Christoph Ruegg
17 years ago
11 changed files with 1175 additions and 1 deletions
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// <copyright file="IntegrationTest.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.UnitTests.IntegrationTests |
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{ |
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using System; |
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using Integration; |
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using Integration.Algorithms; |
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using MbUnit.Framework; |
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[TestFixture] |
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public class IntegrationTest |
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{ |
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/// <summary>
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/// Test Function: f(x) = exp(-x/5) (2 + sin(2 * x))
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/// </summary>
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private static double TargetFunctionA(double x) |
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{ |
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return Math.Exp(-x / 5) * (2 + Math.Sin(2 * x)); |
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} |
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private const double StartA = 0; |
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private const double StopA = 10; |
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private const double TargetAreaA = 9.1082396073229965070; |
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[Test] |
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public void TestIntegratePortal() |
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{ |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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Integrate.OnClosedInterval(TargetFunctionA, StartA, StopA), |
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1e-5, |
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"Basic"); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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Integrate.OnClosedInterval(TargetFunctionA, StartA, StopA, 1e-10), |
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1e-10, |
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"Basic Target 1e-10"); |
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} |
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[Test] |
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[Row(1e-5)] |
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[Row(1e-13)] |
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public void TestDoubleExponentialTransformationAlgorithm(double targetRelativeError) |
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{ |
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var algorithm = new DoubleExponentialTransformation(); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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algorithm.Integrate(TargetFunctionA, StartA, StopA, targetRelativeError), |
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targetRelativeError * TargetAreaA, |
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"DET Adaptive {0}", targetRelativeError); |
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} |
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[Test] |
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public void TrapeziumRuleSupportsTwoPointIntegration() |
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{ |
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var algorithm = new NewtonCotesTrapeziumRule(); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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algorithm.IntegrateTwoPoint(TargetFunctionA, StartA, StopA), |
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0.4 * TargetAreaA, |
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"Direct (1 Partition)"); |
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} |
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[Test] |
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[Row(1, 3.5e-1)] |
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[Row(5, 1e-1)] |
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[Row(10, 2e-2)] |
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[Row(50, 6e-4)] |
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[Row(1000, 1.5e-6)] |
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public void TrapeziumRuleSupportsCompositeIntegration(int partitions, double maxRelativeError) |
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{ |
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var algorithm = new NewtonCotesTrapeziumRule(); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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algorithm.IntegrateComposite(TargetFunctionA, StartA, StopA, partitions), |
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maxRelativeError * TargetAreaA, |
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"Composite {0} Partitions", partitions); |
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} |
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[Test] |
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[Row(1e-1)] |
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[Row(1e-5)] |
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[Row(1e-10)] |
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public void TrapeziumRuleSupportsAdaptiveIntegration(double targetRelativeError) |
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{ |
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var algorithm = new NewtonCotesTrapeziumRule(); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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algorithm.IntegrateAdaptive(TargetFunctionA, StartA, StopA, targetRelativeError), |
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targetRelativeError * TargetAreaA, |
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"Adaptive {0}", targetRelativeError); |
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} |
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[Test] |
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public void SimpsonRuleSupportsThreePointIntegration() |
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{ |
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var algorithm = new SimpsonRule(); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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algorithm.IntegrateThreePoint(TargetFunctionA, StartA, StopA), |
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0.2 * TargetAreaA, |
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"Direct (2 Partitions)"); |
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} |
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[Test] |
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[Row(2, 1.7e-1)] |
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[Row(6, 1.2e-1)] |
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[Row(10, 8e-3)] |
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[Row(50, 8e-6)] |
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[Row(1000, 5e-11)] |
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public void SimpsonRuleSupportsCompositeIntegration(int partitions, double maxRelativeError) |
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{ |
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var algorithm = new SimpsonRule(); |
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Assert.AreApproximatelyEqual( |
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TargetAreaA, |
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algorithm.IntegrateComposite(TargetFunctionA, StartA, StopA, partitions), |
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maxRelativeError * TargetAreaA, |
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"Composite {0} Partitions", partitions); |
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} |
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} |
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} |
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@ -0,0 +1,579 @@ |
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// <copyright file="DoubleExponentialTransformation.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.Integration.Algorithms |
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{ |
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using System; |
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using System.Collections.Generic; |
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/// <summary>
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/// Analytic integration algorithm for smooth functions with no discontinuities
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/// or derivative discontinuities and no poles inside the interval.
