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| date | Mon, 10 Oct 2011 17:52:22 +0200 |
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <!--NewPage--> <HTML> <HEAD> <!-- Generated by javadoc (build 1.6.0_17) on Sat Mar 27 22:46:17 EDT 2010 --> <META http-equiv="Content-Type" content="text/html; charset=UTF-8"> <TITLE> org.apache.commons.math.ode (Commons Math 2.1 API) </TITLE> <META NAME="date" CONTENT="2010-03-27"> <LINK REL ="stylesheet" TYPE="text/css" HREF="../../../../../stylesheet.css" TITLE="Style"> <SCRIPT type="text/javascript"> function windowTitle() { if (location.href.indexOf('is-external=true') == -1) { parent.document.title="org.apache.commons.math.ode (Commons Math 2.1 API)"; } } </SCRIPT> <NOSCRIPT> </NOSCRIPT> </HEAD> <BODY BGCOLOR="white" onload="windowTitle();"> <HR> <!-- ========= START OF TOP NAVBAR ======= --> <A NAME="navbar_top"><!-- --></A> <A HREF="#skip-navbar_top" title="Skip navigation links"></A> <TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY=""> <TR> <TD COLSPAN=2 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HREF="../../../../../index-all.html"><FONT CLASS="NavBarFont1"><B>Index</B></FONT></A> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../../../help-doc.html"><FONT CLASS="NavBarFont1"><B>Help</B></FONT></A> </TD> </TR> </TABLE> </TD> <TD ALIGN="right" VALIGN="top" ROWSPAN=3><EM> </EM> </TD> </TR> <TR> <TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2"> <A HREF="../../../../../org/apache/commons/math/linear/package-summary.html"><B>PREV PACKAGE</B></A> <A HREF="../../../../../org/apache/commons/math/ode/events/package-summary.html"><B>NEXT PACKAGE</B></A></FONT></TD> <TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2"> <A HREF="../../../../../index.html?org/apache/commons/math/ode/package-summary.html" target="_top"><B>FRAMES</B></A> <A HREF="package-summary.html" target="_top"><B>NO FRAMES</B></A> <SCRIPT type="text/javascript"> <!-- if(window==top) { document.writeln('<A HREF="../../../../../allclasses-noframe.html"><B>All Classes</B></A>'); } //--> </SCRIPT> <NOSCRIPT> <A HREF="../../../../../allclasses-noframe.html"><B>All Classes</B></A> </NOSCRIPT> </FONT></TD> </TR> </TABLE> <A NAME="skip-navbar_top"></A> <!-- ========= END OF TOP NAVBAR ========= --> <HR> <H2> Package org.apache.commons.math.ode </H2> This package provides classes to solve Ordinary Differential Equations problems. <P> <B>See:</B> <BR> <A HREF="#package_description"><B>Description</B></A> <P> <TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY=""> <TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor"> <TH ALIGN="left" COLSPAN="2"><FONT SIZE="+2"> <B>Interface Summary</B></FONT></TH> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/FirstOrderDifferentialEquations.html" title="interface in org.apache.commons.math.ode">FirstOrderDifferentialEquations</A></B></TD> <TD>This interface represents a first order differential equations set.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/FirstOrderIntegrator.html" title="interface in org.apache.commons.math.ode">FirstOrderIntegrator</A></B></TD> <TD>This interface represents a first order integrator for differential equations.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/MultistepIntegrator.NordsieckTransformer.html" title="interface in org.apache.commons.math.ode">MultistepIntegrator.NordsieckTransformer</A></B></TD> <TD>Transformer used to convert the first step to Nordsieck representation.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/ODEIntegrator.html" title="interface in org.apache.commons.math.ode">ODEIntegrator</A></B></TD> <TD>This interface defines the common parts shared by integrators for first and second order differential equations.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/SecondOrderDifferentialEquations.html" title="interface in org.apache.commons.math.ode">SecondOrderDifferentialEquations</A></B></TD> <TD>This interface represents a second order differential equations set.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/SecondOrderIntegrator.html" title="interface in org.apache.commons.math.ode">SecondOrderIntegrator</A></B></TD> <TD>This interface represents a second order integrator for differential equations.</TD> </TR> </TABLE> <P> <TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY=""> <TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor"> <TH ALIGN="left" COLSPAN="2"><FONT SIZE="+2"> <B>Class Summary</B></FONT></TH> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/AbstractIntegrator.html" title="class in org.apache.commons.math.ode">AbstractIntegrator</A></B></TD> <TD>Base class managing common boilerplate for all integrators.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/ContinuousOutputModel.html" title="class in org.apache.commons.math.ode">ContinuousOutputModel</A></B></TD> <TD>This class stores all information provided by an ODE integrator during the integration process and build a continuous model of the solution from this.