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

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 T[][]	a Internal weights from Butcher array (without the first empty row).
private final T[]	b External weights for the high order method from Butcher array.
private final T[]	c Time steps from Butcher array (without the first zero).
private final T	exp Stepsize control exponent.
private final int	fsal Index of the pre-computed derivative for fsal methods.
private T	maxGrowth Maximal growth factor for stepsize control.
private T	minReduction Minimal reduction factor for stepsize control.
private T	safety Safety factor for stepsize control.

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

Constructor Summary

Access	Constructor and Description
protected	EmbeddedRungeKuttaFieldIntegrator(final Field<T> field to which the time and state vector elements belong field, final String name of the method name, final int index of the pre-computed derivative for fsal methods or -1 if method is not fsal fsal, 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	EmbeddedRungeKuttaFieldIntegrator(final Field<T> field to which the time and state vector elements belong field, final String name of the method name, final int index of the pre-computed derivative for fsal methods or -1 if method is not fsal fsal, 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

EmbeddedRungeKuttaFieldIntegrator(final Field<T>

field to which the time and state vector elements belong

field, final String

name of the method

name, final int

index of the pre-computed derivative for fsal methods or -1 if method is not fsal

fsal, 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

EmbeddedRungeKuttaFieldIntegrator(final Field<T>

field to which the time and state vector elements belong

field, final String

name of the method

name, final int

index of the pre-computed derivative for fsal methods or -1 if method is not fsal

fsal, 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 RungeKuttaFieldStepInterpolator<T>	Returns: external weights for the high order method from Butcher array createInterpolator(boolean integration direction indicator forward, T[][] slopes at the intermediate points yDotK, final FieldODEStateAndDerivative<T> start of the global step globalPreviousState, final FieldODEStateAndDerivative<T> end of the global step globalCurrentState, FieldEquationsMapper<T> equations mapper for the all equations mapper) Create an interpolator.
protected abstract T	Returns: error ratio, greater than 1 if step should be rejected estimateError(T[][] derivatives computed during the first stages yDotK, T[] estimate of the step at the start of the step y0, T[] estimate of the step at the end of the step y1, T current step h) Compute the error ratio.
protected T	Returns: p/q computed in the instance field fraction(final int numerator p, final int denominator q) Create a fraction.
protected T	Returns: p/q computed in the instance field fraction(final double numerator p, final double denominator q) Create a fraction.
public T	Returns: maximal growth factor getMaxGrowth() Get the maximal growth factor for stepsize control.
public T	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 T	Returns: safety factor getSafety() Get the safety factor for stepsize control.
public FieldODEStateAndDerivative<T>	integrate(final FieldExpandableODE<T> differential equations to integrate equations, final FieldODEState<T> initial state (time, primary and secondary state vectors) initialState, final T target time for the integration (can be set to a value smaller than `t0` for backward integration) finalTime) Implements org.apache.commons.math3.ode.FirstOrderFieldIntegrator.integrate. Integrate the differential equations up to the given time.
public void	setMaxGrowth(final T maximal growth factor maxGrowth) Set the maximal growth factor for stepsize control.
public void	setMinReduction(final T minimal reduction factor minReduction) Set the minimal reduction factor for stepsize control.
public void	setSafety(final T safety factor safety) Set the safety factor for stepsize control.

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

Field Detail

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

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

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

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

fsal	back to summary
private final int fsal Index of the pre-computed derivative for fsal methods.

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

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

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

Constructor Detail

EmbeddedRungeKuttaFieldIntegrator	back to summary
protected EmbeddedRungeKuttaFieldIntegrator(final Field<T> field, final String name, final int fsal, final double minStep, final double maxStep, final double scalAbsoluteTolerance, final double scalRelativeTolerance) Build a Runge-Kutta integrator with the given Butcher array. Parameters field:Field<T> field to which the time and state vector elements belong name:String name of the method fsal:int index of the pre-computed derivative for fsal methods or -1 if method is not fsal 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

EmbeddedRungeKuttaFieldIntegrator

back to summary

protected EmbeddedRungeKuttaFieldIntegrator(final Field<T> field, final String name, final int fsal, final double minStep, final double maxStep, final double scalAbsoluteTolerance, final double scalRelativeTolerance)

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

Parameters

field:Field<T>: field to which the time and state vector elements belong
name:String: name of the method
fsal:int: index of the pre-computed derivative for fsal methods or -1 if method is not fsal
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

EmbeddedRungeKuttaFieldIntegrator	back to summary
protected EmbeddedRungeKuttaFieldIntegrator(final Field<T> field, final String name, final int fsal, final double minStep, final double maxStep, final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance) Build a Runge-Kutta integrator with the given Butcher array. Parameters field:Field<T> field to which the time and state vector elements belong name:String name of the method fsal:int index of the pre-computed derivative for fsal methods or -1 if method is not fsal 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

EmbeddedRungeKuttaFieldIntegrator

back to summary

protected EmbeddedRungeKuttaFieldIntegrator(final Field<T> field, final String name, final int fsal, final double minStep, final double maxStep, final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance)

