| <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> |
| <!-- NewPage --> |
| <html lang="en"> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html" charset="UTF-8"> |
| <title>org.apache.commons.math3.ode (Apache Commons Math 3.3 API)</title> |
| <link rel="stylesheet" type="text/css" href="../../../../../stylesheet.css" title="Style"> |
| </head> |
| <body> |
| <script type="text/javascript"><!-- |
| if (location.href.indexOf('is-external=true') == -1) { |
| parent.document.title="org.apache.commons.math3.ode (Apache Commons Math 3.3 API)"; |
| } |
| //--> |
| </script> |
| <noscript> |
| <div>JavaScript is disabled on your browser.</div> |
| </noscript> |
| <!-- ========= START OF TOP NAVBAR ======= --> |
| <div class="topNav"><a name="navbar_top"> |
| <!-- --> |
| </a><a href="#skip-navbar_top" title="Skip navigation links"></a><a name="navbar_top_firstrow"> |
| <!-- --> |
| </a> |
| <ul class="navList" title="Navigation"> |
| <li><a href="../../../../../overview-summary.html">Overview</a></li> |
| <li class="navBarCell1Rev">Package</li> |
| <li>Class</li> |
| <li><a href="package-use.html">Use</a></li> |
| <li><a href="package-tree.html">Tree</a></li> |
| <li><a href="../../../../../deprecated-list.html">Deprecated</a></li> |
| <li><a href="../../../../../index-all.html">Index</a></li> |
| <li><a href="../../../../../help-doc.html">Help</a></li> |
| </ul> |
| <div class="aboutLanguage"><em><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script></em></div> |
| </div> |
| <div class="subNav"> |
| <ul class="navList"> |
| <li><a href="../../../../../org/apache/commons/math3/ml/neuralnet/twod/package-summary.html">Prev Package</a></li> |
| <li><a href="../../../../../org/apache/commons/math3/ode/events/package-summary.html">Next Package</a></li> |
| </ul> |
| <ul class="navList"> |
| <li><a href="../../../../../index.html?org/apache/commons/math3/ode/package-summary.html" target="_top">Frames</a></li> |
| <li><a href="package-summary.html" target="_top">No Frames</a></li> |
| </ul> |
| <ul class="navList" id="allclasses_navbar_top"> |
| <li><a href="../../../../../allclasses-noframe.html">All Classes</a></li> |
| </ul> |
| <div> |
| <script type="text/javascript"><!-- |
| allClassesLink = document.getElementById("allclasses_navbar_top"); |
| if(window==top) { |
| allClassesLink.style.display = "block"; |
| } |
| else { |
| allClassesLink.style.display = "none"; |
| } |
| //--> |
| </script> |
| </div> |
| <a name="skip-navbar_top"> |
| <!-- --> |
| </a></div> |
| <!-- ========= END OF TOP NAVBAR ========= --> |
| <div class="header"> |
| <h1 title="Package" class="title">Package org.apache.commons.math3.ode</h1> |
| <div class="docSummary"> |
| <div class="block"> |
| This package provides classes to solve Ordinary Differential Equations problems.</div> |
| </div> |
| <p>See: <a href="#package_description">Description</a></p> |
| </div> |
| <div class="contentContainer"> |
| <ul class="blockList"> |
| <li class="blockList"> |
| <table class="packageSummary" border="0" cellpadding="3" cellspacing="0" summary="Interface Summary table, listing interfaces, and an explanation"> |
| <caption><span>Interface Summary</span><span class="tabEnd"> </span></caption> |
| <tr> |
| <th class="colFirst" scope="col">Interface</th> |
| <th class="colLast" scope="col">Description</th> |
| </tr> |
| <tbody> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/FirstOrderDifferentialEquations.html" title="interface in org.apache.commons.math3.ode">FirstOrderDifferentialEquations</a></td> |
| <td class="colLast"> |
| <div class="block">This interface represents a first order differential equations set.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/FirstOrderIntegrator.html" title="interface in org.apache.commons.math3.ode">FirstOrderIntegrator</a></td> |
| <td class="colLast"> |
| <div class="block">This interface represents a first order integrator for |
| differential equations.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/MainStateJacobianProvider.html" title="interface in org.apache.commons.math3.ode">MainStateJacobianProvider</a></td> |
| <td class="colLast"> |
| <div class="block">Interface expanding <a href="../../../../../org/apache/commons/math3/ode/FirstOrderDifferentialEquations.html" title="interface in org.apache.commons.math3.ode"><code>first order |
| differential equations</code></a> in order to compute exactly the main state jacobian |
| matrix for <a href="../../../../../org/apache/commons/math3/ode/JacobianMatrices.html" title="class in org.apache.commons.math3.ode"><code>partial derivatives equations</code></a>.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/MultistepIntegrator.NordsieckTransformer.html" title="interface in org.apache.commons.math3.ode">MultistepIntegrator.NordsieckTransformer</a></td> |
