org.apache.commons.math.ode.nonstiff
Class DormandPrince853Integrator

java.lang.Object
  extended by org.apache.commons.math.ode.AbstractIntegrator
      extended by org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator
          extended by org.apache.commons.math.ode.nonstiff.EmbeddedRungeKuttaIntegrator
              extended by org.apache.commons.math.ode.nonstiff.DormandPrince853Integrator
All Implemented Interfaces:
FirstOrderIntegrator, ODEIntegrator

public class DormandPrince853Integrator
extends EmbeddedRungeKuttaIntegrator

This class implements the 8(5,3) Dormand-Prince integrator for Ordinary Differential Equations.

This integrator is an embedded Runge-Kutta integrator of order 8(5,3) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 12 functions evaluations per step for integration and 4 evaluations for interpolation. However, since the first interpolation evaluation is the same as the first integration evaluation of the next step, we have included it in the integrator rather than in the interpolator and specified the method was an fsal. Hence, despite we have 13 stages here, the cost is really 12 evaluations per step even if no interpolation is done, and the overcost of interpolation is only 3 evaluations.

This method is based on an 8(6) method by Dormand and Prince (i.e. order 8 for the integration and order 6 for error estimation) modified by Hairer and Wanner to use a 5th order error estimator with 3rd order correction. This modification was introduced because the original method failed in some cases (wrong steps can be accepted when step size is too large, for example in the Brusselator problem) and also had severe difficulties when applied to problems with discontinuities. This modification is explained in the second edition of the first volume (Nonstiff Problems) of the reference book by Hairer, Norsett and Wanner: Solving Ordinary Differential Equations (Springer-Verlag, ISBN 3-540-56670-8).

Since:
1.2
Version:
$Revision: 786881 $ $Date: 2009-06-20 14:53:08 -0400 (Sat, 20 Jun 2009) $

Field Summary
private static double e1_01
          First error weights array, element 1.
private static double e1_06
          First error weights array, element 6.
private static double e1_07
          First error weights array, element 7.
private static double e1_08
          First error weights array, element 8.
private static double e1_09
          First error weights array, element 9.
private static double e1_10
          First error weights array, element 10.
private static double e1_11
          First error weights array, element 11.
private static double e1_12
          First error weights array, element 12.
private static double e2_01
          Second error weights array, element 1.
private static double e2_06
          Second error weights array, element 6.
private static double e2_07
          Second error weights array, element 7.
private static double e2_08
          Second error weights array, element 8.
private static double e2_09
          Second error weights array, element 9.
private static double e2_10
          Second error weights array, element 10.
private static double e2_11
          Second error weights array, element 11.
private static double e2_12
          Second error weights array, element 12.
private static String METHOD_NAME
          Integrator method name.
private static double[][] staticA
          Internal weights Butcher array.
private static double[] staticB
          Propagation weights Butcher array.
private static double[] staticC
          Time steps Butcher array.
 
Fields inherited from class org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator
scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeTolerance
 
Fields inherited from class org.apache.commons.math.ode.AbstractIntegrator
eventsHandlersManager, stepHandlers, stepSize, stepStart
 
Constructor Summary
DormandPrince853Integrator(double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
          Simple constructor.
DormandPrince853Integrator(double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
          Simple constructor.
 
Method Summary
protected  double estimateError(double[][] yDotK, double[] y0, double[] y1, double h)
          Compute the error ratio.
 int getOrder()
          Get the order of the method.
 
Methods inherited from class org.apache.commons.math.ode.nonstiff.EmbeddedRungeKuttaIntegrator
getMaxGrowth, getMinReduction, getSafety, integrate, setMaxGrowth, setMinReduction, setSafety
 
Methods inherited from class org.apache.commons.math.ode.nonstiff.AdaptiveStepsizeIntegrator
filterStep, getCurrentStepStart, getMaxStep, getMinStep, initializeStep, resetInternalState, sanityChecks, setInitialStepSize
 
Methods inherited from class org.apache.commons.math.ode.AbstractIntegrator
addEndTimeChecker, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEvaluations, getEventHandlers, getMaxEvaluations, getName, getStepHandlers, requiresDenseOutput, resetEvaluations, setEquations, setMaxEvaluations
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Field Detail

METHOD_NAME

private static final String METHOD_NAME
Integrator method name.

See Also:
Constant Field Values

staticC

private static final double[] staticC
Time steps Butcher array.


staticA

private static final double[][] staticA
Internal weights Butcher array.


staticB

private static final double[] staticB
Propagation weights Butcher array.


e1_01

private static final double e1_01
First error weights array, element 1.

See Also:
Constant Field Values

e1_06

private static final double e1_06
First error weights array, element 6.

See Also:
Constant Field Values

e1_07

private static final double e1_07
First error weights array, element 7.

See Also:
Constant Field Values

e1_08

private static final double e1_08
First error weights array, element 8.

See Also:
Constant Field Values

e1_09

private static final double e1_09
First error weights array, element 9.

See Also:
Constant Field Values

e1_10

private static final double e1_10
First error weights array, element 10.

See Also:
Constant Field Values

e1_11

private static final double e1_11
First error weights array, element 11.

See Also:
Constant Field Values

e1_12

private static final double e1_12
First error weights array, element 12.

See Also:
Constant Field Values

e2_01

private static final double e2_01
Second error weights array, element 1.

See Also:
Constant Field Values

e2_06

private static final double e2_06
Second error weights array, element 6.

See Also:
Constant Field Values

e2_07

private static final double e2_07
Second error weights array, element 7.

See Also:
Constant Field Values

e2_08

private static final double e2_08
Second error weights array, element 8.

See Also:
Constant Field Values

e2_09

private static final double e2_09
Second error weights array, element 9.

See Also:
Constant Field Values

e2_10

private static final double e2_10
Second error weights array, element 10.

See Also:
Constant Field Values

e2_11

private static final double e2_11
Second error weights array, element 11.

See Also:
Constant Field Values

e2_12

private static final double e2_12
Second error weights array, element 12.

See Also:
Constant Field Values
Constructor Detail

DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
                                  double maxStep,
                                  double scalAbsoluteTolerance,
                                  double scalRelativeTolerance)
Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds

Parameters:
minStep - minimal step (must be positive even for backward integration), the last step can be smaller than this
maxStep - maximal step (must be positive even for backward integration)
scalAbsoluteTolerance - allowed absolute error
scalRelativeTolerance - allowed relative error

DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
                                  double maxStep,
                                  double[] vecAbsoluteTolerance,
                                  double[] vecRelativeTolerance)
Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds

Parameters:
minStep - minimal step (must be positive even for backward integration), the last step can be smaller than this
maxStep - maximal step (must be positive even for backward integration)
vecAbsoluteTolerance - allowed absolute error
vecRelativeTolerance - allowed relative error
Method Detail

getOrder

public int getOrder()
Get the order of the method.

Specified by:
getOrder in class EmbeddedRungeKuttaIntegrator
Returns:
order of the method

estimateError

protected double estimateError(double[][] yDotK,
                               double[] y0,
                               double[] y1,
                               double h)
Compute the error ratio.

Specified by:
estimateError in class EmbeddedRungeKuttaIntegrator
Parameters:
yDotK - derivatives computed during the first stages
y0 - estimate of the step at the start of the step
y1 - estimate of the step at the end of the step
h - current step
Returns:
error ratio, greater than 1 if step should be rejected


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