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

Parameters

field:Field<T>: field to which the time and state vector elements belong
name:String: name of the method
fsal:int: index of the pre-computed derivative for fsal methods or -1 if method is not fsal
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

createInterpolator	back to summary
protected abstract RungeKuttaFieldStepInterpolator<T> createInterpolator(boolean forward, T[][] yDotK, final FieldODEStateAndDerivative<T> globalPreviousState, final FieldODEStateAndDerivative<T> globalCurrentState, FieldEquationsMapper<T> mapper) Create an interpolator. Parameters forward:boolean integration direction indicator yDotK:T[][] slopes at the intermediate points globalPreviousState:FieldODEStateAndDerivative<T> start of the global step globalCurrentState:FieldODEStateAndDerivative<T> end of the global step mapper:FieldEquationsMapper<T> equations mapper for the all equations Returns:RungeKuttaFieldStepInterpolator<T> external weights for the high order method from Butcher array

createInterpolator

back to summary

protected abstract RungeKuttaFieldStepInterpolator<T> createInterpolator(boolean forward, T[][] yDotK, final FieldODEStateAndDerivative<T> globalPreviousState, final FieldODEStateAndDerivative<T> globalCurrentState, FieldEquationsMapper<T> mapper)

Create an interpolator.

Parameters

forward:boolean: integration direction indicator
yDotK:T[][]: slopes at the intermediate points
globalPreviousState:FieldODEStateAndDerivative<T>: start of the global step
globalCurrentState:FieldODEStateAndDerivative<T>: end of the global step
mapper:FieldEquationsMapper<T>: equations mapper for the all equations

Returns:RungeKuttaFieldStepInterpolator<T>

external weights for the high order method from Butcher array

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

estimateError

back to summary

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

Compute the error ratio.

Parameters

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

Returns:T

error ratio, greater than 1 if step should be rejected

fraction	back to summary
protected T fraction(final int p, final int q) Create a fraction. Parameters p:int numerator q:int denominator Returns:T p/q computed in the instance field

fraction

back to summary

protected T fraction(final int p, final int q)

Create a fraction.

Parameters

p:int: numerator
q:int: denominator

Returns:T

p/q computed in the instance field

fraction	back to summary
protected T fraction(final double p, final double q) Create a fraction. Parameters p:double numerator q:double denominator Returns:T p/q computed in the instance field

fraction

back to summary

protected T fraction(final double p, final double q)

Create a fraction.

Parameters

p:double: numerator
q:double: denominator

Returns:T

p/q computed in the instance field

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

getMaxGrowth

back to summary

public T getMaxGrowth()

Get the maximal growth factor for stepsize control.

Returns:T: maximal growth factor

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

getMinReduction

back to summary

public T getMinReduction()

Get the minimal reduction factor for stepsize control.

Returns:T: 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 T getSafety() Get the safety factor for stepsize control. Returns:T safety factor

getSafety

back to summary

public T getSafety()

Get the safety factor for stepsize control.

Returns:T: safety factor

integrate back to summary

integrate	back to summary
public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations, final FieldODEState<T> initialState, final T finalTime) throws NumberIsTooSmallException, DimensionMismatchException, MaxCountExceededException, NoBracketingException Implements org.apache.commons.math3.ode.FirstOrderFieldIntegrator.integrate. Doc from org.apache.commons.math3.ode.FirstOrderFieldIntegrator.integrate. Integrate the differential equations up to the given time. This method solves an Initial Value Problem (IVP). Since this method stores some internal state variables made available in its public interface during integration (`getCurrentSignedStepsize()`), it is not thread-safe. Parameters equations:FieldExpandableODE<T> differential equations to integrate initialState:FieldODEState<T> initial state (time, primary and secondary state vectors) finalTime:T target time for the integration (can be set to a value smaller than `t0` for backward integration) Returns:FieldODEStateAndDerivative<T> final state, its time will be the same as `finalTime` if integration reached its target, but may be different if some `org.apache.commons.math3.ode.events.FieldEventHandler` stops it at some point. Exceptions 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

public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations, final FieldODEState<T> initialState, final T finalTime) throws NumberIsTooSmallException, DimensionMismatchException, MaxCountExceededException, NoBracketingException

Implements org.apache.commons.math3.ode.FirstOrderFieldIntegrator.integrate.

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

Integrate the differential equations up to the given time.

This method solves an Initial Value Problem (IVP).

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

Parameters

equations:FieldExpandableODE<T>: differential equations to integrate
initialState:FieldODEState<T>: initial state (time, primary and secondary state vectors)
finalTime:T: target time for the integration (can be set to a value smaller than t0 for backward integration)

Returns:FieldODEStateAndDerivative<T>

final state, its time will be the same as finalTime if integration reached its target, but may be different if some org.apache.commons.math3.ode.events.FieldEventHandler stops it at some point.

Exceptions

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

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

setMaxGrowth

back to summary

public void setMaxGrowth(final T maxGrowth)

Set the maximal growth factor for stepsize control.

Parameters

maxGrowth:T: maximal growth factor

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

setMinReduction

back to summary

public void setMinReduction(final T minReduction)

Set the minimal reduction factor for stepsize control.

Parameters

minReduction:T: minimal reduction factor

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

setSafety

back to summary

public void setSafety(final T safety)

Set the safety factor for stepsize control.

Parameters

safety:T: safety factor

public abstract Class EmbeddedRungeKuttaFieldIntegrator<T extends RealFieldElement<T>>

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail

Method Detail