Apache Commons Math 3.6.1 · org.apache.commons.math3.ode.nonstiff.EmbeddedRungeKuttaIntegrator

This class implements the common part of all embedded Runge-Kutta 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

Modifier and Type	Field and Description
private final double[][]	a Internal weights from Butcher array (without the first empty row).
private final double[]	b External weights for the high order method from Butcher array.
private final double[]	c Time steps from Butcher array (without the first zero).
private final double	exp Stepsize control exponent.
private final boolean	fsal Indicator for fsal methods.
private double	maxGrowth Maximal growth factor for stepsize control.
private double	minReduction Minimal reduction factor for stepsize control.
private final RungeKuttaStepInterpolator	prototype Prototype of the step interpolator.
private double	safety Safety factor for stepsize control.

Inherited from org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator:: mainSetDimension scalAbsoluteTolerance scalRelativeTolerance vecAbsoluteTolerance vecRelativeTolerance

Constructor Summary

Access	Constructor and Description
protected	EmbeddedRungeKuttaIntegrator(final String name of the method name, final boolean indicate that the method is an fsal fsal, final double[] time steps from Butcher array (without the first zero) c, final double[][] internal weights from Butcher array (without the first empty row) a, final double[] propagation weights for the high order method from Butcher array b, final RungeKuttaStepInterpolator prototype of the step interpolator to use prototype, final double minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this minStep, final double maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this maxStep, final double allowed absolute error scalAbsoluteTolerance, final double allowed relative error scalRelativeTolerance) Build a Runge-Kutta integrator with the given Butcher array.
protected	EmbeddedRungeKuttaIntegrator(final String name of the method name, final boolean indicate that the method is an fsal fsal, final double[] time steps from Butcher array (without the first zero) c, final double[][] internal weights from Butcher array (without the first empty row) a, final double[] propagation weights for the high order method from Butcher array b, final RungeKuttaStepInterpolator prototype of the step interpolator to use prototype, final double minimal step (must be positive even for backward integration), the last step can be smaller than this minStep, final double maximal step (must be positive even for backward integration) maxStep, final double[] allowed absolute error vecAbsoluteTolerance, final double[] allowed relative error vecRelativeTolerance) Build a Runge-Kutta integrator with the given Butcher array.

Access

Constructor and Description

protected

EmbeddedRungeKuttaIntegrator(final String

name of the method

name, final boolean

indicate that the method is an fsal

fsal, final double[]

time steps from Butcher array (without the first zero)

c, final double[][]

internal weights from Butcher array (without the first empty row)

a, final double[]

propagation weights for the high order method from Butcher array

b, final RungeKuttaStepInterpolator

prototype of the step interpolator to use

prototype, final double

minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this

minStep, final double

maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this

maxStep, final double

allowed absolute error

scalAbsoluteTolerance, final double

allowed relative error

scalRelativeTolerance)

Build a Runge-Kutta integrator with the given Butcher array.

protected

EmbeddedRungeKuttaIntegrator(final String

name of the method

name, final boolean

indicate that the method is an fsal

fsal, final double[]

time steps from Butcher array (without the first zero)

c, final double[][]

internal weights from Butcher array (without the first empty row)

a, final double[]

propagation weights for the high order method from Butcher array

b, final RungeKuttaStepInterpolator

prototype of the step interpolator to use

prototype, final double

minimal step (must be positive even for backward integration), the last step can be smaller than this

minStep, final double

maximal step (must be positive even for backward integration)

maxStep, final double[]

allowed absolute error

vecAbsoluteTolerance, final double[]

allowed relative error

vecRelativeTolerance)

Build a Runge-Kutta integrator with the given Butcher array.

Method Summary

Modifier and Type	Method and Description
protected abstract double	Returns: error ratio, greater than 1 if step should be rejected estimateError(double[][] derivatives computed during the first stages yDotK, double[] estimate of the step at the start of the step y0, double[] estimate of the step at the end of the step y1, double current step h) Compute the error ratio.
public double	Returns: maximal growth factor getMaxGrowth() Get the maximal growth factor for stepsize control.
public double	Returns: minimal reduction factor getMinReduction() Get the minimal reduction factor for stepsize control.
public abstract int	Returns: order of the method getOrder() Get the order of the method.
public double	Returns: safety factor getSafety() Get the safety factor for stepsize control.
public void	integrate(final ExpandableStatefulODE complete set of differential equations to integrate equations, final double target time for the integration (can be set to a value smaller than `t0` for backward integration) t) Implements abstract org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator.integrate. Integrate a set of differential equations up to the given time.
public void	setMaxGrowth(final double maximal growth factor maxGrowth) Set the maximal growth factor for stepsize control.
public void	setMinReduction(final double minimal reduction factor minReduction) Set the minimal reduction factor for stepsize control.
public void	setSafety(final double safety factor safety) Set the safety factor for stepsize control.

