interpolation: ported akima spline

Signed-off-by: Christoph Ruegg <git@cdrnet.ch>

src/Managed.UnitTests/InterpolationTests/InterpolationTest.cs

@@ -116,5 +116,15 @@ namespace MathNet.Numerics.UnitTests.InterpolationTests
             PolynomialBehavior = false,
             RationalBehavior = false
         };
+
+        [VerifyContract]
+        public readonly IContract AkimaSplineContractTests = new InterpolationContract<AkimaSplineInterpolation>()
+        {
+            Factory = (t, x) => new AkimaSplineInterpolation(t, x),
+            MinimumSampleCount = 5,
+            LinearBehavior = true,
+            PolynomialBehavior = false,
+            RationalBehavior = false
+        };
     }
 }

src/Managed/Interpolation/Algorithms/AkimaSplineInterpolation.cs

@@ -0,0 +1,280 @@
+// <copyright file="AkimaSplineInterpolation.cs" company="Math.NET">
+// Math.NET Numerics, part of the Math.NET Project
+// http://mathnet.opensourcedotnet.info
+//
+// Copyright (c) 2009 Math.NET
+//
+// 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
+// OTHER DEALINGS IN THE SOFTWARE.
+// </copyright>
+
+namespace MathNet.Numerics.Interpolation.Algorithms
+{
+    using System;
+    using System.Collections.Generic;
+    using Properties;
+
+    /// <summary>
+    /// Akima Spline Interpolation Algorithm.
+    /// </summary>
+    /// <remarks>
+    /// This algorithm supports both differentiation and integration.
+    /// </remarks>
+    public class AkimaSplineInterpolation : IInterpolation
+    {
+        /// <summary>
+        /// Internal Spline Interpolation
+        /// </summary>
+        private readonly CubicHermiteSplineInterpolation _spline;
+
+        /// <summary>
+        /// Initializes a new instance of the AkimaSplineInterpolation class.
+        /// </summary>
+        public AkimaSplineInterpolation()
+        {
+            _spline = new CubicHermiteSplineInterpolation();
+        }
+
+        /// <summary>
+        /// Initializes a new instance of the AkimaSplineInterpolation class.
+        /// </summary>
+        /// <param name="samplePoints">Sample Points t, sorted ascending.</param>
+        /// <param name="sampleValues">Sample Values x(t)</param>
+        public AkimaSplineInterpolation(
+            IList<double> samplePoints,
+            IList<double> sampleValues)
+        {
+            _spline = new CubicHermiteSplineInterpolation();
+            Initialize(samplePoints, sampleValues);
+        }
+
+        /// <summary>
+        /// Gets a value indicating whether the algorithm supports differentiation (interpolated derivative).
+        /// </summary>
+        /// <seealso cref="Differentiate(double)"/>
+        /// <seealso cref="Differentiate(double, out double, out double)"/>
+        bool IInterpolation.SupportsDifferentiation
+        {
+            get { return true; }
+        }
+
+        /// <summary>
+        /// Gets a value indicating whether the algorithm supports integration (interpolated quadrature).
+        /// </summary>
+        /// <seealso cref="Integrate"/>
+        bool IInterpolation.SupportsIntegration
+        {
+            get { return true; }
+        }
+
+        /// <summary>
+        /// Initialize the interpolation method with the given spline coefficients (sorted by the sample points t).
+        /// </summary>
+        /// <param name="samplePoints">Sample Points t, sorted ascending.</param>
+        /// <param name="sampleValues">Sample Values x(t)</param>
+        public void Initialize(
+            IList<double> samplePoints,
+            IList<double> sampleValues)
+        {
+            double[] derivatives = EvaluateSplineDerivatives(
+                samplePoints,
+                sampleValues);
+
+            _spline.Initialize(samplePoints, sampleValues, derivatives);
+        }
+
+        /// <summary>
+        /// Evaluate the spline derivatives as used
+        /// internally by this interpolation algorithm.
+        /// </summary>
+        /// <param name="samplePoints">Sample Points t, sorted ascending.</param>
+        /// <param name="sampleValues">Sample Values x(t)</param>
+        /// <returns>Spline Derivative Vector</returns>
+        public static double[] EvaluateSplineDerivatives(
+            IList<double> samplePoints,
+            IList<double> sampleValues)
+        {
+            if (null == samplePoints)
+            {
+                throw new ArgumentNullException("samplePoints");
+            }
+
+            if (null == sampleValues)
+            {
+                throw new ArgumentNullException("sampleValues");
+            }
+
+            if (samplePoints.Count < 5)
+            {
+                throw new ArgumentOutOfRangeException("samplePoints");
+            }
+
+            if (samplePoints.Count != sampleValues.Count)
+            {
+                throw new ArgumentException(Resources.ArgumentVectorsSameLengths);
+            }
+
+            int n = samplePoints.Count;
+
+            /* Prepare divided differences (diff) and weights (w) */
+
+            var differences = new double[n - 1];
+            var weights = new double[n - 1];
+
+            for (int i = 0; i < differences.Length; i++)
+            {
+                differences[i] = (sampleValues[i + 1] - sampleValues[i]) / (samplePoints[i + 1] - samplePoints[i]);
+            }
+
+            for (int i = 1; i < weights.Length; i++)
+            {
