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* 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.
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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.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.ode.FieldEquationsMapper;
import org.apache.commons.math3.ode.FieldExpandableODE;
import org.apache.commons.math3.ode.FieldODEState;
import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;
/**
* This class implements the common part of all embedded Runge-Kutta
* integrators for Ordinary Differential Equations.
*
* <p>These methods are embedded explicit Runge-Kutta methods with two
* sets of coefficients allowing to estimate the error, their Butcher
* arrays are as follows :
* <pre>
* 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
* </pre>
* </p>
*
* <p>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.</p>
*
* <p>Some methods are qualified as <i>fsal</i> (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 <i>fsal</i> method, we have cs = 1 and
* asi = bi for all i.</p>
*
* @param <T> the type of the field elements
* @since 3.6
*/
public abstract class EmbeddedRungeKuttaFieldIntegrator<T extends RealFieldElement<T>>
extends AdaptiveStepsizeFieldIntegrator<T>
implements FieldButcherArrayProvider<T> {
/** Index of the pre-computed derivative for <i>fsal</i> methods. */
private final int fsal;
/** Time steps from Butcher array (without the first zero). */
private final T[] c;
/** Internal weights from Butcher array (without the first empty row). */
private final T[][] a;
/** External weights for the high order method from Butcher array. */
private final T[] b;
/** Stepsize control exponent. */
private final T exp;
/** Safety factor for stepsize control. */
private T safety;
/** Minimal reduction factor for stepsize control. */
private T minReduction;
/** Maximal growth factor for stepsize control. */
private T maxGrowth;
/** Build a Runge-Kutta integrator with the given Butcher array.
* @param field field to which the time and state vector elements belong
* @param name name of the method
* @param fsal index of the pre-computed derivative for <i>fsal</i> methods
* or -1 if method is not <i>fsal</i>
* @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
*/
protected EmbeddedRungeKuttaFieldIntegrator(final Field<T> field, final String name, final int fsal,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance) {
super(field, name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
this.fsal = fsal;
this.c = getC();
this.a = getA();
this.b = getB();
exp = field.getOne().divide(-getOrder());
// set the default values of the algorithm control parameters
setSafety(field.getZero().add(0.9));
setMinReduction(field.getZero().add(0.2));
setMaxGrowth(field.getZero().add(10.0));
}
/** Build a Runge-Kutta integrator with the given Butcher array.
* @param field field to which the time and state vector elements belong
* @param name name of the method
* @param fsal index of the pre-computed derivative for <i>fsal</i> methods
* or -1 if method is not <i>fsal</i>
* @param minStep minimal step (must be positive even for backward
* integration), the last step can be smaller than this
* @param maxStep maximal step (must be positive even for backward
* integration)
* @param vecAbsoluteTolerance allowed absolute error
* @param vecRelativeTolerance allowed relative error
*/
protected EmbeddedRungeKuttaFieldIntegrator(final Field<T> field, final String name, final int fsal,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance) {
super(field, name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
this.fsal = fsal;
this.c = getC();
this.a = getA();
this.b = getB();
exp = field.getOne().divide(-getOrder());
// set the default values of the algorithm control parameters
setSafety(field.getZero().add(0.9));
setMinReduction(field.getZero().add(0.2));
setMaxGrowth(field.getZero().add(10.0));
}
/** Create a fraction.
* @param p numerator
* @param q denominator
* @return p/q computed in the instance field
*/
protected T fraction(final int p, final int q) {
return getField().getOne().multiply(p).divide(q);
}
/** Create a fraction.
* @param p numerator
* @param q denominator
* @return p/q computed in the instance field
*/
protected T fraction(final double p, final double q) {
return getField().getOne().multiply(p).divide(q);
}
/** Create an interpolator.
* @param forward integration direction indicator
* @param yDotK slopes at the intermediate points
* @param globalPreviousState start of the global step
* @param globalCurrentState end of the global step
* @param mapper equations mapper for the all equations
* @return external weights for the high order method from Butcher array
*/
protected abstract RungeKuttaFieldStepInterpolator<T> createInterpolator(boolean forward, T[][] yDotK,
final FieldODEStateAndDerivative<T> globalPreviousState,
final FieldODEStateAndDerivative<T> globalCurrentState,
FieldEquationsMapper<T> mapper);
/** Get the order of the method.
* @return order of the method
*/
public abstract int getOrder();
/** Get the safety factor for stepsize control.
* @return safety factor
*/
public T getSafety() {
return safety;
}
/** Set the safety factor for stepsize control.
