EmbeddedRungeKuttaIntegrator.java
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* The Hipparchus project licenses this file to You under the Apache License, Version 2.0
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* https://www.apache.org/licenses/LICENSE-2.0
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package org.hipparchus.ode.nonstiff;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.exception.MathIllegalStateException;
import org.hipparchus.ode.EquationsMapper;
import org.hipparchus.ode.ExpandableODE;
import org.hipparchus.ode.LocalizedODEFormats;
import org.hipparchus.ode.ODEState;
import org.hipparchus.ode.ODEStateAndDerivative;
import org.hipparchus.util.FastMath;
/**
* 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 :</p>
* <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>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>
*
*/
public abstract class EmbeddedRungeKuttaIntegrator
extends AdaptiveStepsizeIntegrator
implements ExplicitRungeKuttaIntegrator {
/** 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 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;
/** 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 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 EmbeddedRungeKuttaIntegrator(final String name, final int fsal,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance) {
super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
this.fsal = fsal;
this.c = getC();
this.a = getA();
this.b = getB();
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 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 EmbeddedRungeKuttaIntegrator(final String name, final int fsal,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance) {
super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
this.fsal = fsal;
this.c = getC();
this.a = getA();
this.b = getB();
exp = -1.0 / getOrder();
// set the default values of the algorithm control parameters
setSafety(0.9);
setMinReduction(0.2);
setMaxGrowth(10.0);
}
/** 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 RungeKuttaStateInterpolator createInterpolator(boolean forward, double[][] yDotK,
ODEStateAndDerivative globalPreviousState,
ODEStateAndDerivative globalCurrentState,
EquationsMapper 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 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 ODEStateAndDerivative integrate(final ExpandableODE equations,
final ODEState initialState, final double finalTime)
throws MathIllegalArgumentException, MathIllegalStateException {
sanityChecks(initialState, finalTime);
setStepStart(initIntegration(equations, initialState, finalTime));
final boolean forward = finalTime > initialState.getTime();
// create some internal working arrays
final int stages = c.length + 1;
final double[][] yDotK = new double[stages][];
double[] yTmp = new double[equations.getMapper().getTotalDimension()];
// set up integration control objects
double hNew = 0;
boolean firstTime = true;
// main integration loop
setIsLastStep(false);
do {
// iterate over step size, ensuring local normalized error is smaller than 1
double error = 10;
while (error >= 1.0) {
// first stage
final double[] y = getStepStart().getCompleteState();
yDotK[0] = getStepStart().getCompleteDerivative();
if (firstTime) {
final StepsizeHelper helper = getStepSizeHelper();
final double[] scale = new double[helper.getMainSetDimension()];
for (int i = 0; i < scale.length; ++i) {
scale[i] = helper.getTolerance(i, FastMath.abs(y[i]));
}
hNew = initializeStep(forward, getOrder(), scale, getStepStart());
firstTime = false;
}
setStepSize(hNew);
if (forward) {
if (getStepStart().getTime() + getStepSize() >= finalTime) {
setStepSize(finalTime - getStepStart().getTime());
}
} else {
if (getStepStart().getTime() + getStepSize() <= finalTime) {
setStepSize(finalTime - getStepStart().getTime());
}
}
// next stages
ExplicitRungeKuttaIntegrator.applyInternalButcherWeights(getEquations(), getStepStart().getTime(), y,
getStepSize(), a, c, yDotK);
yTmp = ExplicitRungeKuttaIntegrator.applyExternalButcherWeights(y, yDotK, getStepSize(), b);
incrementEvaluations(stages - 1);
// estimate the error at the end of the step
error = estimateError(yDotK, y, yTmp, getStepSize());
if (Double.isNaN(error)) {
throw new MathIllegalStateException(LocalizedODEFormats.NAN_APPEARING_DURING_INTEGRATION,
getStepStart().getTime() + getStepSize());
}
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 = getStepSizeHelper().filterStep(getStepSize() * factor, forward, false);
}
}
final double stepEnd = getStepStart().getTime() + getStepSize();
final double[] yDotTmp = (fsal >= 0) ? yDotK[fsal] : computeDerivatives(stepEnd, yTmp);
final ODEStateAndDerivative stateTmp = equations.getMapper().mapStateAndDerivative(stepEnd, yTmp, yDotTmp);
// local error is small enough: accept the step, trigger events and step handlers
setStepStart(acceptStep(createInterpolator(forward, yDotK, getStepStart(), stateTmp, equations.getMapper()), finalTime));
if (!isLastStep()) {
// stepsize control for next step
final double factor =
FastMath.min(maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp)));
final double scaledH = getStepSize() * factor;
final double nextT = getStepStart().getTime() + scaledH;
final boolean nextIsLast = forward ? (nextT >= finalTime) : (nextT <= finalTime);
hNew = getStepSizeHelper().filterStep(scaledH, forward, nextIsLast);
final double filteredNextT = getStepStart().getTime() + hNew;
final boolean filteredNextIsLast = forward ? (filteredNextT >= finalTime) : (filteredNextT <= finalTime);
if (filteredNextIsLast) {
hNew = finalTime - getStepStart().getTime();
}
}
} while (!isLastStep());
final ODEStateAndDerivative finalState = getStepStart();
resetInternalState();
return finalState;
}
/** 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);
}