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With inspiration from other libraries
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.ode.nonstiff;
import org.apache.commons.math3.Field;
import org.apache.commons.math3.RealFieldElement;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.ode.AbstractFieldIntegrator;
import org.apache.commons.math3.ode.FieldEquationsMapper;
import org.apache.commons.math3.ode.FieldODEState;
import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;
/**
* This abstract class holds the common part of all adaptive
* stepsize integrators for Ordinary Differential Equations.
*
* These algorithms perform integration with stepsize control, which
* means the user does not specify the integration step but rather a
* tolerance on error. The error threshold is computed as
*
* threshold_i = absTol_i + relTol_i * max (abs (ym), abs (ym+1))
*
* where absTol_i is the absolute tolerance for component i of the
* state vector and relTol_i is the relative tolerance for the same
* component. The user can also use only two scalar values absTol and
* relTol which will be used for all components.
*
*
* Note that only the {@link FieldODEState#getState() main part}
* of the state vector is used for stepsize control. The {@link
* FieldODEState#getSecondaryState(int) secondary parts} of the state
* vector are explicitly ignored for stepsize control.
*
*
* If the estimated error for ym+1 is such that
*
* sqrt((sum (errEst_i / threshold_i)^2 ) / n) < 1
*
*
* (where n is the main set dimension) then the step is accepted,
* otherwise the step is rejected and a new attempt is made with a new
* stepsize.
*
* @param the type of the field elements
* @since 3.6
*
*/
public abstract class AdaptiveStepsizeFieldIntegrator>
extends AbstractFieldIntegrator {
/** Allowed absolute scalar error. */
protected double scalAbsoluteTolerance;
/** Allowed relative scalar error. */
protected double scalRelativeTolerance;
/** Allowed absolute vectorial error. */
protected double[] vecAbsoluteTolerance;
/** Allowed relative vectorial error. */
protected double[] vecRelativeTolerance;
/** Main set dimension. */
protected int mainSetDimension;
/** User supplied initial step. */
private T initialStep;
/** Minimal step. */
private T minStep;
/** Maximal step. */
private T maxStep;
/** Build an integrator with the given stepsize bounds.
* The default step handler does nothing.
* @param field field to which the time and state vector elements belong
* @param name name of the method
* @param minStep minimal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param maxStep maximal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param scalAbsoluteTolerance allowed absolute error
* @param scalRelativeTolerance allowed relative error
*/
public AdaptiveStepsizeFieldIntegrator(final Field field, final String name,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance) {
super(field, name);
setStepSizeControl(minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
resetInternalState();
}
/** Build an integrator with the given stepsize bounds.
* The default step handler does nothing.
* @param field field to which the time and state vector elements belong
* @param name name of the method
* @param minStep minimal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param maxStep maximal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param vecAbsoluteTolerance allowed absolute error
* @param vecRelativeTolerance allowed relative error
*/
public AdaptiveStepsizeFieldIntegrator(final Field field, final String name,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance) {
super(field, name);
setStepSizeControl(minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
resetInternalState();
}
/** Set the adaptive step size control parameters.
*
* A side effect of this method is to also reset the initial
* step so it will be automatically computed by the integrator
* if {@link #setInitialStepSize(RealFieldElement) setInitialStepSize}
* is not called by the user.
*
* @param minimalStep minimal step (must be positive even for backward
* integration), the last step can be smaller than this
* @param maximalStep maximal step (must be positive even for backward
* integration)
* @param absoluteTolerance allowed absolute error
* @param relativeTolerance allowed relative error
*/
public void setStepSizeControl(final double minimalStep, final double maximalStep,
final double absoluteTolerance,
final double relativeTolerance) {
minStep = getField().getZero().add(FastMath.abs(minimalStep));
maxStep = getField().getZero().add(FastMath.abs(maximalStep));
initialStep = getField().getOne().negate();
scalAbsoluteTolerance = absoluteTolerance;
scalRelativeTolerance = relativeTolerance;
vecAbsoluteTolerance = null;
vecRelativeTolerance = null;
}
/** Set the adaptive step size control parameters.
*
* A side effect of this method is to also reset the initial
* step so it will be automatically computed by the integrator
* if {@link #setInitialStepSize(RealFieldElement) setInitialStepSize}
* is not called by the user.
*
* @param minimalStep minimal step (must be positive even for backward
* integration), the last step can be smaller than this
* @param maximalStep maximal step (must be positive even for backward
* integration)
* @param absoluteTolerance allowed absolute error
* @param relativeTolerance allowed relative error
*/
public void setStepSizeControl(final double minimalStep, final double maximalStep,
final double[] absoluteTolerance,
final double[] relativeTolerance) {
minStep = getField().getZero().add(FastMath.abs(minimalStep));
maxStep = getField().getZero().add(FastMath.abs(maximalStep));
initialStep = getField().getOne().negate();
scalAbsoluteTolerance = 0;
scalRelativeTolerance = 0;
vecAbsoluteTolerance = absoluteTolerance.clone();
vecRelativeTolerance = relativeTolerance.clone();
}
/** Set the initial step size.
* This method allows the user to specify an initial positive
* step size instead of letting the integrator guess it by
* itself. If this method is not called before integration is
* started, the initial step size will be estimated by the
* integrator.