+                weights[i] = Math.Abs(differences[i] - differences[i - 1]);
+            }
+
+            /* Prepare Hermite interpolation scheme */
+
+            var derivatives = new double[n];
+
+            for (int i = 2; i < derivatives.Length - 2; i++)
+            {
+                derivatives[i] =
+                    weights[i - 1].AlmostZero() && weights[i + 1].AlmostZero()
+                        ? (((samplePoints[i + 1] - samplePoints[i]) * differences[i - 1])
+                           + ((samplePoints[i] - samplePoints[i - 1]) * differences[i]))
+                          / (samplePoints[i + 1] - samplePoints[i - 1])
+                        : ((weights[i + 1] * differences[i - 1])
+                           + (weights[i - 1] * differences[i]))
+                          / (weights[i + 1] + weights[i - 1]);
+            }
+
+            derivatives[0] = DifferentiateThreePoint(samplePoints, sampleValues, 0, 0, 1, 2);
+            derivatives[1] = DifferentiateThreePoint(samplePoints, sampleValues, 1, 0, 1, 2); 
+            derivatives[n - 2] = DifferentiateThreePoint(samplePoints, sampleValues, n - 2, n - 3, n - 2, n - 1);
+            derivatives[n - 1] = DifferentiateThreePoint(samplePoints, sampleValues, n - 1, n - 3, n - 2, n - 1);
+
+            /* Build Akima spline using Hermite interpolation scheme */
+
+            return derivatives;
+        }
+
+        /// <summary>
+        /// Evaluate the spline coefficients as used
+        /// internally by this interpolation algorithm.
+        /// </summary>
+        /// <param name="samplePoints">Sample Points t, sorted ascending.</param>
+        /// <param name="sampleValues">Sample Values x(t)</param>
+        /// <returns>Spline Coefficient Vector</returns>
+        public static double[] EvaluateSplineCoefficients(
+            IList<double> samplePoints,
+            IList<double> sampleValues)
+        {
+            double[] derivatives = EvaluateSplineDerivatives(
+                samplePoints,
+                sampleValues);
+
+            return CubicHermiteSplineInterpolation.EvaluateSplineCoefficients(
+                samplePoints,
+                sampleValues,
+                derivatives);
+        }
+
+        /// <summary>
+        /// Three-Point Differentiation Helper.
+        /// </summary>
+        /// <param name="samplePoints">Sample Points t.</param>
+        /// <param name="sampleValues">Sample Values x(t).</param>
+        /// <param name="indexT">Index of the point of the differentiation.</param>
+        /// <param name="index0">Index of the first sample.</param>
+        /// <param name="index1">Index of the second sample.</param>
+        /// <param name="index2">Index of the third sample.</param>
+        /// <returns>The derivative approximation.</returns>
+        private static double DifferentiateThreePoint(
+            IList<double> samplePoints,
+            IList<double> sampleValues,
+            int indexT,
+            int index0,
+            int index1,
+            int index2)
+        {
+            double x0 = sampleValues[index0];
+            double x1 = sampleValues[index1];
+            double x2 = sampleValues[index2];
+
+            double t = samplePoints[indexT] - samplePoints[index0];
+            double t1 = samplePoints[index1] - samplePoints[index0];
+            double t2 = samplePoints[index2] - samplePoints[index0];
+
+            double a = (x2 - x0 - (t2 / t1 * (x1 - x0))) / ((t2 * t2) - (t1 * t2));
+            double b = (x1 - x0 - (a * t1 * t1)) / t1;
+            return (2 * a * t) + b;
+        }
+
+        /// <summary>
+        /// 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)
+        {
+            return _spline.Interpolate(t);
+        }
+
+        /// <summary>
+        /// Differentiate at point t.
+        /// </summary>
+        /// <param name="t">Point t to interpolate at.</param>
+        /// <returns>Interpolated first derivative at point t.</returns>
+        /// <seealso cref="IInterpolation.SupportsDifferentiation"/>
+        /// <seealso cref="Differentiate(double, out double, out double)"/>
+        public double Differentiate(double t)
+        {
+            return _spline.Differentiate(t);
+        }
+
+        /// <summary>
+        /// Differentiate at point t.
+        /// </summary>
+        /// <param name="t">Point t to interpolate at.</param>
+        /// <param name="interpolatedValue">Interpolated value x(t)</param>
+        /// <param name="secondDerivative">Interpolated second derivative at point t.</param>
+        /// <returns>Interpolated first derivative at point t.</returns>
+        /// <seealso cref="IInterpolation.SupportsDifferentiation"/>
+        /// <seealso cref="Differentiate(double)"/>
+        public double Differentiate(
+            double t,
+            out double interpolatedValue,
+            out double secondDerivative)
+        {
+            return _spline.Differentiate(t, out interpolatedValue, out secondDerivative);
+        }
+
+        /// <summary>
+        /// Integrate up to point t.
+        /// </summary>
+        /// <param name="t">Right bound of the integration interval [a,t].</param>
+        /// <returns>Interpolated definite integral over the interval [a,t].</returns>
+        /// <seealso cref="IInterpolation.SupportsIntegration"/>
+        public double Integrate(double t)
+        {
+            return _spline.Integrate(t);
+        }
+    }
+}