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/// </summary>
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public class DoubleExponentialTransformation |
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{ |
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private const int NumberOfMaximumLevels = 10; |
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private readonly NewtonCotesTrapeziumRule _trapezium = new NewtonCotesTrapeziumRule(); |
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private IEnumerable<double[]> _levelAbcissas; |
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private IEnumerable<double[]> _levelWeights; |
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#region Precomputed Abcissas and Weights
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private static readonly double[][] PrecomputedAbscissas = |
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new[] |
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{ |
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new[] |
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{ |
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0.00000000000000000000, |
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0.95136796407274694574, |
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0.99997747719246159286, |
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0.99999999999995705839 |
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}, |
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new[] |
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{ |
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0.67427149224843582608, |
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0.99751485645722438683, |
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0.99999998887566488198 |
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}, |
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new[] |
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{ |
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0.37720973816403417379, |
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0.85956905868989663517, |
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0.98704056050737689169, |
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0.99968826402835320905, |
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0.99999920473711471266, |
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0.99999999995285644818 |
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}, |
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new[] |
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{ |
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0.19435700332493543161, |
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0.53914670538796776905, |
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0.78060743898320029925, |
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0.91487926326457461091, |
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0.97396686819567744856, |
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0.99405550663140214329, |
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0.99906519645578584642, |
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0.99990938469514399984, |
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0.99999531604122052843, |
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0.99999989278161241838, |
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0.99999999914270509218, |
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0.99999999999823216531 |
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}, |
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new[] |
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{ |
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0.097923885287832333262, |
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0.28787993274271591456, |
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0.46125354393958570440, |
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0.61027365750063894488, |
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0.73101803479256151149, |
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0.82331700550640237006, |
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0.88989140278426019808, |
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0.93516085752198468323, |
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0.96411216422354729193, |
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0.98145482667733517003, |
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0.99112699244169880223, |
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0.99610866543750854254, |
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0.99845420876769773751, |
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0.99945143443527460584, |
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0.99982882207287494166, |
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0.99995387100562796075, |
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0.99998948201481850361, |
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0.99999801714059543208, |
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0.99999969889415261122, |
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0.99999996423908091534, |
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0.99999999678719909830, |
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0.99999999978973286224, |
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0.99999999999039393352, |
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0.99999999999970809734 |
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}, |
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new[] |
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{ |
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0.049055967305077886315, |
