Package org.apache.commons.math4.legacy.ode

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 the derivatives 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 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 FirstOrderDifferentialEquations interface. Then he should pass it to the integrator he prefers among all the classes that implement the 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 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 StepHandler interface or a StepNormalizer object wrapping a user-specified object implementing the FixedStepHandler interface into the integrator before calling the 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.

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 file system 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 StepHandler interface are available for general needs (DummyStepHandler, 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 AdaptiveStepsizeIntegrator abstract class). In this case, the step handler which is called after each successful step shows up the variable stepsize. The StepNormalizer class can be used to convert the variable stepsize into a fixed stepsize that can be handled by classes implementing the 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

Fixed Step Integrators
Name	Order
Euler	1
Midpoint	2
Classical Runge-Kutta	4
Gill	4
3/8	4
Luther	6

Adaptive Stepsize Integrators

Adaptive Stepsize Integrators
Name	Integration Order	Error Estimation Order
Higham and Hall	5	4
Dormand-Prince 5(4)	5	4
Dormand-Prince 8(5,3)	8	5 and 3
Gragg-Bulirsch-Stoer	variable (up to 18 by default)	variable
Adams-Bashforth	variable	variable
Adams-Moulton	variable	variable

In the table above, the Adams-Bashforth and Adams-Moulton integrators appear as variable-step ones. This is an experimental extension to the classical algorithms using the Nordsieck vector representation.

Interface Summary

Interface	Description
FieldSecondaryEquations<T extends RealFieldElement<T>>	This interface allows users to add secondary differential equations to a primary set of differential equations.
FirstOrderDifferentialEquations	This interface represents a first order differential equations set.
FirstOrderFieldDifferentialEquations<T extends RealFieldElement<T>>	This interface represents a first order differential equations set.
FirstOrderFieldIntegrator<T extends RealFieldElement<T>>	This interface represents a first order integrator for differential equations.
FirstOrderIntegrator	This interface represents a first order integrator for differential equations.
MainStateJacobianProvider	Interface expanding first order differential equations in order to compute exactly the main state jacobian matrix for partial derivatives equations.
MultistepIntegrator.NordsieckTransformer	Deprecated. as of 3.6 this unused interface is deprecated
ODEIntegrator	This interface defines the common parts shared by integrators for first and second order differential equations.
Parameterizable	This interface enables to process any parameterizable object.
ParameterizedODE	Interface to compute by finite difference Jacobian matrix for some parameter when computing partial derivatives equations.
ParameterJacobianProvider	Interface to compute exactly Jacobian matrix for some parameter when computing partial derivatives equations.
SecondaryEquations	This interface allows users to add secondary differential equations to a primary set of differential equations.
SecondOrderDifferentialEquations	This interface represents a second order differential equations set.
SecondOrderIntegrator	This interface represents a second order integrator for differential equations.

Class Summary

Class	Description
AbstractFieldIntegrator<T extends RealFieldElement<T>>	Base class managing common boilerplate for all integrators.
AbstractIntegrator	Base class managing common boilerplate for all integrators.
AbstractParameterizable	This abstract class provides boilerplate parameters list.
ContinuousOutputFieldModel<T extends RealFieldElement<T>>	This class stores all information provided by an ODE integrator during the integration process and build a continuous model of the solution from this.
ContinuousOutputModel	This class stores all information provided by an ODE integrator during the integration process and build a continuous model of the solution from this.
EquationsMapper	Class mapping the part of a complete state or derivative that pertains to a specific differential equation.
ExpandableStatefulODE	This class represents a combined set of first order differential equations, with at least a primary set of equations expandable by some sets of secondary equations.
FieldEquationsMapper<T extends RealFieldElement<T>>	Class mapping the part of a complete state or derivative that pertains to a set of differential equations.
FieldExpandableODE<T extends RealFieldElement<T>>	This class represents a combined set of first order differential equations, with at least a primary set of equations expandable by some sets of secondary equations.
FieldODEState<T extends RealFieldElement<T>>	Container for time, main and secondary state vectors.
FieldODEStateAndDerivative<T extends RealFieldElement<T>>	Container for time, main and secondary state vectors as well as their derivatives.
FirstOrderConverter	This class converts second order differential equations to first order ones.
JacobianMatrices	This class defines a set of secondary equations to compute the Jacobian matrices with respect to the initial state vector and, if any, to some parameters of the primary ODE set.
MultistepFieldIntegrator<T extends RealFieldElement<T>>	This class is the base class for multistep integrators for Ordinary Differential Equations.
MultistepIntegrator	This class is the base class for multistep integrators for Ordinary Differential Equations.

Exception Summary

Exception	Description
JacobianMatrices.MismatchedEquations	Special exception for equations mismatch.
UnknownParameterException	Exception to be thrown when a parameter is unknown.