src/Managed/Managed.csproj

@@ -58,6 +58,7 @@
     <Compile Include="Integration\Algorithms\SimpsonRule.cs" />
     <Compile Include="Integration\Algorithms\NewtonCotesTrapeziumRule.cs" />
     <Compile Include="Integration\Integrate.cs" />
+    <Compile Include="Interpolation\Algorithms\AkimaSplineInterpolation.cs" />
     <Compile Include="Interpolation\Algorithms\BarycentricInterpolation.cs" />
     <Compile Include="Interpolation\Algorithms\BulirschStoerRationalInterpolation.cs" />
     <Compile Include="Interpolation\Algorithms\CubicSplineInterpolation.cs" />

src/Native/Native.csproj

@@ -83,6 +83,9 @@
     <Compile Include="..\Managed\Integration\Integrate.cs">
       <Link>Integration\Integrate.cs</Link>
     </Compile>
+    <Compile Include="..\Managed\Interpolation\Algorithms\AkimaSplineInterpolation.cs">
+      <Link>Interpolation\Algorithms\AkimaSplineInterpolation.cs</Link>
+    </Compile>
     <Compile Include="..\Managed\Interpolation\Algorithms\BarycentricInterpolation.cs">
       <Link>Interpolation\Algorithms\BarycentricInterpolation.cs</Link>
     </Compile>