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0.14641798429058794053, |
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0.24156631953888365838, |
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0.33314226457763809244, |
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0.41995211127844715849, |
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0.50101338937930910152, |
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0.57558449063515165995, |
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0.64317675898520470128, |
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0.70355000514714201566, |
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0.75669390863372994941, |
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0.80279874134324126576, |
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0.84221924635075686382, |
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0.87543539763040867837, |
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0.90301328151357387064, |
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0.92556863406861266645, |
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0.94373478605275715685, |
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0.95813602271021369012, |
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0.96936673289691733517, |
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0.97797623518666497298, |
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0.98445883116743083087, |
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0.98924843109013389601, |
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0.99271699719682728538, |
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0.99517602615532735426, |
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0.99688031812819187372, |
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0.99803333631543375402, |
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0.99879353429880589929, |
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0.99928111192179195541, |
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0.99958475035151758732, |
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0.99976797159956083506, |
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0.99987486504878034648, |
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0.99993501992508242369, |
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0.99996759306794345976, |
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0.99998451990227082442, |
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0.99999293787666288565, |
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0.99999693244919035751, |
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0.99999873547186590954, |
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0.99999950700571943689, |
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0.99999981889371276701, |
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0.99999993755407837378, |
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0.99999997987450320175, |
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0.99999999396413420165, |
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0.99999999832336194826, |
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0.99999999957078777261, |
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0.99999999989927772326, |
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0.99999999997845533741, |
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0.99999999999582460688, |
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0.99999999999927152627, |
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0.99999999999988636130 |
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}, |
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new[] |
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{ |
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0.024539763574649160379, |
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0.073525122985671294475, |
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0.12222912220155764235, |
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0.17046797238201051811, |
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0.21806347346971200463, |
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0.26484507658344795046, |
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0.31065178055284596083, |
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0.35533382516507453330, |
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0.39875415046723775644, |
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0.44078959903390086627, |
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0.48133184611690504422, |
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0.52028805069123015958, |
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0.55758122826077823080, |
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0.59315035359195315880, |
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0.62695020805104287950, |
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0.65895099174335012438, |
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0.68913772506166767176, |
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0.71750946748732412721, |
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0.74407838354734739913, |
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0.76886868676824658459, |
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0.79191549237614211447, |
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0.81326360850297385168, |
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0.83296629391941087564, |
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0.85108400798784873261, |
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0.86768317577564598669, |
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0.88283498824466895513, |