| <td class="colLast"> |
| <div class="block">Transformer used to convert the first step to Nordsieck representation.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/ODEIntegrator.html" title="interface in org.apache.commons.math3.ode">ODEIntegrator</a></td> |
| <td class="colLast"> |
| <div class="block">This interface defines the common parts shared by integrators |
| for first and second order differential equations.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/Parameterizable.html" title="interface in org.apache.commons.math3.ode">Parameterizable</a></td> |
| <td class="colLast"> |
| <div class="block">This interface enables to process any parameterizable object.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/ParameterizedODE.html" title="interface in org.apache.commons.math3.ode">ParameterizedODE</a></td> |
| <td class="colLast"> |
| <div class="block">Interface to compute by finite difference Jacobian matrix for some parameter |
| when computing <a href="../../../../../org/apache/commons/math3/ode/JacobianMatrices.html" title="class in org.apache.commons.math3.ode"><code>partial derivatives equations</code></a>.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/ParameterJacobianProvider.html" title="interface in org.apache.commons.math3.ode">ParameterJacobianProvider</a></td> |
| <td class="colLast"> |
| <div class="block">Interface to compute exactly Jacobian matrix for some parameter |
| when computing <a href="../../../../../org/apache/commons/math3/ode/JacobianMatrices.html" title="class in org.apache.commons.math3.ode"><code>partial derivatives equations</code></a>.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/SecondaryEquations.html" title="interface in org.apache.commons.math3.ode">SecondaryEquations</a></td> |
| <td class="colLast"> |
| <div class="block">This interface allows users to add secondary differential equations to a primary |
| set of differential equations.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/SecondOrderDifferentialEquations.html" title="interface in org.apache.commons.math3.ode">SecondOrderDifferentialEquations</a></td> |
| <td class="colLast"> |
| <div class="block">This interface represents a second order differential equations set.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/SecondOrderIntegrator.html" title="interface in org.apache.commons.math3.ode">SecondOrderIntegrator</a></td> |
| <td class="colLast"> |
| <div class="block">This interface represents a second order integrator for |
| differential equations.</div> |
| </td> |
| </tr> |
| </tbody> |
| </table> |
| </li> |
| <li class="blockList"> |
| <table class="packageSummary" border="0" cellpadding="3" cellspacing="0" summary="Class Summary table, listing classes, and an explanation"> |
| <caption><span>Class Summary</span><span class="tabEnd"> </span></caption> |
| <tr> |
| <th class="colFirst" scope="col">Class</th> |
| <th class="colLast" scope="col">Description</th> |
| </tr> |
| <tbody> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/AbstractIntegrator.html" title="class in org.apache.commons.math3.ode">AbstractIntegrator</a></td> |
| <td class="colLast"> |
| <div class="block">Base class managing common boilerplate for all integrators.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/AbstractParameterizable.html" title="class in org.apache.commons.math3.ode">AbstractParameterizable</a></td> |
| <td class="colLast"> |
| <div class="block">This abstract class provides boilerplate parameters list.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/ContinuousOutputModel.html" title="class in org.apache.commons.math3.ode">ContinuousOutputModel</a></td> |
| <td class="colLast"> |
| <div class="block">This class stores all information provided by an ODE integrator |
| during the integration process and build a continuous model of the |
| solution from this.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/EquationsMapper.html" title="class in org.apache.commons.math3.ode">EquationsMapper</a></td> |
| <td class="colLast"> |
| <div class="block">Class mapping the part of a complete state or derivative that pertains |
| to a specific differential equation.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/ExpandableStatefulODE.html" title="class in org.apache.commons.math3.ode">ExpandableStatefulODE</a></td> |
| <td class="colLast"> |