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/FirstOrderConverter.html" title="class in org.apache.commons.math.ode">FirstOrderConverter</A></B></TD> <TD>This class converts second order differential equations to first order ones.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/MultistepIntegrator.html" title="class in org.apache.commons.math.ode">MultistepIntegrator</A></B></TD> <TD>This class is the base class for multistep integrators for Ordinary Differential Equations.</TD> </TR> </TABLE> <P> <TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY=""> <TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor"> <TH ALIGN="left" COLSPAN="2"><FONT SIZE="+2"> <B>Exception Summary</B></FONT></TH> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/DerivativeException.html" title="class in org.apache.commons.math.ode">DerivativeException</A></B></TD> <TD>This exception is made available to users to report the error conditions that are triggered while computing the differential equations.</TD> </TR> <TR BGCOLOR="white" CLASS="TableRowColor"> <TD WIDTH="15%"><B><A HREF="../../../../../org/apache/commons/math/ode/IntegratorException.html" title="class in org.apache.commons.math.ode">IntegratorException</A></B></TD> <TD>This exception is made available to users to report the error conditions that are triggered during integration</TD> </TR> </TABLE> <P> <A NAME="package_description"><!-- --></A><H2> Package org.apache.commons.math.ode Description </H2> <P> <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 <A HREF="../../../../../org/apache/commons/math/ode/sampling/StepInterpolator.html" title="interface in org.apache.commons.math.ode.sampling"><CODE>StepInterpolator</CODE></A> 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 <A HREF="../../../../../org/apache/commons/math/ode/FirstOrderDifferentialEquations.html" title="interface in org.apache.commons.math.ode"><CODE>FirstOrderDifferentialEquations</CODE></A> interface. Then he should pass it to the integrator he prefers among all the classes that implement the <A HREF="../../../../../org/apache/commons/math/ode/FirstOrderIntegrator.html" title="interface in org.apache.commons.math.ode"><CODE>FirstOrderIntegrator</CODE></A> 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 <A HREF="../../../../../org/apache/commons/math/ode/FirstOrderIntegrator.html#integrate(org.apache.commons.math.ode.FirstOrderDifferentialEquations, double, double[], double, double[])"><CODE>FirstOrderIntegrator.integrate</CODE></A> 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 <A HREF="../../../../../org/apache/commons/math/ode/sampling/StepHandler.html" title="interface in org.apache.commons.math.ode.sampling"><CODE>StepHandler</CODE></A> interface or a <A HREF="../../../../../org/apache/commons/math/ode/sampling/StepNormalizer.html" title="class in org.apache.commons.math.ode.sampling"><CODE>StepNormalizer</CODE></A> object wrapping a user-specified object implementing the <A HREF="../../../../../org/apache/commons/math/ode/sampling/FixedStepHandler.html" title="interface in org.apache.commons.math.ode.sampling"><CODE>FixedStepHandler</CODE></A> interface into the integrator before calling the <A HREF="../../../../../org/apache/commons/math/ode/FirstOrderIntegrator.html#integrate(org.apache.commons.math.ode.FirstOrderDifferentialEquations, double, double[], double, double[])"><CODE>FirstOrderIntegrator.integrate</CODE></A> 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> <A HREF="../../../../../org/apache/commons/math/ode/ContinuousOutputModel.html" title="class in org.apache.commons.math.ode"><CODE>ContinuousOutputModel</CODE></A> 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 <A HREF="../../../../../org/apache/commons/math/ode/sampling/StepHandler.html" title="interface in org.apache.commons.math.ode.sampling"><CODE>StepHandler</CODE></A> interface are available for general needs (<A HREF="../../../../../org/apache/commons/math/ode/sampling/DummyStepHandler.html" title="class in org.apache.commons.math.ode.sampling"><CODE>DummyStepHandler</CODE></A>, <A HREF="../../../../../org/apache/commons/math/ode/sampling/StepNormalizer.html" title="class in org.apache.commons.math.ode.sampling"><CODE>StepNormalizer</CODE></A>) 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 <A HREF="../../../../../org/apache/commons/math/ode/nonstiff/AdaptiveStepsizeIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>AdaptiveStepsizeIntegrator</CODE></A> abstract class). In this case, the step handler which is called after each successful step shows up the variable stepsize. The <A HREF="../../../../../org/apache/commons/math/ode/sampling/StepNormalizer.html" title="class in org.apache.commons.math.ode.sampling"><CODE>StepNormalizer</CODE></A> class can be used to convert the variable stepsize into a fixed stepsize that can be handled by classes implementing the <A HREF="../../../../../org/apache/commons/math/ode/sampling/FixedStepHandler.html" title="interface in org.apache.commons.math.ode.sampling"><CODE>FixedStepHandler</CODE></A> 