/*
 * 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.
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/**
 * This package provides classes to solve Ordinary Differential Equations problems.
 *
 * <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>. It is also possible to get thederivatives 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>.
 *
 * <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.math3.ode.sampling.StepInterpolator
 * StepInterpolator} abstract class, which are made available to the user at the end of each step.
 *
 * <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>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.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.
 *
 * <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.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.
 *
 * <p>{@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 <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>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.
 *
 * <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.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.
 *
 * <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.math3.ode.nonstiff.EulerIntegrator Euler}</td><td>1</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.MidpointIntegrator Midpoint}</td><td>2</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.ClassicalRungeKuttaIntegrator Classical Runge-Kutta}</td><td>4</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.GillIntegrator Gill}</td><td>4</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.ThreeEighthesIntegrator 3/8}</td><td>4</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.LutherIntegrator Luther}</td><td>6</td></tr>
 * </table>
 *
 * <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.math3.ode.nonstiff.HighamHall54Integrator Higham and Hall}</td><td>5</td><td>4</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.DormandPrince54Integrator Dormand-Prince 5(4)}</td><td>5</td><td>4</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.DormandPrince853Integrator Dormand-Prince 8(5,3)}</td><td>8</td><td>5 and 3</td></tr>
 * <tr><td>{@link org.apache.commons.math3.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.math3.ode.nonstiff.AdamsBashforthIntegrator Adams-Bashforth}</td><td>variable</td><td>variable</td></tr>
 * <tr><td>{@link org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator Adams-Moulton}</td><td>variable</td><td>variable</td></tr>
 * </table>
 *
 * <p>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;