| <div class="block">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.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/FirstOrderConverter.html" title="class in org.apache.commons.math3.ode">FirstOrderConverter</a></td> |
| <td class="colLast"> |
| <div class="block">This class converts second order differential equations to first |
| order ones.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/JacobianMatrices.html" title="class in org.apache.commons.math3.ode">JacobianMatrices</a></td> |
| <td class="colLast"> |
| <div class="block">This class defines a set of <a href="../../../../../org/apache/commons/math3/ode/SecondaryEquations.html" title="interface in org.apache.commons.math3.ode"><code>secondary equations</code></a> to |
| compute the Jacobian matrices with respect to the initial state vector and, if |
| any, to some parameters of the primary ODE set.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/MultistepIntegrator.html" title="class in org.apache.commons.math3.ode">MultistepIntegrator</a></td> |
| <td class="colLast"> |
| <div class="block">This class is the base class for multistep integrators for Ordinary |
| Differential Equations.</div> |
| </td> |
| </tr> |
| </tbody> |
| </table> |
| </li> |
| <li class="blockList"> |
| <table class="packageSummary" border="0" cellpadding="3" cellspacing="0" summary="Exception Summary table, listing exceptions, and an explanation"> |
| <caption><span>Exception Summary</span><span class="tabEnd"> </span></caption> |
| <tr> |
| <th class="colFirst" scope="col">Exception</th> |
| <th class="colLast" scope="col">Description</th> |
| </tr> |
| <tbody> |
| <tr class="altColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/JacobianMatrices.MismatchedEquations.html" title="class in org.apache.commons.math3.ode">JacobianMatrices.MismatchedEquations</a></td> |
| <td class="colLast"> |
| <div class="block">Special exception for equations mismatch.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><a href="../../../../../org/apache/commons/math3/ode/UnknownParameterException.html" title="class in org.apache.commons.math3.ode">UnknownParameterException</a></td> |
| <td class="colLast"> |
| <div class="block">Exception to be thrown when a parameter is unknown.</div> |
| </td> |
| </tr> |
| </tbody> |
| </table> |
| </li> |
| </ul> |
| <a name="package_description"> |
| <!-- --> |
| </a> |
| <h2 title="Package org.apache.commons.math3.ode Description">Package org.apache.commons.math3.ode Description</h2> |
| <div class="block"><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>. |
| 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> |
| |
| <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/math3/ode/sampling/StepInterpolator.html" title="interface in org.apache.commons.math3.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/math3/ode/FirstOrderDifferentialEquations.html" title="interface in org.apache.commons.math3.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/math3/ode/FirstOrderIntegrator.html" title="interface in org.apache.commons.math3.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/math3/ode/FirstOrderIntegrator.html#integrate(org.apache.commons.math3.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/math3/ode/sampling/StepHandler.html" title="interface in org.apache.commons.math3.ode.sampling"><code>StepHandler</code></a> interface or a |
| <a href="../../../../../org/apache/commons/math3/ode/sampling/StepNormalizer.html" title="class in org.apache.commons.math3.ode.sampling"><code>StepNormalizer</code></a> |
| object wrapping a user-specified object implementing the <a href="../../../../../org/apache/commons/math3/ode/sampling/FixedStepHandler.html" title="interface in org.apache.commons.math3.ode.sampling"><code>FixedStepHandler</code></a> |
| interface into the integrator before calling the <a href="../../../../../org/apache/commons/math3/ode/FirstOrderIntegrator.html#integrate(org.apache.commons.math3.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/math3/ode/ContinuousOutputModel.html" title="class in org.apache.commons.math3.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/math3/ode/sampling/StepHandler.html" title="interface in org.apache.commons.math3.ode.sampling"><code>StepHandler</code></a> interface are |