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><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/EulerIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Euler</CODE></A></td><td>1</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/MidpointIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Midpoint</CODE></A></td><td>2</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/ClassicalRungeKuttaIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Classical Runge-Kutta</CODE></A></td><td>4</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/GillIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Gill</CODE></A></td><td>4</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/ThreeEighthesIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>3/8</CODE></A></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><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/HighamHall54Integrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Higham and Hall</CODE></A></td><td>5</td><td>4</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/DormandPrince54Integrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Dormand-Prince 5(4)</CODE></A></td><td>5</td><td>4</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/DormandPrince853Integrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Dormand-Prince 8(5,3)</CODE></A></td><td>8</td><td>5 and 3</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/GraggBulirschStoerIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Gragg-Bulirsch-Stoer</CODE></A></td><td>variable (up to 18 by default)</td><td>variable</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/AdamsBashforthIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Adams-Bashforth</CODE></A></td><td>variable</td><td>variable</td></tr> <tr><td><A HREF="../../../../../org/apache/commons/math/ode/nonstiff/AdamsMoultonIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Adams-Moulton</CODE></A></td><td>variable</td><td>variable</td></tr> </table> </p> <p> In the table above, the <A HREF="../../../../../org/apache/commons/math/ode/nonstiff/AdamsBashforthIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Adams-Bashforth</CODE></A> and <A HREF="../../../../../org/apache/commons/math/ode/nonstiff/AdamsMoultonIntegrator.html" title="class in org.apache.commons.math.ode.nonstiff"><CODE>Adams-Moulton</CODE></A> integrators appear as variable-step ones. This is an experimental extension to the classical algorithms using the Nordsieck vector representation. </p> <P> <P> <DL> </DL> <HR> <!-- ======= START OF BOTTOM NAVBAR ====== --> <A NAME="navbar_bottom"><!-- --></A> <A HREF="#skip-navbar_bottom" title="Skip navigation links"></A> <TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY=""> <TR> <TD COLSPAN=2 BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A NAME="navbar_bottom_firstrow"><!-- --></A> <TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY=""> <TR ALIGN="center" VALIGN="top"> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../../../overview-summary.html"><FONT CLASS="NavBarFont1"><B>Overview</B></FONT></A> </TD> <TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> <FONT CLASS="NavBarFont1Rev"><B>Package</B></FONT> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Class</FONT> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-use.html"><FONT CLASS="NavBarFont1"><B>Use</B></FONT></A> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-tree.html"><FONT CLASS="NavBarFont1"><B>Tree</B></FONT></A> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../../../deprecated-list.html"><FONT CLASS="NavBarFont1"><B>Deprecated</B></FONT></A> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../../../index-all.html"><FONT CLASS="NavBarFont1"><B>Index</B></FONT></A> </TD> <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../../../help-doc.html"><FONT CLASS="NavBarFont1"><B>Help</B></FONT></A> </TD> </TR> </TABLE> </TD> <TD ALIGN="right" VALIGN="top" ROWSPAN=3><EM> </EM> </TD> </TR> <TR> <TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2"> <A HREF="../../../../../org/apache/commons/math/linear/package-summary.html"><B>PREV PACKAGE</B></A> <A HREF="../../../../../org/apache/commons/math/ode/events/package-summary.html"><B>NEXT PACKAGE</B></A></FONT></TD> <TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2"> <A HREF="../../../../../index.html?org/apache/commons/math/ode/package-summary.html" target="_top"><B>FRAMES</B></A> <A HREF="package-summary.html" target="_top"><B>NO FRAMES</B></A> <SCRIPT type="text/javascript"> <!-- if(window==top) { document.writeln('<A HREF="../../../../../allclasses-noframe.html"><B>All Classes</B></A>'); } //--> </SCRIPT> <NOSCRIPT> <A HREF="../../../../../allclasses-noframe.html"><B>All Classes</B></A> </NOSCRIPT> </FONT></TD> </TR> </TABLE> <A NAME="skip-navbar_bottom"></A> <!-- ======== END OF BOTTOM NAVBAR ======= --> <HR> Copyright © 2003-2010 <a href="http://www.apache.org/">The Apache Software Foundation</a>. 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