3, b); |
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} |
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/// <summary>
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/// Evaluate the spline coefficients as used
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/// internally by this interpolation algorithm.
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/// </summary>
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/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
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/// <param name="sampleValues">Sample Values x(t)</param>
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/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
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/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
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/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
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/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
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/// <returns>Spline Coefficient Vector</returns>
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public static double[] EvaluateSplineCoefficients( |
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IList<double> samplePoints, |
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IList<double> sampleValues, |
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SplineBoundaryCondition leftBoundaryCondition, |
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double leftBoundary, |
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SplineBoundaryCondition rightBoundaryCondition, |
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double rightBoundary) |
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{ |
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double[] derivatives = EvaluateSplineDerivatives( |
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samplePoints, |
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sampleValues, |
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leftBoundaryCondition, |
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leftBoundary, |
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rightBoundaryCondition, |
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rightBoundary); |
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return CubicHermiteSplineInterpolation.EvaluateSplineCoefficients( |
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samplePoints, |
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sampleValues, |
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derivatives); |
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} |
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/// <summary>
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/// Tridiagonal Solve Helper.
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/// </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>
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private static double[] SolveTridiagonal( |
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double[] a, |
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double[] b, |
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double[] c, |
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double[] d) |
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{ |
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double[] x = new double[a.Length]; |
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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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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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{ |
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x[k] = (d[k] - (c[k] * x[k + 1])) / b[k]; |
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} |
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return x; |
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} |
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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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return _spline.Interpolate(t); |
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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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/// <seealso cref="IInterpolation.SupportsDifferentiation"/>
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/// <seealso cref="Differentiate(double, out double, out double)"/>
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public double Differentiate(double t) |
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{ |
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return _spline.Differentiate(t); |
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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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/// <param name="interpolatedValue">Interpolated value x(t)</param>
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/// <param name="secondDerivative">Interpolated second derivative at point t.</param>
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/// <returns>Interpolated first derivative at point t.</returns>
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/// <seealso cref="IInterpolation.SupportsDifferentiation"/>
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/// <seealso cref="Differentiate(double)"/>
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public double Differentiate( |
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double t, |
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out double interpolatedValue, |
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out double secondDerivative) |
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{ |
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return _spline.Differentiate(t, out interpolatedValue, out secondDerivative); |
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} |
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/// <summary>
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/// Integrate up to point t.
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/// </summary>
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/// <param name="t">Right bound of the integration interval [a,t].</param>
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/// <returns>Interpolated definite integral over the interval [a,t].</returns>
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/// <seealso cref="IInterpolation.SupportsIntegration"/>
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public double Integrate(double t) |
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{ |
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return _spline.Integrate(t); |
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} |
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} |
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} |
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@ -0,0 +1,56 @@ |
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// <copyright file="SplineBoundaryCondition.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://mathnet.opensourcedotnet.info
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//
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// Copyright (c) 2009 Math.NET
|
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//
|
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// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
|||
//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
|||
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// 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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namespace MathNet.Numerics.Interpolation |
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{ |
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/// <summary>
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/// Left and right boundary conditions.
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/// </summary>
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public enum SplineBoundaryCondition |
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{ |
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/// <summary>
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/// Natural Boundary (Zero second derivative).
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/// </summary>
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Natural = 0, |
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/// <summary>
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/// Parabolically Terminated boundary.
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/// </summary>
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ParabolicallyTerminated, |
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/// <summary>
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/// Fixed first derivative at the boundary.
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/// </summary>
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FirstDerivative, |
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/// <summary>
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/// Fixed second derivative at the boundary.
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/// </summary>
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SecondDerivative |
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} |
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} |
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