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0.89661425428007602579, |
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0.90909831816302043511, |
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0.92036605303195280235, |
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0.93049693799715340631, |
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0.93957022393327475539, |
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0.94766419061515309734, |
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0.95485549580502268541, |
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0.96121861515111640753, |
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0.96682537031235585284, |
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0.97174454156548730892, |
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0.97604156025657673933, |
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0.97977827580061576265, |
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0.98301279148110110558, |
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0.98579936302528343597, |
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0.98818835380074264243, |
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0.99022624046752774694, |
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0.99195566300267761562, |
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0.99341551316926403900, |
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0.99464105571251119672, |
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0.99566407681695316965, |
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0.99651305464025377317, |
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0.99721334704346870224, |
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0.99778739195890653083, |
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0.99825491617199629344, |
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0.99863314864067747762, |
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0.99893703483351217373, |
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0.99917944893488591716, |
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0.99937140114093768690, |
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0.99952223765121720422, |
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0.99963983134560036519, |
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0.99973076151980848263, |
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0.99980048143113838630, |
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0.99985347277311141171, |
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0.99989338654759256426, |
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0.99992317012928932869, |
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0.99994518061445869309, |
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0.99996128480785666613, |
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0.99997294642523223656, |
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0.99998130127012072679, |
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0.99998722128200062811, |
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0.99999136844834487344, |
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0.99999423962761663478, |
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|
0.99999620334716617675, |
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|
0.99999752962380516793, |
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0.99999841381096473542, |
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0.99999899541068996962, |
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0.99999937270733536947, |
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0.99999961398855024275, |
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|
0.99999976602333243312, |
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0.99999986037121459941, |
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0.99999991800479471056, |
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|
0.99999995264266446185, |
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0.99999997311323594362, |
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0.99999998500307631173, |
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0.99999999178645609907, |
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|
0.99999999558563361584, |
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0.99999999767323673790, |
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0.99999999879798350040, |
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|
0.99999999939177687583, |
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|
0.99999999969875436925, |
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|
0.99999999985405611550, |
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|
0.99999999993088839501, |
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0.99999999996803321674, |
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0.99999999998556879008, |
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0.99999999999364632387, |
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0.99999999999727404948, |
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0.99999999999886126543, |
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0.99999999999953722654, |
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0.99999999999981720098, |
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0.99999999999992987953 |
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} |
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}; |
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private static readonly double[][] PrecomputedWeights = |
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new[] |
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{ |
