Package org.apache.commons.math3.ode

Class MultistepIntegrator

java.lang.Object
org.apache.commons.math3.ode.AbstractIntegrator
org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator
org.apache.commons.math3.ode.MultistepIntegrator
All Implemented Interfaces:
FirstOrderIntegrator, ODEIntegrator
Direct Known Subclasses:
AdamsIntegrator
public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator
This class is the base class for multistep integrators for Ordinary Differential Equations.

We define scaled derivatives si(n) at step n as: 

 s1(n) = h y'n for first derivative
 s2(n) = h2/2 y''n for second derivative
 s3(n) = h3/6 y'''n for third derivative
 ...
 sk(n) = hk/k! y(k)n for kth derivative

Rather than storing several previous steps separately, this implementation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step (yn, s1(n) and rn) where rn is defined as: 

 rn = [ s2(n), s3(n) ... sk(n) ]T
(we omit the k index in the notation for clarity)

Multistep integrators with Nordsieck representation are highly sensitive to large step changes because when the step is multiplied by factor a, the kth component of the Nordsieck vector is multiplied by ak and the last components are the least accurate ones. The default max growth factor is therefore set to a quite low value: 21/order.

Since:
2.0
See Also:

  • Field Details

    • scaled

      protected double[] scaled
      First scaled derivative (h y').

    • nordsieck

      protected Array2DRowRealMatrix nordsieck
      Nordsieck matrix of the higher scaled derivatives.

      (h2/2 y'', h3/6 y''' ..., hk/k! y(k))

  • Constructor Details

    • MultistepIntegrator

      protected MultistepIntegrator(String name, int nSteps, int order, double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance) throws NumberIsTooSmallException
      Build a multistep integrator with the given stepsize bounds.

      The default starter integrator is set to the Dormand-Prince 8(5,3) integrator with some defaults settings.

      The default max growth factor is set to a quite low value: 21/order.

      Parameters:
      name - name of the method
      nSteps - number of steps of the multistep method (excluding the one being computed)
      order - order of the method
      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
      Throws:
      NumberIsTooSmallException - if number of steps is smaller than 2

    • MultistepIntegrator

      protected MultistepIntegrator(String name, int nSteps, int order, double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
      Build a multistep integrator with the given stepsize bounds.

      The default starter integrator is set to the Dormand-Prince 8(5,3) integrator with some defaults settings.

      The default max growth factor is set to a quite low value: 21/order.

      Parameters:
      name - name of the method
      nSteps - number of steps of the multistep method (excluding the one being computed)
      order - order of the method
      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 Details

    • getStarterIntegrator

      public ODEIntegrator getStarterIntegrator()
      Get the starter integrator.
      Returns:
      starter integrator

    • setStarterIntegrator

      public void setStarterIntegrator(FirstOrderIntegrator starterIntegrator)
      Set the starter integrator.

      The various step and event handlers for this starter integrator will be managed automatically by the multi-step integrator. Any user configuration for these elements will be cleared before use.

      Parameters:
      starterIntegrator - starter integrator

    • start

      protected void start(double t0, double[] y0, double t) throws DimensionMismatchException, NumberIsTooSmallException, MaxCountExceededException, NoBracketingException
      Start the integration.

      This method computes one step using the underlying starter integrator, and initializes the Nordsieck vector at step start. The starter integrator purpose is only to establish initial conditions, it does not really change time by itself. The top level multistep integrator remains in charge of handling time propagation and events handling as it will starts its own computation right from the beginning. In a sense, the starter integrator can be seen as a dummy one and so it will never trigger any user event nor call any user step handler.

      Parameters:
      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 than t0 for backward integration)
      Throws:
      DimensionMismatchException - if arrays dimension do not match equations settings
      NumberIsTooSmallException - if integration step is too small
      MaxCountExceededException - if the number of functions evaluations is exceeded
      NoBracketingException - if the location of an event cannot be bracketed

    • initializeHighOrderDerivatives

      protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives(double h, double[] t, double[][] y, double[][] yDot)
      Initialize the high order scaled derivatives at step start.
      Parameters:
      h - step size to use for scaling
      t - first steps times
      y - first steps states
      yDot - first steps derivatives
      Returns:
      Nordieck vector at first step (h2/2 y''n, h3/6 y'''n ... hk/k! y(k)n)

    • 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

    • 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

    • getNSteps

      public int getNSteps()
      Get the number of steps of the multistep method (excluding the one being computed).
      Returns:
      number of steps of the multistep method (excluding the one being computed)

    • computeStepGrowShrinkFactor

      protected double computeStepGrowShrinkFactor(double error)
      Compute step grow/shrink factor according to normalized error.
      Parameters:
      error - normalized error of the current step
      Returns:
      grow/shrink factor for next step