Inherited from org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator:: filterStep getCurrentStepStart getMaxStep getMinStep initializeStep resetInternalState sanityChecks setInitialStepSize setStepSizeControl setStepSizeControl

Field Detail

a	back to summary
private final double[][] a Internal weights from Butcher array (without the first empty row).

b	back to summary
private final double[] b External weights for the high order method from Butcher array.

c	back to summary
private final double[] c Time steps from Butcher array (without the first zero).

exp	back to summary
private final double exp Stepsize control exponent.

fsal	back to summary
private final boolean fsal Indicator for fsal methods.

maxGrowth	back to summary
private double maxGrowth Maximal growth factor for stepsize control.

minReduction	back to summary
private double minReduction Minimal reduction factor for stepsize control.

prototype	back to summary
private final RungeKuttaStepInterpolator prototype Prototype of the step interpolator.

safety	back to summary
private double safety Safety factor for stepsize control.

Constructor Detail

EmbeddedRungeKuttaIntegrator	back to summary
protected EmbeddedRungeKuttaIntegrator(final String name, final boolean fsal, final double[] c, final double[][] a, final double[] b, final RungeKuttaStepInterpolator prototype, final double minStep, final double maxStep, final double scalAbsoluteTolerance, final double scalRelativeTolerance) Build a Runge-Kutta integrator with the given Butcher array. Parameters name:String name of the method fsal:boolean indicate that the method is an fsal c:double[] time steps from Butcher array (without the first zero) a:double[][] internal weights from Butcher array (without the first empty row) b:double[] propagation weights for the high order method from Butcher array prototype:RungeKuttaStepInterpolator prototype of the step interpolator to use minStep:double minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this maxStep:double maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this scalAbsoluteTolerance:double allowed absolute error scalRelativeTolerance:double allowed relative error

EmbeddedRungeKuttaIntegrator

back to summary

protected EmbeddedRungeKuttaIntegrator(final String name, final boolean fsal, final double[] c, final double[][] a, final double[] b, final RungeKuttaStepInterpolator prototype, final double minStep, final double maxStep, final double scalAbsoluteTolerance, final double scalRelativeTolerance)

Build a Runge-Kutta integrator with the given Butcher array.

Parameters

name:String: name of the method
fsal:boolean: indicate that the method is an fsal
c:double[]: time steps from Butcher array (without the first zero)
a:double[][]: internal weights from Butcher array (without the first empty row)
b:double[]: propagation weights for the high order method from Butcher array
prototype:RungeKuttaStepInterpolator: prototype of the step interpolator to use
minStep:double: minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
maxStep:double: maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
scalAbsoluteTolerance:double: allowed absolute error
scalRelativeTolerance:double: allowed relative error

EmbeddedRungeKuttaIntegrator	back to summary
protected EmbeddedRungeKuttaIntegrator(final String name, final boolean fsal, final double[] c, final double[][] a, final double[] b, final RungeKuttaStepInterpolator prototype, final double minStep, final double maxStep, final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance) Build a Runge-Kutta integrator with the given Butcher array. Parameters name:String name of the method fsal:boolean indicate that the method is an fsal c:double[] time steps from Butcher array (without the first zero) a:double[][] internal weights from Butcher array (without the first empty row) b:double[] propagation weights for the high order method from Butcher array prototype:RungeKuttaStepInterpolator prototype of the step interpolator to use minStep:double minimal step (must be positive even for backward integration), the last step can be smaller than this maxStep:double maximal step (must be positive even for backward integration) vecAbsoluteTolerance:double[] allowed absolute error vecRelativeTolerance:double[] allowed relative error

EmbeddedRungeKuttaIntegrator

back to summary

protected EmbeddedRungeKuttaIntegrator(final String name, final boolean fsal, final double[] c, final double[][] a, final double[] b, final RungeKuttaStepInterpolator prototype, final double minStep, final double maxStep, final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance)

Build a Runge-Kutta integrator with the given Butcher array.

Parameters

name:String: name of the method
fsal:boolean: indicate that the method is an fsal
c:double[]: time steps from Butcher array (without the first zero)
a:double[][]: internal weights from Butcher array (without the first empty row)
b:double[]: propagation weights for the high order method from Butcher array
prototype:RungeKuttaStepInterpolator: prototype of the step interpolator to use
minStep:double: minimal step (must be positive even for backward integration), the last step can be smaller than this
maxStep:double: maximal step (must be positive even for backward integration)
vecAbsoluteTolerance:double[]: allowed absolute error
vecRelativeTolerance:double[]: allowed relative error

Method Detail

estimateError	back to summary
protected abstract double estimateError(double[][] yDotK, double[] y0, double[] y1, double h) Compute the error ratio. Parameters yDotK:double[][] derivatives computed during the first stages y0:double[] estimate of the step at the start of the step y1:double[] estimate of the step at the end of the step h:double current step Returns:double error ratio, greater than 1 if step should be rejected

estimateError

back to summary

protected abstract double estimateError(double[][] yDotK, double[] y0, double[] y1, double h)

Compute the error ratio.

