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/*
 * 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.
 */
/**
 *
 * 

* This package provides classes to solve Ordinary Differential Equations problems. *

* *

* This package solves Initial Value Problems of the form * y'=f(t,y) with t0 and * y(t0)=y0 known. The provided * integrators compute an estimate of y(t) from * t=t0 to t=t1. * It is also possible to get thederivatives with respect to the initial state * dy(t)/dy(t0) or the derivatives with * respect to some ODE parameters dy(t)/dp. *

* *

* 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.math3.ode.sampling.StepInterpolator StepInterpolator} * abstract class, which are made available to the user at the end of * each step. *

* *

* 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. *

* *

* The user should describe his problem in his own classes * (UserProblem in the diagram below) which should implement * the {@link org.apache.commons.math3.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.math3.ode.FirstOrderIntegrator * FirstOrderIntegrator} interface. *

* *

* 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 y array of the * {@link org.apache.commons.math3.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.math3.ode.sampling.StepHandler StepHandler} interface or a * {@link org.apache.commons.math3.ode.sampling.StepNormalizer StepNormalizer} * object wrapping a user-specified object implementing the {@link * org.apache.commons.math3.ode.sampling.FixedStepHandler FixedStepHandler} * interface into the integrator before calling the {@link * org.apache.commons.math3.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. *

* *

* {@link org.apache.commons.math3.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 Serializable * 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. *

* *

* Other default implementations of the {@link * org.apache.commons.math3.ode.sampling.StepHandler StepHandler} interface are * available for general needs ({@link * org.apache.commons.math3.ode.sampling.DummyStepHandler DummyStepHandler}, {@link * org.apache.commons.math3.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. *

* *

* 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.math3.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.math3.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.math3.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. *

* *

*

* * * * * * * * *
Fixed Step Integrators
NameOrder
{@link org.apache.commons.math3.ode.nonstiff.EulerIntegrator Euler}1
{@link org.apache.commons.math3.ode.nonstiff.MidpointIntegrator Midpoint}2
{@link org.apache.commons.math3.ode.nonstiff.ClassicalRungeKuttaIntegrator Classical Runge-Kutta}4
{@link org.apache.commons.math3.ode.nonstiff.GillIntegrator Gill}4
{@link org.apache.commons.math3.ode.nonstiff.ThreeEighthesIntegrator 3/8}4
{@link org.apache.commons.math3.ode.nonstiff.LutherIntegrator Luther}6
*

* * * * * * * * * * *
Adaptive Stepsize Integrators
NameIntegration OrderError Estimation Order
{@link org.apache.commons.math3.ode.nonstiff.HighamHall54Integrator Higham and Hall}54
{@link org.apache.commons.math3.ode.nonstiff.DormandPrince54Integrator Dormand-Prince 5(4)}54
{@link org.apache.commons.math3.ode.nonstiff.DormandPrince853Integrator Dormand-Prince 8(5,3)}85 and 3
{@link org.apache.commons.math3.ode.nonstiff.GraggBulirschStoerIntegrator Gragg-Bulirsch-Stoer}variable (up to 18 by default)variable
{@link org.apache.commons.math3.ode.nonstiff.AdamsBashforthIntegrator Adams-Bashforth}variablevariable
{@link org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator Adams-Moulton}variablevariable
*

* *

* In the table above, the {@link org.apache.commons.math3.ode.nonstiff.AdamsBashforthIntegrator * Adams-Bashforth} and {@link org.apache.commons.math3.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. *

* * */ package org.apache.commons.math3.ode;




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