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new[] |
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{ |
||||
|
1.5707963267948966192, |
||||
|
0.230022394514788685, |
||||
|
0.00026620051375271690866, |
||||
|
1.3581784274539090834e-12 |
||||
|
}, |
||||
|
new[] |
||||
|
{ |
||||
|
0.96597657941230114801, |
||||
|
0.018343166989927842087, |
||||
|
2.1431204556943039358e-7 |
||||
|
}, |
||||
|
new[] |
||||
|
{ |
||||
|
1.3896147592472563229, |
||||
|
0.53107827542805397476, |
||||
|
0.076385743570832304188, |
||||
|
0.0029025177479013135936, |
||||
|
0.000011983701363170720047, |
||||
|
1.1631165814255782766e-9 |
||||
|
}, |
||||
|
new[] |
||||
|
{ |
||||
|
1.5232837186347052132, |
||||
|
1.1934630258491569639, |
||||
|
0.73743784836154784136, |
||||
|
0.36046141846934367417, |
||||
|
0.13742210773316772341, |
||||
|
0.039175005493600779072, |
||||
|
0.0077426010260642407123, |
||||
|
0.00094994680428346871691, |
||||
|
0.000062482559240744082891, |
||||
|
1.8263320593710659699e-6, |
||||
|
1.8687282268736410132e-8, |
||||
|
4.9378538776631926964e-11 |
||||
|
}, |
||||
|
new[] |
||||
|
{ |
||||
|
1.5587733555333301451, |
||||
|
1.466014426716965781, |
||||
|
1.297475750424977998, |
||||
|
1.0816349854900704074, |
||||
|
0.85017285645662006895, |
||||
|
0.63040513516474369106, |
||||
|
0.44083323627385823707, |
||||
|
0.290240679312454185, |
||||
|
0.17932441211072829296, |
||||
|
0.10343215422333290062, |
||||
|
0.055289683742240583845, |
||||
|
0.027133510013712003219, |
||||
|
0.012083543599157953493, |
||||
|
0.0048162981439284630173, |
||||
|
0.0016908739981426396472, |
||||
|
0.00051339382406790336017, |
||||
|
0.00013205234125609974879, |
||||
|
0.000028110164327940134749, |
||||
|
4.8237182032615502124e-6, |
||||
|
6.4777566035929719908e-7, |
||||
|
6.5835185127183396672e-8, |
||||
|
4.8760060974240625869e-9, |
||||
|
2.5216347918530148572e-10, |
||||
|
8.6759314149796046502e-12 |
||||
|
}, |
||||
|
new[] |
||||
|
{ |
||||
|
1.5677814313072218572, |
||||
|
1.5438811161769592204, |
||||
|
1.4972262225410362896, |
||||
|
1.4300083548722996676, |
||||
|
1.3452788847662516615, |
||||
|
1.2467012074518577048, |
||||
|
1.1382722433763053734, |
||||
|
1.0240449331118114483, |
||||
|
0.90787937915489531693, |
||||
|
0.79324270082051671787, |
||||
|
0.68306851634426375464, |
||||
|
0.57967810308778764708, |
||||
|
0.48475809121475539287, |
||||
|
0.39938474152571713515, |
||||
|
0.32408253961152890402, |
||||
|
0.258904639514053516, |
||||
|
0.20352399885860174519, |
||||
|
0.15732620348436615027, |
||||
|
0.11949741128869592428, |
||||
|
0.089103139240941462841, |
||||
|
0.065155533432536205042, |
||||
|
0.046668208054846613644, |
||||
|
0.032698732726609031113, |
||||
|
0.022379471063648476483, |
||||
|
0.014937835096050129696, |
||||
|
0.0097072237393916892692, |
||||
|
0.0061300376320830301252, |
||||
|
0.0037542509774318343023, |
||||
|
0.0022250827064786427022, |
||||
|
0.0012733279447082382027, |
||||
|
0.0007018595156842422708, |
||||
|
0.00037166693621677760301, |
||||
|
0.00018856442976700318572, |
||||
|
0.000091390817490710122732, |
||||
|
0.000042183183841757600604, |
||||
|
0.000018481813599879217116, |
||||
|
7.6595758525203162562e-6, |
||||
|
2.9916615878138787094e-6, |
||||
|
1.0968835125901264732e-6, |
||||
|
3.7595411862360630091e-7, |
||||
|
1.1992442782902770219e-7, |
||||
|
3.5434777171421953043e-8, |
||||
|
9.6498888961089633609e-9, |
||||
|
2.4091773256475940779e-9, |
||||
|
5.482835779709497755e-10, |
||||
|
1.1306055347494680536e-10, |
||||
|
2.0989335404511469109e-11, |
||||
|
3.4841937670261059685e-12 |
||||
|
}, |
||||
|
new[] |
||||
|
{ |
||||
|
1.5700420292795931467, |
||||
|
1.5640214037732320999, |
||||
|
1.5520531698454121192, |
||||
|
1.5342817381543034316, |
||||
|
1.5109197230741697127, |
||||
|
1.48224329788553807, |
||||
|
1.4485862549613225916, |
||||
|
1.4103329714462590129, |
||||
|
1.3679105116808964881, |
||||
|
1.3217801174437728579, |
||||
|
1.2724283455378627082, |
||||
|
1.2203581095793582207, |
||||
|
1.1660798699324345766, |
||||
|
1.1101031939653403796, |
||||
|
1.0529288799552666556, |
||||
|
0.99504180404613271514, |
||||
|
0.93690461274566793366, |
||||
|
0.87895234555278212039, |
||||
|
0.82158803526696470334, |
||||
|
0.7651792989089561367, |
||||
|
0.71005590120546898385, |
||||
|
0.65650824613162753076, |
||||
|
0.60478673057840362158, |
||||
|
0.55510187800363350959, |
||||
|
0.5076251588319080997, |
||||
|
0.4624903980553677613, |
||||
|
0.41979566844501548066, |
||||
|
0.37960556938665160999, |
||||
|
0.3419537959230168323, |
||||
|
0.30684590941791694932, |
||||
|
0.27426222968906810637, |
||||
|
0.24416077786983990868, |
||||
|
0.21648020911729617038, |
||||
|
0.19114268413342749532, |
||||
|
0.16805663794826916233, |
||||
|
0.14711941325785693248, |
||||
|
0.12821973363120098675, |
||||
|
0.11123999898874453035, |
||||
|
0.096058391865189467849, |
||||
|
0.082550788110701737654, |
||||
|
0.070592469906866999352, |
||||
|
0.060059642358636300319, |
||||
|
0.05083075757257047107, |
||||
|
0.042787652157725676034, |
||||
|
0.035816505604196436523, |
||||
|
0.029808628117310126969, |
||||
|
0.024661087314753282511, |
||||
|
0.020277183817500123926, |
||||
|
0.016566786254247575375, |
||||
|
0.013446536605285730674, |
||||
|
0.010839937168255907211, |
||||
|
0.0086773307495391815854, |
||||
|
0.0068957859690660035329, |
||||
|
0.0054388997976239984331, |
||||
|
0.0042565295990178580165, |
||||
|
0.0033044669940348302363, |
||||
|
0.0025440657675291729678, |
||||
|
0.0019418357759843675814, |
||||
|
0.0014690143599429791058, |
||||
|
0.0011011261134519383862, |
||||
|
0.00081754101332469493115, |
||||
|
0.00060103987991147422573, |
||||
|
0.00043739495615911687786, |
||||
|
0.00031497209186021200274, |
||||
|
0.00022435965205008549104, |
||||
|
0.00015802788400701191949, |
||||
|
0.00011002112846666697224, |
||||
|
0.000075683996586201477788, |
||||
|
0.000051421497447658802092, |
