org.jscience.mathematics.analysis.ode
Class RungeKuttaFehlbergIntegrator

java.lang.Object
  extended by org.jscience.mathematics.analysis.ode.AdaptiveStepsizeIntegrator
      extended by org.jscience.mathematics.analysis.ode.RungeKuttaFehlbergIntegrator
All Implemented Interfaces:
FirstOrderIntegrator, Named
Direct Known Subclasses:
DormandPrince54Integrator, DormandPrince853Integrator, HighamHall54Integrator

public abstract class RungeKuttaFehlbergIntegrator
extends AdaptiveStepsizeIntegrator
implements Named

This class implements the common part of all Runge-Kutta-Fehlberg integrators for Ordinary Differential Equations.

These methods are embedded explicit Runge-Kutta methods with two sets of coefficients allowing to estimate the error, their Butcher arrays are as follows :

    0  |
   c2  | a21
   c3  | a31  a32
   ... |        ...
   cs  | as1  as2  ...  ass-1
       |--------------------------
       |  b1   b2  ...   bs-1  bs
       |  b'1  b'2 ...   b's-1 b's
 

In fact, we rather use the array defined by ej = bj - b'j to compute directly the error rather than computing two estimates and then comparing them.

Some methods are qualified as fsal (first same as last) methods. This means the last evaluation of the derivatives in one step is the same as the first in the next step. Then, this evaluation can be reused from one step to the next one and the cost of such a method is really s-1 evaluations despite the method still has s stages. This behaviour is true only for successful steps, if the step is rejected after the error estimation phase, no evaluation is saved. For an fsal method, we have cs = 1 and asi = bi for all i.


Field Summary
 
Fields inherited from class org.jscience.mathematics.analysis.ode.AdaptiveStepsizeIntegrator
handler, scalAbsoluteTolerance, scalRelativeTolerance, switchesHandler, vecAbsoluteTolerance, vecRelativeTolerance
 
Constructor Summary
protected RungeKuttaFehlbergIntegrator(boolean fsal, double[] c, double[][] a, double[] b, org.jscience.mathematics.analysis.ode.RungeKuttaStepInterpolator prototype, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
          Build a Runge-Kutta integrator with the given Butcher array.
protected RungeKuttaFehlbergIntegrator(boolean fsal, double[] c, double[][] a, double[] b, org.jscience.mathematics.analysis.ode.RungeKuttaStepInterpolator prototype, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
          Build a Runge-Kutta integrator with the given Butcher array.
 
Method Summary
protected abstract  double estimateError(double[][] yDotK, double[] y0, double[] y1, double h)
          Compute the error ratio.
 double getMaxGrowth()
          Get the maximal growth factor for stepsize control.
 double getMinReduction()
          Get the minimal reduction factor for stepsize control.
abstract  java.lang.String getName()
          Get the name of the method.
abstract  int getOrder()
          Get the order of the method.
 double getSafety()
          Get the safety factor for stepsize control.
 void integrate(FirstOrderDifferentialEquations equations, double t0, double[] y0, double t, double[] y)
          Integrate the differential equations up to the given time
 void setMaxGrowth(double maxGrowth)
          Set the maximal growth factor for stepsize control.
 void setMinReduction(double minReduction)
          Set the minimal reduction factor for stepsize control.
 void setSafety(double safety)
          Set the safety factor for stepsize control.
 
Methods inherited from class org.jscience.mathematics.analysis.ode.AdaptiveStepsizeIntegrator
addSwitchingFunction, filterStep, getMaxStep, getMinStep, getStepHandler, initializeStep, setInitialStepSize, setStepHandler
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

RungeKuttaFehlbergIntegrator

protected RungeKuttaFehlbergIntegrator(boolean fsal,
                                       double[] c,
                                       double[][] a,
                                       double[] b,
                                       org.jscience.mathematics.analysis.ode.RungeKuttaStepInterpolator prototype,
                                       double minStep,
                                       double maxStep,
                                       double scalAbsoluteTolerance,
                                       double scalRelativeTolerance)
Build a Runge-Kutta integrator with the given Butcher array.

Parameters:
fsal - indicate that the method is an fsal
c - time steps from Butcher array (without the first zero)
a - internal weights from Butcher array (without the first empty row)
b - external weights for the high order method from Butcher array
prototype - prototype of the step interpolator to use
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

RungeKuttaFehlbergIntegrator

protected RungeKuttaFehlbergIntegrator(boolean fsal,
                                       double[] c,
                                       double[][] a,
                                       double[] b,
                                       org.jscience.mathematics.analysis.ode.RungeKuttaStepInterpolator prototype,
                                       double minStep,
                                       double maxStep,
                                       double[] vecAbsoluteTolerance,
                                       double[] vecRelativeTolerance)
Build a Runge-Kutta integrator with the given Butcher array.

Parameters:
fsal - indicate that the method is an fsal
c - time steps from Butcher array (without the first zero)
a - internal weights from Butcher array (without the first empty row)
b - external weights for the high order method from Butcher array
prototype - prototype of the step interpolator to use
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

getName

public abstract java.lang.String getName()
Get the name of the method.

Specified by:
getName in interface FirstOrderIntegrator
Specified by:
getName in interface Named
Returns:
name of the method

getOrder

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

Returns:
order of the method

getSafety

public double getSafety()
Get the safety factor for stepsize control.

Returns:
safety factor

setSafety

public void setSafety(double safety)
Set the safety factor for stepsize control.

Parameters:
safety - safety factor

integrate

public void integrate(FirstOrderDifferentialEquations equations,
                      double t0,
                      double[] y0,
                      double t,
                      double[] y)
               throws DerivativeException,
                      IntegrationException
Description copied from interface: FirstOrderIntegrator
Integrate the differential equations up to the given time

Specified by:
integrate in interface FirstOrderIntegrator
Specified by:
integrate in class AdaptiveStepsizeIntegrator
Parameters:
equations - differential equations to integrate
t0 - initial time
y0 - initial value of the state vector at t0
t - target time for the integration (can be set to a value smaller thant t0 for backward integration)
y - placeholder where to put the state vector at each successful step (and hence at the end of integration), can be the same object as y0
Throws:
DerivativeException - this exception is propagated to the caller if the underlying user function triggers one
IntegrationException - if the integrator cannot perform integration

getMinReduction

public double getMinReduction()
Get the minimal reduction factor for stepsize control.

Returns:
minimal reduction factor

setMinReduction

public void setMinReduction(double minReduction)
Set the minimal reduction factor for stepsize control.

Parameters:
minReduction - minimal reduction factor

getMaxGrowth

public double getMaxGrowth()
Get the maximal growth factor for stepsize control.

Returns:
maximal growth factor

setMaxGrowth

public void setMaxGrowth(double maxGrowth)
Set the maximal growth factor for stepsize control.

Parameters:
maxGrowth - maximal growth factor

estimateError

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

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