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`. If in addition to `y(t)` users need to get the derivatives with respect to the initial state `dy(t)/dy(t0)` or the derivatives with respect to some ODE parameters `dy(t)/dp`, then the classes from the org.apache.commons.math.ode.jacobians package must be used instead of the classes in this package.

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.

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.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.

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.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.

{@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 `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.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.

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.

Fixed Step Integrators
Name | Order
{@link org.apache.commons.math.ode.nonstiff.EulerIntegrator Euler} | 1
{@link org.apache.commons.math.ode.nonstiff.MidpointIntegrator Midpoint} | 2
{@link org.apache.commons.math.ode.nonstiff.ClassicalRungeKuttaIntegrator Classical Runge-Kutta} | 4
{@link org.apache.commons.math.ode.nonstiff.GillIntegrator Gill} | 4
{@link org.apache.commons.math.ode.nonstiff.ThreeEighthesIntegrator 3/8} | 4

Adaptive Stepsize Integrators
Name | Integration Order | Error Estimation Order
{@link org.apache.commons.math.ode.nonstiff.HighamHall54Integrator Higham and Hall} | 5 | 4
{@link org.apache.commons.math.ode.nonstiff.DormandPrince54Integrator Dormand-Prince 5(4)} | 5 | 4
{@link org.apache.commons.math.ode.nonstiff.DormandPrince853Integrator Dormand-Prince 8(5,3)} | 8 | 5 and 3
{@link org.apache.commons.math.ode.nonstiff.GraggBulirschStoerIntegrator Gragg-Bulirsch-Stoer} | variable (up to 18 by default) | variable
{@link org.apache.commons.math.ode.nonstiff.AdamsBashforthIntegrator Adams-Bashforth} | variable | variable
{@link org.apache.commons.math.ode.nonstiff.AdamsMoultonIntegrator Adams-Moulton} | variable | variable

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.