||||
|
0.0000344921247593431977, |
||||
|
0.000022832118109036146591, |
||||
|
0.000014908514031870608449, |
||||
|
9.5981941283784710776e-6, |
||||
|
6.0899100320949039256e-6, |
||||
|
3.8061983264644899045e-6, |
||||
|
2.3421667208528096843e-6, |
||||
|
1.4183067155493917523e-6, |
||||
|
8.4473756384859863469e-7, |
||||
|
4.9458288702754198508e-7, |
||||
|
2.8449923659159806339e-7, |
||||
|
1.6069394579076224911e-7, |
||||
|
8.9071395140242387124e-8, |
||||
|
4.8420950198072369669e-8, |
||||
|
2.579956822953589238e-8, |
||||
|
1.3464645522302038796e-8, |
||||
|
6.8784610955899001111e-9, |
||||
|
3.4371856744650090511e-9, |
||||
|
1.6788897682161906807e-9, |
||||
|
8.0099784479729665356e-10, |
||||
|
3.7299501843052790038e-10, |
||||
|
1.6939457789411646876e-10, |
||||
|
7.4967397573818224522e-11, |
||||
|
3.230446433325236576e-11, |
||||
|
1.3542512912336274432e-11, |
||||
|
5.5182369468174885821e-12, |
||||
|
2.1835922099233609052e-12 |
||||
|
} |
||||
|
}; |
||||
|
|
||||
|
#endregion
|
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Approximate the integral by the double exponential transformation
|
||||
|
/// </summary>
|
||||
|
public double Integrate( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd, |
||||
|
double targetRelativeError) |
||||
|
{ |
||||
|
if (_levelAbcissas == null) |
||||
|
{ |
||||
|
_levelAbcissas = ProvideLevelAbcissas(); |
||||
|
_levelWeights = ProvideLevelWeights(); |
||||
|
} |
||||
|
|
||||
|
return _trapezium.IntegrateAdaptiveTransformedOdd( |
||||
|
f, |
||||
|
intervalBegin, |
||||
|
intervalEnd, |
||||
|
_levelAbcissas, |
||||
|
_levelWeights, |
||||
|
1, |
||||
|
targetRelativeError); |
||||
|
} |
||||
|
|
||||
|
private static IEnumerable<double[]> ProvideLevelAbcissas() |
||||
|
{ |
||||
|
for (int i = 0; i < NumberOfMaximumLevels; i++) |
||||
|
{ |
||||
|
yield return EvaluateAbcissas(i); |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
private static IEnumerable<double[]> ProvideLevelWeights() |
||||
|
{ |
||||
|
for (int i = 0; i < NumberOfMaximumLevels; i++) |
||||
|
{ |
||||
|
yield return EvaluateWeights(i); |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
private static double[] EvaluateAbcissas(int level) |
||||
|
{ |
||||
|
if (level < PrecomputedAbscissas.Length) |
||||
|
{ |
||||
|
return PrecomputedAbscissas[level]; |
||||
|
} |
||||
|
|
||||
|
double step = level <= 1 ? 1.0 : (1.0 / (2 << (level - 2))); |
||||
|
double offset = level == 0 ? 0.0 : (1.0 / (2 << (level - 1))); |
||||
|
int length = level == 0 ? 4 : (3 << (level - 1)); |
||||
|
|
||||
|
double t = 0; |
||||
|
double[] abcissas = new double[length]; |
||||
|
for (int i = 0; i < abcissas.Length; i++) |
||||
|
{ |
||||
|
double arg = offset + t; |
||||
|
t += step; |
||||
|
|
||||
|
abcissas[i] = Math.Tanh(Constants.PiOver2 * Math.Sinh(arg)); |
||||
|
} |
||||
|
|
||||
|
return abcissas; |
||||
|
} |
||||
|
|
||||
|
private static double[] EvaluateWeights(int level) |
||||
|
{ |
||||
|
if (level < PrecomputedWeights.Length) |
||||
|
{ |
||||
|
return PrecomputedWeights[level]; |
||||
|
} |
||||
|
|
||||
|
double step = level <= 1 ? 1.0 : (1.0 / (2 << (level - 2))); |
||||
|
double offset = level == 0 ? 0.0 : (1.0 / (2 << (level - 1))); |
||||
|
int length = level == 0 ? 4 : (3 << (level - 1)); |
||||
|
|
||||
|
double t = 0; |
||||
|
double[] weights = new double[length]; |
||||
|
for (int i = 0; i < weights.Length; i++) |
||||
|
{ |
||||
|
double arg = offset + t; |
||||
|
t += step; |
||||
|
|
||||
|
// TODO: reuse abcissas as computed in EvaluateAbcissas
|
||||
|
double abcissa = Math.Tanh(Constants.PiOver2 * Math.Sinh(arg)); |
||||
|
weights[i] = Constants.PiOver2 * (1 - (abcissa * abcissa)) * Math.Cosh(arg); |
||||
|
} |
||||
|
|
||||
|
return weights; |
||||
|
} |
||||
|
} |
||||
|
} |
||||
@ -0,0 +1,216 @@ |
|||||
|
// <copyright file="NewtonCotesTrapeziumRule.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.Integration.Algorithms |
||||
|
{ |
||||
|
using System; |
||||
|
using System.Collections.Generic; |
||||
|
using Properties; |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Approximation algorithm for definite integrals by the Trapezium rule of the Newton-Cotes family.
|
||||
|
/// </summary>
|
||||
|
/// <remarks>
|
||||
|
/// <a href="http://en.wikipedia.org/wiki/Trapezium_rule">Wikipedia - Trapezium Rule</a>
|
||||
|
/// </remarks>
|
||||
|
public class NewtonCotesTrapeziumRule |
||||
|
{ |
||||
|
/// <summary>
|
||||
|
/// Direct 2-point approximation of the definite integral in the provided interval by the trapezium rule.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <returns>approximation of the area in the given interval.</returns>
|
||||
|
public double IntegrateTwoPoint( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd) |
||||
|
{ |
||||
|
return (intervalEnd - intervalBegin) / 2 * (f(intervalBegin) + f(intervalEnd)); |
||||
|
} |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Composite N-point approximation of the definite integral in the provided interval by the trapezium rule.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <param name="numberOfPartitions">Number of composite subdivision partitions.</param>
|
||||
|
/// <returns>approximation of the area in the given interval.</returns>
|
||||
|
public double IntegrateComposite( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd, |
||||
|
int numberOfPartitions) |
||||
|
{ |
||||
|
if (numberOfPartitions <= 0) |
||||
|
{ |
||||
|
throw new ArgumentOutOfRangeException("numberOfPartitions", Resources.ArgumentPositive); |
||||
|
} |
||||
|
|
||||
|
double step = (intervalEnd - intervalBegin) / numberOfPartitions; |
||||
|
|
||||
|
double offset = step; |
||||
|
double sum = 0.5 * (f(intervalBegin) + f(intervalEnd)); |
||||
|
for (int i = 0; i < numberOfPartitions - 1; i++) |
||||
|
{ |
||||
|
// NOTE (ruegg, 2009-01-07): Do not combine intervalBegin and offset (numerical stability!)