* @param initialStepSize initial step size to use (must be positive even
* for backward integration ; providing a negative value or a value
* outside of the min/max step interval will lead the integrator to
* ignore the value and compute the initial step size by itself)
*/
public void setInitialStepSize(final T initialStepSize) {
if (initialStepSize.subtract(minStep).getReal() < 0 ||
initialStepSize.subtract(maxStep).getReal() > 0) {
initialStep = getField().getOne().negate();
} else {
initialStep = initialStepSize;
}
}
/** {@inheritDoc} */
@Override
protected void sanityChecks(final FieldODEState eqn, final T t)
throws DimensionMismatchException, NumberIsTooSmallException {
super.sanityChecks(eqn, t);
mainSetDimension = eqn.getStateDimension();
if (vecAbsoluteTolerance != null && vecAbsoluteTolerance.length != mainSetDimension) {
throw new DimensionMismatchException(mainSetDimension, vecAbsoluteTolerance.length);
}
if (vecRelativeTolerance != null && vecRelativeTolerance.length != mainSetDimension) {
throw new DimensionMismatchException(mainSetDimension, vecRelativeTolerance.length);
}
}
/** Initialize the integration step.
* @param forward forward integration indicator
* @param order order of the method
* @param scale scaling vector for the state vector (can be shorter than state vector)
* @param state0 state at integration start time
* @param mapper mapper for all the equations
* @return first integration step
* @exception MaxCountExceededException if the number of functions evaluations is exceeded
* @exception DimensionMismatchException if arrays dimensions do not match equations settings
*/
public T initializeStep(final boolean forward, final int order, final T[] scale,
final FieldODEStateAndDerivative state0,
final FieldEquationsMapper mapper)
throws MaxCountExceededException, DimensionMismatchException {
if (initialStep.getReal() > 0) {
// use the user provided value
return forward ? initialStep : initialStep.negate();
}
// very rough first guess : h = 0.01 * ||y/scale|| / ||y'/scale||
// this guess will be used to perform an Euler step
final T[] y0 = mapper.mapState(state0);
final T[] yDot0 = mapper.mapDerivative(state0);
T yOnScale2 = getField().getZero();
T yDotOnScale2 = getField().getZero();
for (int j = 0; j < scale.length; ++j) {
final T ratio = y0[j].divide(scale[j]);
yOnScale2 = yOnScale2.add(ratio.multiply(ratio));
final T ratioDot = yDot0[j].divide(scale[j]);
yDotOnScale2 = yDotOnScale2.add(ratioDot.multiply(ratioDot));
}
T h = (yOnScale2.getReal() < 1.0e-10 || yDotOnScale2.getReal() < 1.0e-10) ?
getField().getZero().add(1.0e-6) :
yOnScale2.divide(yDotOnScale2).sqrt().multiply(0.01);
if (! forward) {
h = h.negate();
}
// perform an Euler step using the preceding rough guess
final T[] y1 = MathArrays.buildArray(getField(), y0.length);
for (int j = 0; j < y0.length; ++j) {
y1[j] = y0[j].add(yDot0[j].multiply(h));
}
final T[] yDot1 = computeDerivatives(state0.getTime().add(h), y1);
// estimate the second derivative of the solution
T yDDotOnScale = getField().getZero();
for (int j = 0; j < scale.length; ++j) {
final T ratioDotDot = yDot1[j].subtract(yDot0[j]).divide(scale[j]);
yDDotOnScale = yDDotOnScale.add(ratioDotDot.multiply(ratioDotDot));
}
yDDotOnScale = yDDotOnScale.sqrt().divide(h);
// step size is computed such that
// h^order * max (||y'/tol||, ||y''/tol||) = 0.01
final T maxInv2 = MathUtils.max(yDotOnScale2.sqrt(), yDDotOnScale);
final T h1 = maxInv2.getReal() < 1.0e-15 ?
MathUtils.max(getField().getZero().add(1.0e-6), h.abs().multiply(0.001)) :
maxInv2.multiply(100).reciprocal().pow(1.0 / order);
h = MathUtils.min(h.abs().multiply(100), h1);
h = MathUtils.max(h, state0.getTime().abs().multiply(1.0e-12)); // avoids cancellation when computing t1 - t0
h = MathUtils.max(minStep, MathUtils.min(maxStep, h));
if (! forward) {
h = h.negate();
}
return h;
}
/** Filter the integration step.
* @param h signed step
* @param forward forward integration indicator
* @param acceptSmall if true, steps smaller than the minimal value
* are silently increased up to this value, if false such small
* steps generate an exception
* @return a bounded integration step (h if no bound is reach, or a bounded value)
* @exception NumberIsTooSmallException if the step is too small and acceptSmall is false
*/
protected T filterStep(final T h, final boolean forward, final boolean acceptSmall)
throws NumberIsTooSmallException {
T filteredH = h;
if (h.abs().subtract(minStep).getReal() < 0) {
if (acceptSmall) {
filteredH = forward ? minStep : minStep.negate();
} else {
throw new NumberIsTooSmallException(LocalizedFormats.MINIMAL_STEPSIZE_REACHED_DURING_INTEGRATION,
h.abs().getReal(), minStep.getReal(), true);
}
}
if (filteredH.subtract(maxStep).getReal() > 0) {
filteredH = maxStep;
} else if (filteredH.add(maxStep).getReal() < 0) {
filteredH = maxStep.negate();
}
return filteredH;
}
/** Reset internal state to dummy values. */
protected void resetInternalState() {
setStepStart(null);
setStepSize(minStep.multiply(maxStep).sqrt());
}
/** Get the minimal step.
* @return minimal step
*/
public T getMinStep() {
return minStep;
}
/** Get the maximal step.
* @return maximal step
*/
public T getMaxStep() {
return maxStep;
}
}