add RK2 and its test to verify RK2 erro convergence order

src/Numerics/Numerics.csproj

@@ -112,6 +112,7 @@
     <Compile Include="LinearAlgebra\MatrixExtensions.cs" />
     <Compile Include="LinearAlgebra\VectorExtensions.cs" />
     <Compile Include="LinearRegression\Options.cs" />
+    <Compile Include="OdeSolvers\OdeSolvers.cs" />
     <Compile Include="Precision.Comparison.cs" />
     <Compile Include="Precision.Equality.cs" />
     <Compile Include="Distributions\Bernoulli.cs" />

src/Numerics/OdeSolvers/OdeSolvers.cs

@@ -0,0 +1,59 @@
+// <copyright file="OdeSolvers.cs" company="Math.NET">
+// Math.NET Numerics, part of the Math.NET Project
+// http://numerics.mathdotnet.com
+// http://github.com/mathnet/mathnet-numerics
+//
+// Copyright (c) 2009-2013 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>
+
+using System;
+using MathNet.Numerics.Properties;
+
+namespace MathNet.Numerics.OdeSolvers
+{
+    /// <summary>
+    /// ODE Solver Algorithms
+    /// </summary>
+    public static class RungeKutta
+    {
+        public static double[] SecondOrder(double y0, double start, double end, int N, Func<double, double, double> f)
+        {
+            double dt = (end - start) / (N - 1);
+            double k1 = 0;
+            double k2 = 0;
+            double t = start;
+            double[] y = new double[N];
+            y[0] = y0;
+            for (int i = 1; i < N; i++)
+            {
+                k1 = f(t, y0);
+                k2 = f(t + dt, y0 + k1 * dt);
+                y[i] = y0 + dt * 0.5 * (k1 + k2);
+                t += dt;
+                y0 = y[i];
+            }
+            return y;
+        }
+    }
+}

src/UnitTests/OdeSolvers/OdeSolverTest.cs

@@ -0,0 +1,61 @@
+// <copyright file="OdeSolverTest.cs" company="Math.NET">
+// Math.NET Numerics, part of the Math.NET Project
+// http://numerics.mathdotnet.com
+// http://github.com/mathnet/mathnet-numerics
+//
+// Copyright (c) 2009-2016 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>
+
+extern alias NUnitFramework;
+
+using System;
+using MathNet.Numerics.OdeSolvers;
+using NUnitFramework.NUnit.Framework;
+using System.Linq;
+
+namespace MathNet.Numerics.UnitTests.OdeSolvers
+{
+    /// <summary>
+    /// ODE Solver tests.
+    /// </summary>
+    [TestFixture, Category("OdeSolver")]
+    public class OdeSolverTest
+    {
+        /// <summary>
+        /// Runge-Kutta second order method for first order ODE.
+        /// </summary>
+        [Test]
+        public void RK2Test()
+        {
+            Func<double, double, double> ode = (t, y) => t + 2 * y * t;
+            Func<double, double> sol = (t) => 0.5 * (Math.Exp(t * t) - 1);
+            for (int k = 0; k < 4; k++)
+            {
+                double y0 = 0;
+                double[] y_t = RungeKutta.SecondOrder(y0, 0, 2, Convert.ToInt32(Math.Pow(2, k + 6)), ode);
+                Console.WriteLine(Math.Abs(sol(2) - y_t.Last()));
+            }
+        }
+    }
+}

src/UnitTests/UnitTests.csproj

@@ -362,6 +362,7 @@
     <Compile Include="LinearAlgebraTests\MatrixStorageCombinatorsTests.cs" />
     <Compile Include="LinearAlgebraTests\VectorStorageCombinatorsTests.cs" />
     <Compile Include="LinearAlgebraTests\VectorToStringTests.cs" />
+    <Compile Include="OdeSolvers\OdeSolverTest.cs" />
     <Compile Include="Random\SystemRandomSourceTests.cs" />
     <Compile Include="RootFindingTests\BisectionTest.cs" />
     <Compile Include="PermutationTest.cs" />