|
||||
|
sum += f(intervalBegin + offset); |
||||
|
offset += step; |
||||
|
} |
||||
|
|
||||
|
return step * sum; |
||||
|
} |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Adaptive approximation of the definite integral in the provided interval by the trapezium rule.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <param name="targetRelativeError">The expected relative accuracy of the approximation.</param>
|
||||
|
/// <returns>approximation of the area in the given interval.</returns>
|
||||
|
public double IntegrateAdaptive( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd, |
||||
|
double targetRelativeError) |
||||
|
{ |
||||
|
int numberOfPartitions = 1; |
||||
|
double step = intervalEnd - intervalBegin; |
||||
|
double sum = 0.5 * step * (f(intervalBegin) + f(intervalEnd)); |
||||
|
for (int k = 0; k < 20; k++) |
||||
|
{ |
||||
|
double midpointsum = 0; |
||||
|
for (int i = 0; i < numberOfPartitions; i++) |
||||
|
{ |
||||
|
midpointsum += f(intervalBegin + ((i + 0.5) * step)); |
||||
|
} |
||||
|
|
||||
|
midpointsum *= step; |
||||
|
sum = 0.5 * (sum + midpointsum); |
||||
|
step *= 0.5; |
||||
|
numberOfPartitions *= 2; |
||||
|
|
||||
|
if (sum.AlmostEqualWithRelativeError(midpointsum, targetRelativeError)) |
||||
|
{ |
||||
|
break; |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
return sum; |
||||
|
} |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Adaptive approximation of the definite integral by the trapezium rule.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <param name="levelAbscissas">Abscissa vector per level provider.</param>
|
||||
|
/// <param name="levelWeights">Weight vector per level provider.</param>
|
||||
|
/// <param name="levelOneStep">First Level Step</param>
|
||||
|
/// <param name="targetRelativeError">The expected relative accuracy of the approximation.</param>
|
||||
|
/// <returns>approximation of the area in the given interval.</returns>
|
||||
|
public double IntegrateAdaptiveTransformedOdd( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd, |
||||
|
IEnumerable<double[]> levelAbscissas, |
||||
|
IEnumerable<double[]> levelWeights, |
||||
|
double levelOneStep, |
||||
|
double targetRelativeError) |
||||
|
{ |
||||
|
double linearSlope = 0.5 * (intervalEnd - intervalBegin); |
||||
|
double linearOffset = 0.5 * (intervalEnd + intervalBegin); |
||||
|
targetRelativeError /= 5 * linearSlope; |
||||
|
|
||||
|
var abcissasIterator = levelAbscissas.GetEnumerator(); |
||||
|
var weightsIterator = levelWeights.GetEnumerator(); |
||||
|
|
||||
|
double step = levelOneStep; |
||||
|
|
||||
|
// First Level
|
||||
|
abcissasIterator.MoveNext(); |
||||
|
weightsIterator.MoveNext(); |
||||
|
double[] abcissasL1 = abcissasIterator.Current; |
||||
|
double[] weightsL1 = weightsIterator.Current; |
||||
|
|
||||
|
double sum = f(linearOffset) * weightsL1[0]; |
||||
|
for (int i = 1; i < abcissasL1.Length; i++) |
||||
|
{ |
||||
|
sum += weightsL1[i] * (f((linearSlope * abcissasL1[i]) + linearOffset) + f(-(linearSlope * abcissasL1[i]) + linearOffset)); |
||||
|
} |
||||
|
|
||||
|
sum *= step; |
||||
|
|
||||
|
// Additional Levels
|
||||
|
double previousDelta = double.MaxValue; |
||||
|
for (int level = 1; abcissasIterator.MoveNext() && weightsIterator.MoveNext(); level++) |
||||
|
{ |
||||
|
double[] abcissas = abcissasIterator.Current; |
||||
|
double[] weights = weightsIterator.Current; |
||||
|
|
||||
|
double midpointsum = 0; |
||||
|
for (int i = 0; i < abcissas.Length; i++) |
||||
|
{ |
||||
|
midpointsum += weights[i] * (f((linearSlope * abcissas[i]) + linearOffset) + f(-(linearSlope * abcissas[i]) + linearOffset)); |
||||
|
} |
||||
|
|
||||
|
midpointsum *= step; |
||||
|
sum = 0.5 * (sum + midpointsum); |
||||
|
step *= 0.5; |
||||
|
|
||||
|
double delta = Math.Abs(sum - midpointsum); |
||||
|
|
||||
|
if (level == 1) |
||||
|
{ |
||||
|
previousDelta = delta; |
||||
|
continue; |
||||
|
} |
||||
|
|
||||
|
double r = Math.Log(delta) / Math.Log(previousDelta); |
||||
|
previousDelta = delta; |
||||
|
|
||||
|
if (r > 1.9 && r < 2.1) |
||||
|
{ |
||||
|
// convergence region
|
||||
|
delta = Math.Sqrt(delta); |
||||
|
} |
||||
|
|
||||
|
if (sum.AlmostEqualWithRelativeError(midpointsum, delta, targetRelativeError)) |
||||
|
{ |
||||
|
break; |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
return sum * linearSlope; |
||||
|
} |
||||
|
} |
||||
|
} |
||||
@ -0,0 +1,97 @@ |
|||||
|
// <copyright file="SimpsonRule.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.Integration.Algorithms |
||||
|
{ |
||||
|
using System; |
||||
|
using NumberTheory; |
||||
|
using Properties; |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Approximation algorithm for definite integrals by Simpson's rule.
