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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.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.ExpandableStatefulODE;
import org.apache.commons.math3.util.FastMath;
/**
* 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.
*
* @since 1.2
*/
public abstract class EmbeddedRungeKuttaIntegrator
extends AdaptiveStepsizeIntegrator {
/** Indicator for fsal methods. */
private final boolean fsal;
/** Time steps from Butcher array (without the first zero). */
private final double[] c;
/** Internal weights from Butcher array (without the first empty row). */
private final double[][] a;
/** External weights for the high order method from Butcher array. */
private final double[] b;
/** Prototype of the step interpolator. */
private final RungeKuttaStepInterpolator prototype;
/** Stepsize control exponent. */
private final double exp;
/** Safety factor for stepsize control. */
private double safety;
/** Minimal reduction factor for stepsize control. */
private double minReduction;
/** Maximal growth factor for stepsize control. */
private double maxGrowth;
/** Build a Runge-Kutta integrator with the given Butcher array.
* @param name name of the method
* @param fsal indicate that the method is an fsal
* @param c time steps from Butcher array (without the first zero)
* @param a internal weights from Butcher array (without the first empty row)
* @param b propagation weights for the high order method from Butcher array
* @param prototype prototype of the step interpolator to use
* @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 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) {
super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
this.fsal = fsal;
this.c = c;
this.a = a;
this.b = b;
this.prototype = prototype;
exp = -1.0 / getOrder();
// set the default values of the algorithm control parameters
setSafety(0.9);
setMinReduction(0.2);
setMaxGrowth(10.0);
}
/** Build a Runge-Kutta integrator with the given Butcher array.
* @param name name of the method
* @param fsal indicate that the method is an fsal
* @param c time steps from Butcher array (without the first zero)
* @param a internal weights from Butcher array (without the first empty row)
* @param b propagation weights for the high order method from Butcher array
* @param prototype prototype of the step interpolator to use
* @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 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) {
super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
this.fsal = fsal;
this.c = c;
this.a = a;
this.b = b;
this.prototype = prototype;
exp = -1.0 / getOrder();
// set the default values of the algorithm control parameters
setSafety(0.9);
setMinReduction(0.2);
setMaxGrowth(10.0);
}
/** 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 double getSafety() {
return safety;
}
/** Set the safety factor for stepsize control.
* @param safety safety factor
*/
public void setSafety(final double safety) {
this.safety = safety;
}
/** {@inheritDoc} */
@Override
public void integrate(final ExpandableStatefulODE equations, final double t)
throws NumberIsTooSmallException, DimensionMismatchException,
MaxCountExceededException, NoBracketingException {
sanityChecks(equations, t);
setEquations(equations);
final boolean forward = t > equations.getTime();
// create some internal working arrays
final double[] y0 = equations.getCompleteState();
final double[] y = y0.clone();
final int stages = c.length + 1;
final double[][] yDotK = new double[stages][y.length];
final double[] yTmp = y0.clone();
final double[] yDotTmp = new double[y.length];
// set up an interpolator sharing the integrator arrays
final RungeKuttaStepInterpolator interpolator = (RungeKuttaStepInterpolator) prototype.copy();
interpolator.reinitialize(this, yTmp, yDotK, forward,
equations.getPrimaryMapper(), equations.getSecondaryMappers());
interpolator.storeTime(equations.getTime());
// set up integration control objects
stepStart = equations.getTime();
double hNew = 0;
boolean firstTime = true;
initIntegration(equations.getTime(), y0, t);
// main integration loop
isLastStep = false;
do {
interpolator.shift();
// iterate over step size, ensuring local normalized error is smaller than 1
double error = 10;
while (error >= 1.0) {
if (firstTime || !fsal) {
// first stage
computeDerivatives(stepStart, y, yDotK[0]);
}
if (firstTime) {
final double[] scale = new double[mainSetDimension];
if (vecAbsoluteTolerance == null) {
for (int i = 0; i < scale.length; ++i) {
scale[i] = scalAbsoluteTolerance + scalRelativeTolerance * FastMath.abs(y[i]);
}
} else {
for (int i = 0; i < scale.length; ++i) {
scale[i] = vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * FastMath.abs(y[i]);
}
}
hNew = initializeStep(forward, getOrder(), scale,
stepStart, y, yDotK[0], yTmp, yDotK[1]);
firstTime = false;
}
stepSize = hNew;
if (forward) {
if (stepStart + stepSize >= t) {
stepSize = t - stepStart;
}
} else {
if (stepStart + stepSize <= t) {
stepSize = t - stepStart;
}
}
// next stages
for (int k = 1; k < stages; ++k) {
for (int j = 0; j < y0.length; ++j) {
double sum = a[k-1][0] * yDotK[0][j];
for (int l = 1; l < k; ++l) {
sum += a[k-1][l] * yDotK[l][j];
}
yTmp[j] = y[j] + stepSize * sum;
}
computeDerivatives(stepStart + c[k-1] * stepSize, yTmp, yDotK[k]);
}
// estimate the state at the end of the step
for (int j = 0; j < y0.length; ++j) {
double sum = b[0] * yDotK[0][j];
for (int l = 1; l < stages; ++l) {
sum += b[l] * yDotK[l][j];
}
yTmp[j] = y[j] + stepSize * sum;
}
// estimate the error at the end of the step
error = estimateError(yDotK, y, yTmp, stepSize);
if (error >= 1.0) {
// reject the step and attempt to reduce error by stepsize control
final double factor =
FastMath.min(maxGrowth,
FastMath.max(minReduction, safety * FastMath.pow(error, exp)));
hNew = filterStep(stepSize * factor, forward, false);
}
}
// local error is small enough: accept the step, trigger events and step handlers
interpolator.storeTime(stepStart + stepSize);
System.arraycopy(yTmp, 0, y, 0, y0.length);
System.arraycopy(yDotK[stages - 1], 0, yDotTmp, 0, y0.length);
stepStart = acceptStep(interpolator, y, yDotTmp, t);
System.arraycopy(y, 0, yTmp, 0, y.length);
if (!isLastStep) {
// prepare next step
interpolator.storeTime(stepStart);
if (fsal) {
// save the last evaluation for the next step
System.arraycopy(yDotTmp, 0, yDotK[0], 0, y0.length);
}
// stepsize control for next step
final double factor =
FastMath.min(maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp)));
final double scaledH = stepSize * factor;
final double nextT = stepStart + scaledH;
final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
hNew = filterStep(scaledH, forward, nextIsLast);
final double filteredNextT = stepStart + hNew;
final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
if (filteredNextIsLast) {
hNew = t - stepStart;
}
}
} while (!isLastStep);
// dispatch results
equations.setTime(stepStart);
equations.setCompleteState(y);
resetInternalState();
}
/** Get the minimal reduction factor for stepsize control.
* @return minimal reduction factor
*/
public double getMinReduction() {
return minReduction;
}
/** Set the minimal reduction factor for stepsize control.
* @param minReduction minimal reduction factor
*/
public void setMinReduction(final double minReduction) {
this.minReduction = minReduction;
}
/** Get the maximal growth factor for stepsize control.
* @return maximal growth factor
*/
public double getMaxGrowth() {
return maxGrowth;
}
/** Set the maximal growth factor for stepsize control.
* @param maxGrowth maximal growth factor
*/
public void setMaxGrowth(final double 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 double estimateError(double[][] yDotK,
double[] y0, double[] y1,
double h);
}