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diff --git a/src/main/java/org/apache/commons/math/ode/package.html b/src/main/java/org/apache/commons/math/ode/package.html new file mode 100644 index 0000000..d390204 --- /dev/null +++ b/src/main/java/org/apache/commons/math/ode/package.html @@ -0,0 +1,167 @@ +<html> +<!-- + Licensed to the Apache Software Foundation (ASF) under one or more + contributor license agreements. See the NOTICE file distributed with + this work for additional information regarding copyright ownership. + The ASF licenses this file to You under the Apache License, Version 2.0 + (the "License"); you may not use this file except in compliance with + the License. You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. + --> + <!-- $Revision: 920131 $ --> +<body> +<p> +This package provides classes to solve Ordinary Differential Equations problems. +</p> + +<p> +This package solves Initial Value Problems of the form +<code>y'=f(t,y)</code> with <code>t<sub>0</sub></code> and +<code>y(t<sub>0</sub>)=y<sub>0</sub></code> known. The provided +integrators compute an estimate of <code>y(t)</code> from +<code>t=t<sub>0</sub></code> to <code>t=t<sub>1</sub></code>. +If in addition to <code>y(t)</code> users need to get the +derivatives with respect to the initial state +<code>dy(t)/dy(t<sub>0</sub>)</code> or the derivatives with +respect to some ODE parameters <code>dy(t)/dp</code>, then the +classes from the <a href="./jacobians/package-summary.html"> +org.apache.commons.math.ode.jacobians</a> package must be used +instead of the classes in this package. +</p> + +<p> +All integrators provide dense output. This means that besides +computing the state vector at discrete times, they also provide a +cheap mean to get the state between the time steps. They do so through +classes extending the {@link +org.apache.commons.math.ode.sampling.StepInterpolator StepInterpolator} +abstract class, which are made available to the user at the end of +each step. +</p> + +<p> +All integrators handle multiple discrete events detection based on switching +functions. This means that the integrator can be driven by user specified +discrete events. The steps are shortened as needed to ensure the events occur +at step boundaries (even if the integrator is a fixed-step +integrator). When the events are triggered, integration can be stopped +(this is called a G-stop facility), the state vector can be changed, +or integration can simply go on. The latter case is useful to handle +discontinuities in the differential equations gracefully and get +accurate dense output even close to the discontinuity. +</p> + +<p> +The user should describe his problem in his own classes +(<code>UserProblem</code> in the diagram below) which should implement +the {@link org.apache.commons.math.ode.FirstOrderDifferentialEquations +FirstOrderDifferentialEquations} interface. Then he should pass it to +the integrator he prefers among all the classes that implement the +{@link org.apache.commons.math.ode.FirstOrderIntegrator +FirstOrderIntegrator} interface. +</p> + +<p> +The solution of the integration problem is provided by two means. The +first one is aimed towards simple use: the state vector at the end of +the integration process is copied in the <code>y</code> array of the +{@link org.apache.commons.math.ode.FirstOrderIntegrator#integrate +FirstOrderIntegrator.integrate} method. The second one should be used +when more in-depth information is needed throughout the integration +process. The user can register an object implementing the {@link +org.apache.commons.math.ode.sampling.StepHandler StepHandler} interface or a +{@link org.apache.commons.math.ode.sampling.StepNormalizer StepNormalizer} +object wrapping a user-specified object implementing the {@link +org.apache.commons.math.ode.sampling.FixedStepHandler FixedStepHandler} +interface into the integrator before calling the {@link +org.apache.commons.math.ode.FirstOrderIntegrator#integrate +FirstOrderIntegrator.integrate} method. The user object will be called +appropriately during the integration process, allowing the user to +process intermediate results. The default step handler does nothing. +</p> + +<p> +{@link org.apache.commons.math.ode.ContinuousOutputModel +ContinuousOutputModel} is a special-purpose step handler that is able +to store all steps and to provide transparent access to any +intermediate result once the integration is over. An important feature +of this class is that it implements the <code>Serializable</code> +interface. This means that a complete continuous model of the +integrated function throughout the integration range can be serialized +and reused later (if stored into a persistent medium like a filesystem +or a database) or elsewhere (if sent to another application). Only the +result of the integration is stored, there is no reference to the +integrated problem by itself. +</p> + +<p> +Other default implementations of the {@link +org.apache.commons.math.ode.sampling.StepHandler StepHandler} interface are +available for general needs ({@link +org.apache.commons.math.ode.sampling.DummyStepHandler DummyStepHandler}, {@link +org.apache.commons.math.ode.sampling.StepNormalizer StepNormalizer}) and custom +implementations can be developed for specific needs. As an example, +if an application is to be completely driven by the integration +process, then most of the application code will be run inside a step +handler specific to this application. +</p> + +<p> +Some integrators (the simple ones) use fixed steps that are set at +creation time. The more efficient integrators use variable steps that +are handled internally in order to control the integration error with +respect to a specified accuracy (these integrators extend the {@link +org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator +AdaptiveStepsizeIntegrator} abstract class). In this case, the step +handler which is called after each successful step shows up the +variable stepsize. The {@link +org.apache.commons.math.ode.sampling.StepNormalizer StepNormalizer} class can +be used to convert the variable stepsize into a fixed stepsize that +can be handled by classes implementing the {@link +org.apache.commons.math.ode.sampling.FixedStepHandler FixedStepHandler} +interface. Adaptive stepsize integrators can automatically compute the +initial stepsize by themselves, however the user can specify it if he +prefers to retain full control over the integration or if the +automatic guess is wrong. +</p> + +<p> +<table border="1" align="center"> +<tr BGCOLOR="#CCCCFF"><td colspan=2><font size="+2">Fixed Step Integrators</font></td></tr> +<tr BGCOLOR="#EEEEFF"><font size="+1"><td>Name</td><td>Order</td></font></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.EulerIntegrator Euler}</td><td>1</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.MidpointIntegrator Midpoint}</td><td>2</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.ClassicalRungeKuttaIntegrator Classical Runge-Kutta}</td><td>4</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.GillIntegrator Gill}</td><td>4</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.ThreeEighthesIntegrator 3/8}</td><td>4</td></tr> +</table> +</p> + +<table border="1" align="center"> +<tr BGCOLOR="#CCCCFF"><td colspan=3><font size="+2">Adaptive Stepsize Integrators</font></td></tr> +<tr BGCOLOR="#EEEEFF"><font size="+1"><td>Name</td><td>Integration Order</td><td>Error Estimation Order</td></font></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.HighamHall54Integrator Higham and Hall}</td><td>5</td><td>4</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.DormandPrince54Integrator Dormand-Prince 5(4)}</td><td>5</td><td>4</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.DormandPrince853Integrator Dormand-Prince 8(5,3)}</td><td>8</td><td>5 and 3</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.GraggBulirschStoerIntegrator Gragg-Bulirsch-Stoer}</td><td>variable (up to 18 by default)</td><td>variable</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.AdamsBashforthIntegrator Adams-Bashforth}</td><td>variable</td><td>variable</td></tr> +<tr><td>{@link org.apache.commons.math.ode.nonstiff.AdamsMoultonIntegrator Adams-Moulton}</td><td>variable</td><td>variable</td></tr> +</table> +</p> + +<p> +In the table above, the {@link org.apache.commons.math.ode.nonstiff.AdamsBashforthIntegrator +Adams-Bashforth} and {@link org.apache.commons.math.ode.nonstiff.AdamsMoultonIntegrator +Adams-Moulton} integrators appear as variable-step ones. This is an experimental extension +to the classical algorithms using the Nordsieck vector representation. +</p> + +</body> +</html> |