|
||||
|
/// </summary>
|
||||
|
public class SimpsonRule |
||||
|
{ |
||||
|
/// <summary>
|
||||
|
/// Direct 3-point approximation of the definite integral in the provided interval by Simpson's rule.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <returns>approximation of the area in the given interval.</returns>
|
||||
|
public double IntegrateThreePoint( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd) |
||||
|
{ |
||||
|
double midpoint = (intervalEnd + intervalBegin) / 2; |
||||
|
return (intervalEnd - intervalBegin) / 6 * (f(intervalBegin) + f(intervalEnd) + (4 * f(midpoint))); |
||||
|
} |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Composite N-point approximation of the definite integral in the provided interval by Simpson's rule.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <param name="numberOfPartitions">Even number of composite subdivision partitions.</param>
|
||||
|
/// <returns>approximation of the area in the given interval.</returns>
|
||||
|
public double IntegrateComposite( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd, |
||||
|
int numberOfPartitions) |
||||
|
{ |
||||
|
if (numberOfPartitions <= 0) |
||||
|
{ |
||||
|
throw new ArgumentOutOfRangeException("numberOfPartitions", Resources.ArgumentPositive); |
||||
|
} |
||||
|
|
||||
|
if (numberOfPartitions.IsOdd()) |
||||
|
{ |
||||
|
throw new ArgumentException(Resources.ArgumentEven, "numberOfPartitions"); |
||||
|
} |
||||
|
|
||||
|
double step = (intervalEnd - intervalBegin) / numberOfPartitions; |
||||
|
double factor = step / 3; |
||||
|
|
||||
|
double offset = step; |
||||
|
int m = 4; |
||||
|
double sum = f(intervalBegin) + f(intervalEnd); |
||||
|
for (int i = 0; i < numberOfPartitions - 1; i++) |
||||
|
{ |
||||
|
// NOTE (ruegg, 2009-01-07): Do not combine intervalBegin and offset (numerical stability)
|
||||
|
sum += m * f(intervalBegin + offset); |
||||
|
m = 6 - m; |
||||
|
offset += step; |
||||
|
} |
||||
|
|
||||
|
return factor * sum; |
||||
|
} |
||||
|
} |
||||
|
} |
||||
@ -0,0 +1,83 @@ |
|||||
|
// <copyright file="Integrate.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.Integration |
||||
|
{ |
||||
|
using System; |
||||
|
using Algorithms; |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Numeric Integration (Quadrature).
|
||||
|
/// </summary>
|
||||
|
public static class Integrate |
||||
|
{ |
||||
|
static readonly DoubleExponentialTransformation Det = new DoubleExponentialTransformation(); |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Approximation of the definite interval of an analytic smooth function on a closed interval.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
/// <param name="targetAbsoluteError">The expected relative accuracy of the approximation.</param>
|
||||
|
public static |
||||
|
double |
||||
|
OnClosedInterval( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd, |
||||
|
double targetAbsoluteError) |
||||
|
{ |
||||
|
return Det.Integrate( |
||||
|
f, |
||||
|
intervalBegin, |
||||
|
intervalEnd, |
||||
|
targetAbsoluteError); |
||||
|
} |
||||
|
|
||||
|
/// <summary>
|
||||
|
/// Approximation of the definite interval of an analytic smooth function on a closed interval.
|
||||
|
/// </summary>
|
||||
|
/// <param name="f">The analytic smooth function to integrate.</param>
|
||||
|
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
|
||||
|
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
|
||||
|
public static |
||||
|
double |
||||
|
OnClosedInterval( |
||||
|
Func<double, double> f, |
||||
|
double intervalBegin, |
||||
|
double intervalEnd) |
||||
|
{ |
||||
|
return Det.Integrate( |
||||
|
f, |
||||
|
intervalBegin, |
||||
|
intervalEnd, |
||||
|
1e-8); |
||||
|
} |
||||
|
} |
||||
|
} |
||||
Loading…
Reference in new issue