| available for general needs (<a href="../../../../../org/apache/commons/math3/ode/sampling/DummyStepHandler.html" title="class in org.apache.commons.math3.ode.sampling"><code>DummyStepHandler</code></a>, <a href="../../../../../org/apache/commons/math3/ode/sampling/StepNormalizer.html" title="class in org.apache.commons.math3.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/math3/ode/nonstiff/AdaptiveStepsizeIntegrator.html" title="class in org.apache.commons.math3.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/math3/ode/sampling/StepNormalizer.html" title="class in org.apache.commons.math3.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/math3/ode/sampling/FixedStepHandler.html" title="interface in org.apache.commons.math3.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/math3/ode/nonstiff/EulerIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Euler</code></a></td><td>1</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/MidpointIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Midpoint</code></a></td><td>2</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/ClassicalRungeKuttaIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Classical Runge-Kutta</code></a></td><td>4</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/GillIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Gill</code></a></td><td>4</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/ThreeEighthesIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>3/8</code></a></td><td>4</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/LutherIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Luther</code></a></td><td>6</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/math3/ode/nonstiff/HighamHall54Integrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Higham and Hall</code></a></td><td>5</td><td>4</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/DormandPrince54Integrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Dormand-Prince 5(4)</code></a></td><td>5</td><td>4</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/DormandPrince853Integrator.html" title="class in org.apache.commons.math3.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/math3/ode/nonstiff/GraggBulirschStoerIntegrator.html" title="class in org.apache.commons.math3.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/math3/ode/nonstiff/AdamsBashforthIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Adams-Bashforth</code></a></td><td>variable</td><td>variable</td></tr> |
| <tr><td><a href="../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html" title="class in org.apache.commons.math3.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/math3/ode/nonstiff/AdamsBashforthIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Adams-Bashforth</code></a> and <a href="../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html" title="class in org.apache.commons.math3.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></div> |
| </div> |
| <!-- ======= START OF BOTTOM NAVBAR ====== --> |
| <div class="bottomNav"><a name="navbar_bottom"> |
| <!-- --> |
| </a><a href="#skip-navbar_bottom" title="Skip navigation links"></a><a name="navbar_bottom_firstrow"> |
| <!-- --> |
| </a> |
| <ul class="navList" title="Navigation"> |
| <li><a href="../../../../../overview-summary.html">Overview</a></li> |
| <li class="navBarCell1Rev">Package</li> |
| <li>Class</li> |
| <li><a href="package-use.html">Use</a></li> |
| <li><a href="package-tree.html">Tree</a></li> |
| <li><a href="../../../../../deprecated-list.html">Deprecated</a></li> |
| <li><a href="../../../../../index-all.html">Index</a></li> |
| <li><a href="../../../../../help-doc.html">Help</a></li> |
| </ul> |
| <div class="aboutLanguage"><em><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script></em></div> |
| </div> |
| <div class="subNav"> |
| <ul class="navList"> |
| <li><a href="../../../../../org/apache/commons/math3/ml/neuralnet/twod/package-summary.html">Prev Package</a></li> |
| <li><a href="../../../../../org/apache/commons/math3/ode/events/package-summary.html">Next Package</a></li> |
| </ul> |
| <ul class="navList"> |
| <li><a href="../../../../../index.html?org/apache/commons/math3/ode/package-summary.html" target="_top">Frames</a></li> |
| <li><a href="package-summary.html" target="_top">No Frames</a></li> |
| </ul> |
| <ul class="navList" id="allclasses_navbar_bottom"> |
| <li><a href="../../../../../allclasses-noframe.html">All Classes</a></li> |
| </ul> |
| <div> |
| <script type="text/javascript"><!-- |
| allClassesLink = document.getElementById("allclasses_navbar_bottom"); |
| if(window==top) { |
| allClassesLink.style.display = "block"; |
| } |
| else { |
| allClassesLink.style.display = "none"; |
| } |
| //--> |
| </script> |
| </div> |
| <a name="skip-navbar_bottom"> |
| <!-- --> |
| </a></div> |
| <!-- ======== END OF BOTTOM NAVBAR ======= --> |
| <p class="legalCopy"><small>Copyright © 2003–2014 <a href="http://www.apache.org/">The Apache Software Foundation</a>. All rights reserved.</small></p> |
| </body> |
| </html> |
