add Adams-Bashforth First Order(same as Euler method) solver and test

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\AdamsBashforth.cs" />
     <Compile Include="OdeSolvers\RungeKutta.cs" />
     <Compile Include="Precision.Comparison.cs" />
     <Compile Include="Precision.Equality.cs" />

src/Numerics/OdeSolvers/AdamsBashforth.cs

@@ -0,0 +1,78 @@
+// <copyright file="AdamsBashforth.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>
+using System;
+using System.Collections.Generic;
+using System.Linq;
+using System.Text;
+
+namespace MathNet.Numerics.OdeSolvers
+{
+    public static class AdamsBashforth
+    {
+
+        /// <summary>
+        /// First Order AB method(same as Forward Euler)
+        /// </summary>
+        /// <param name="y0">Initial value</param>
+        /// <param name="start">Start Time</param>
+        /// <param name="end">End Time</param>
+        /// <param name="N">Number of subintervals</param>
+        /// <param name="f">ode model</param>
+        /// <returns></returns>
+        public static double[] FirstOrder(double y0, double start, double end, int N, Func<double, double, double> f)
+        {
+            double dt = (end - start) / (N - 1);
+            double t = start;
+            double[] y = new double[N];
+            y[0] = y0;
+            for (int i = 1; i < N; i++)
+            {
+                y[i] = y0 + dt * f(t, y0);
+                t += dt;
+                y0 = y[i];
+            }
+            return y;
+        }
+
+        public static double[] SecondOrder()
+        {
+            return null;
+        }
+
+        public static double[] ThirdOrder()
+        {
+            return null;
+        }
+
+        public static double[] FourthOrder()
+        {
+            return null;
+        }
+    }
+}

src/UnitTests/OdeSolvers/OdeSolverTest.cs

@@ -40,6 +40,26 @@ namespace MathNet.Numerics.UnitTests.OdeSolvers
     [TestFixture, Category("OdeSolver")]
     public class OdeSolverTest
     {
+        [Test]
+        public void AB1Test()
+        {
+            Func<double, double, double> ode = (t, y) => t + 2 * y * t;
+            Func<double, double> sol = (t) => 0.5 * (Math.Exp(t * t) - 1);
+            double ratio = double.NaN;
+            double error = 0;
+            double oldError = 0;
+            for (int k = 0; k < 4; k++)
+            {
+                double y0 = 0;
+                double[] y_t = AdamsBashforth.FirstOrder(y0, 0, 2, Convert.ToInt32(Math.Pow(2, k + 8)), ode);
+                error = Math.Abs(sol(2) - y_t.Last());
+                if (oldError != 0)
+                    ratio = Math.Log(oldError / error, 2);
+                oldError = error;
+                Console.WriteLine(string.Format("{0}, {1}", error, ratio));
+            }
+            Assert.AreEqual(1, ratio, 0.01);// Check error convergence order
+        }
         /// <summary>
         /// Runge-Kutta second order method for first order ODE.
         /// </summary>