* @param safety safety factor
*/
public void setSafety(final T safety) {
this.safety = safety;
}
/** {@inheritDoc} */
public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations,
final FieldODEState<T> initialState, final T finalTime)
throws NumberIsTooSmallException, DimensionMismatchException,
MaxCountExceededException, NoBracketingException {
sanityChecks(initialState, finalTime);
final T t0 = initialState.getTime();
final T[] y0 = equations.getMapper().mapState(initialState);
setStepStart(initIntegration(equations, t0, y0, finalTime));
final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;
// create some internal working arrays
final int stages = c.length + 1;
T[] y = y0;
final T[][] yDotK = MathArrays.buildArray(getField(), stages, -1);
final T[] yTmp = MathArrays.buildArray(getField(), y0.length);
// set up integration control objects
T hNew = getField().getZero();
boolean firstTime = true;
// main integration loop
setIsLastStep(false);
do {
// iterate over step size, ensuring local normalized error is smaller than 1
T error = getField().getZero().add(10);
while (error.subtract(1.0).getReal() >= 0) {
// first stage
y = equations.getMapper().mapState(getStepStart());
yDotK[0] = equations.getMapper().mapDerivative(getStepStart());
if (firstTime) {
final T[] scale = MathArrays.buildArray(getField(), mainSetDimension);
if (vecAbsoluteTolerance == null) {
for (int i = 0; i < scale.length; ++i) {
scale[i] = y[i].abs().multiply(scalRelativeTolerance).add(scalAbsoluteTolerance);
}
} else {
for (int i = 0; i < scale.length; ++i) {
scale[i] = y[i].abs().multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]);
}
}
hNew = initializeStep(forward, getOrder(), scale, getStepStart(), equations.getMapper());
firstTime = false;
}
setStepSize(hNew);
if (forward) {
if (getStepStart().getTime().add(getStepSize()).subtract(finalTime).getReal() >= 0) {
setStepSize(finalTime.subtract(getStepStart().getTime()));
}
} else {
if (getStepStart().getTime().add(getStepSize()).subtract(finalTime).getReal() <= 0) {
setStepSize(finalTime.subtract(getStepStart().getTime()));
}
}
// next stages
for (int k = 1; k < stages; ++k) {
for (int j = 0; j < y0.length; ++j) {
T sum = yDotK[0][j].multiply(a[k-1][0]);
for (int l = 1; l < k; ++l) {
sum = sum.add(yDotK[l][j].multiply(a[k-1][l]));
}
yTmp[j] = y[j].add(getStepSize().multiply(sum));
}
yDotK[k] = computeDerivatives(getStepStart().getTime().add(getStepSize().multiply(c[k-1])), yTmp);
}
// estimate the state at the end of the step
for (int j = 0; j < y0.length; ++j) {
T sum = yDotK[0][j].multiply(b[0]);
for (int l = 1; l < stages; ++l) {
sum = sum.add(yDotK[l][j].multiply(b[l]));
}
yTmp[j] = y[j].add(getStepSize().multiply(sum));
}
// estimate the error at the end of the step
error = estimateError(yDotK, y, yTmp, getStepSize());
if (error.subtract(1.0).getReal() >= 0) {
// reject the step and attempt to reduce error by stepsize control
final T factor = MathUtils.min(maxGrowth,
MathUtils.max(minReduction, safety.multiply(error.pow(exp))));
hNew = filterStep(getStepSize().multiply(factor), forward, false);
}
}
final T stepEnd = getStepStart().getTime().add(getStepSize());
final T[] yDotTmp = (fsal >= 0) ? yDotK[fsal] : computeDerivatives(stepEnd, yTmp);
final FieldODEStateAndDerivative<T> stateTmp = new FieldODEStateAndDerivative<T>(stepEnd, yTmp, yDotTmp);
// local error is small enough: accept the step, trigger events and step handlers
System.arraycopy(yTmp, 0, y, 0, y0.length);
setStepStart(acceptStep(createInterpolator(forward, yDotK, getStepStart(), stateTmp, equations.getMapper()),
finalTime));
if (!isLastStep()) {
// stepsize control for next step
final T factor = MathUtils.min(maxGrowth,
MathUtils.max(minReduction, safety.multiply(error.pow(exp))));
final T scaledH = getStepSize().multiply(factor);
final T nextT = getStepStart().getTime().add(scaledH);
final boolean nextIsLast = forward ?
nextT.subtract(finalTime).getReal() >= 0 :
nextT.subtract(finalTime).getReal() <= 0;
hNew = filterStep(scaledH, forward, nextIsLast);
final T filteredNextT = getStepStart().getTime().add(hNew);
final boolean filteredNextIsLast = forward ?
filteredNextT.subtract(finalTime).getReal() >= 0 :
filteredNextT.subtract(finalTime).getReal() <= 0;
if (filteredNextIsLast) {
hNew = finalTime.subtract(getStepStart().getTime());
}
}
} while (!isLastStep());
final FieldODEStateAndDerivative<T> finalState = getStepStart();
resetInternalState();
return finalState;
}
/** Get the minimal reduction factor for stepsize control.
* @return minimal reduction factor
*/
public T getMinReduction() {
return minReduction;
}
/** Set the minimal reduction factor for stepsize control.
* @param minReduction minimal reduction factor
*/
public void setMinReduction(final T minReduction) {
this.minReduction = minReduction;
}
/** Get the maximal growth factor for stepsize control.
* @return maximal growth factor
*/
public T getMaxGrowth() {
return maxGrowth;
}
/** Set the maximal growth factor for stepsize control.
* @param maxGrowth maximal growth factor
*/
public void setMaxGrowth(final T maxGrowth) {
this.maxGrowth = maxGrowth;
}
/** Compute the error ratio.
* @param yDotK derivatives computed during the first stages
* @param y0 estimate of the step at the start of the step
* @param y1 estimate of the step at the end of the step
* @param h current step
* @return error ratio, greater than 1 if step should be rejected
*/
protected abstract T estimateError(T[][] yDotK, T[] y0, T[] y1, T h);
}