Parameters

yDotK:double[][]: derivatives computed during the first stages
y0:double[]: estimate of the step at the start of the step
y1:double[]: estimate of the step at the end of the step
h:double: current step

Returns:double

error ratio, greater than 1 if step should be rejected

getMaxGrowth	back to summary
public double getMaxGrowth() Get the maximal growth factor for stepsize control. Returns:double maximal growth factor

getMaxGrowth

back to summary

public double getMaxGrowth()

Get the maximal growth factor for stepsize control.

Returns:double: maximal growth factor

getMinReduction	back to summary
public double getMinReduction() Get the minimal reduction factor for stepsize control. Returns:double minimal reduction factor

getMinReduction

back to summary

public double getMinReduction()

Get the minimal reduction factor for stepsize control.

Returns:double: minimal reduction factor

getOrder	back to summary
public abstract int getOrder() Get the order of the method. Returns:int order of the method

getOrder

back to summary

public abstract int getOrder()

Get the order of the method.

Returns:int: order of the method

getSafety	back to summary
public double getSafety() Get the safety factor for stepsize control. Returns:double safety factor

getSafety

back to summary

public double getSafety()

Get the safety factor for stepsize control.

Returns:double: safety factor

integrate back to summary

integrate	back to summary
public void integrate(final ExpandableStatefulODE equations, final double t) throws NumberIsTooSmallException, DimensionMismatchException, MaxCountExceededException, NoBracketingException Implements abstract org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator.integrate. Doc from org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator.integrate. Integrate a set of differential equations up to the given time. This method solves an Initial Value Problem (IVP). The set of differential equations is composed of a main set, which can be extended by some sets of secondary equations. The set of equations must be already set up with initial time and partial states. At integration completion, the final time and partial states will be available in the same object. Since this method stores some internal state variables made available in its public interface during integration (`getCurrentSignedStepsize()`), it is not thread-safe. Parameters equations:ExpandableStatefulODE Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate. complete set of differential equations to integrate t:double Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate. target time for the integration (can be set to a value smaller than `t0` for backward integration) Annotations @Override Exceptions NumberIsTooSmallException: Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate. if integration step is too small DimensionMismatchException: Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate. if the dimension of the complete state does not match the complete equations sets dimension MaxCountExceededException: Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate. if the number of functions evaluations is exceeded NoBracketingException: Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate. if the location of an event cannot be bracketed

public void integrate(final ExpandableStatefulODE equations, final double t) throws NumberIsTooSmallException, DimensionMismatchException, MaxCountExceededException, NoBracketingException

Implements abstract org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator.integrate.

Doc from org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator.integrate.

Integrate a set of differential equations up to the given time.

This method solves an Initial Value Problem (IVP).

The set of differential equations is composed of a main set, which can be extended by some sets of secondary equations. The set of equations must be already set up with initial time and partial states. At integration completion, the final time and partial states will be available in the same object.

Since this method stores some internal state variables made available in its public interface during integration (getCurrentSignedStepsize()), it is not thread-safe.

Parameters

equations:ExpandableStatefulODE

Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate.

complete set of differential equations to integrate

t:double

Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate.

target time for the integration (can be set to a value smaller than t0 for backward integration)

Annotations

@Override

Exceptions

NumberIsTooSmallException:

Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate.

if integration step is too small

DimensionMismatchException:

Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate.

if the dimension of the complete state does not match the complete equations sets dimension

MaxCountExceededException:

Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate.

if the number of functions evaluations is exceeded

NoBracketingException:

Doc from org.apache.commons.math3.ode.AbstractIntegrator.integrate.

if the location of an event cannot be bracketed

setMaxGrowth	back to summary
public void setMaxGrowth(final double maxGrowth) Set the maximal growth factor for stepsize control. Parameters maxGrowth:double maximal growth factor

setMaxGrowth

back to summary

public void setMaxGrowth(final double maxGrowth)

Set the maximal growth factor for stepsize control.

Parameters

maxGrowth:double: maximal growth factor

setMinReduction	back to summary
public void setMinReduction(final double minReduction) Set the minimal reduction factor for stepsize control. Parameters minReduction:double minimal reduction factor

setMinReduction

back to summary

public void setMinReduction(final double minReduction)

Set the minimal reduction factor for stepsize control.

Parameters

minReduction:double: minimal reduction factor

setSafety	back to summary
public void setSafety(final double safety) Set the safety factor for stepsize control. Parameters safety:double safety factor

setSafety

back to summary

public void setSafety(final double safety)

Set the safety factor for stepsize control.

Parameters

safety:double: safety factor

public abstract Class EmbeddedRungeKuttaIntegrator

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail

Method Detail