package org.spaceroots.mantissa.ode;
/**
* This class implements the common part of all Runge-Kutta-Fehlberg
* 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.
* @version $Id: RungeKuttaFehlbergIntegrator.java 1719 2007-09-26 19:46:57Z luc $
* @author L. Maisonobe
*/
public abstract class RungeKuttaFehlbergIntegrator
extends AdaptiveStepsizeIntegrator {
/** Build a Runge-Kutta integrator with the given Butcher array.
* @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 external 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 scalAbsoluteTolerance allowed absolute error
* @param scalRelativeTolerance allowed relative error
*/
protected RungeKuttaFehlbergIntegrator(boolean fsal,
double[] c, double[][] a, double[] b,
RungeKuttaStepInterpolator prototype,
double minStep, double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance) {
super(minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
this.fsal = fsal;
this.c = c;
this.a = a;
this.b = b;
this.prototype = prototype;
exp = -1.0 / getOrder();
this.safety = 0.9;
// set the default values of the algorithm control parameters
setMinReduction(0.2);
setMaxGrowth(10.0);
}
/** Build a Runge-Kutta integrator with the given Butcher array.
* @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 external 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 RungeKuttaFehlbergIntegrator(boolean fsal,
double[] c, double[][] a, double[] b,
RungeKuttaStepInterpolator prototype,
double minStep, double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance) {
super(minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
this.fsal = fsal;
this.c = c;
this.a = a;
this.b = b;
this.prototype = prototype;
exp = -1.0 / getOrder();
this.safety = 0.9;
// set the default values of the algorithm control parameters
setMinReduction(0.2);
setMaxGrowth(10.0);
}
/** Get the name of the method.
* @return name of the method
*/
public abstract String getName();
/** 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(double safety) {
this.safety = safety;
}
public void integrate(FirstOrderDifferentialEquations equations,
double t0, double[] y0,
double t, double[] y)
throws DerivativeException, IntegratorException {
// sanity check
if (equations.getDimension() != y0.length) {
throw new IntegratorException("dimensions mismatch: ODE problem has dimension {0},"
+ " state vector has dimension {1}",
new String[] {
Integer.toString(equations.getDimension()),
Integer.toString(y0.length)
});
}
if (Math.abs(t - t0) <= 1.0e-12 * Math.max(Math.abs(t0), Math.abs(t))) {
throw new IntegratorException("too small integration interval: length = {0}",
new String[] {
Double.toString(Math.abs(t - t0))
});
}
boolean forward = (t > t0);
// create some internal working arrays
int stages = c.length + 1;
if (y != y0) {
System.arraycopy(y0, 0, y, 0, y0.length);
}
double[][] yDotK = new double[stages][];
for (int i = 0; i < stages; ++i) {
yDotK [i] = new double[y0.length];
}
double[] yTmp = new double[y0.length];
// set up an interpolator sharing the integrator arrays
AbstractStepInterpolator interpolator;
if (handler.requiresDenseOutput() || (! switchesHandler.isEmpty())) {
RungeKuttaStepInterpolator rki = (RungeKuttaStepInterpolator) prototype.clone();
rki.reinitialize(equations, yTmp, yDotK, forward);
interpolator = rki;
} else {
interpolator = new DummyStepInterpolator(yTmp, forward);
}
interpolator.storeTime(t0);
stepStart = t0;
double hNew = 0;
boolean firstTime = true;
boolean lastStep;
handler.reset();
do {
interpolator.shift();
double error = 0;
for (boolean loop = true; loop;) {
if (firstTime || !fsal) {
// first stage
equations.computeDerivatives(stepStart, y, yDotK[0]);
}
if (firstTime) {
double[] scale;
if (vecAbsoluteTolerance != null) {
scale = vecAbsoluteTolerance;
} else {
scale = new double[y0.length];
for (int i = 0; i < scale.length; ++i) {
scale[i] = scalAbsoluteTolerance;
}
}
hNew = initializeStep(equations, forward, getOrder(), scale,
stepStart, y, yDotK[0], yTmp, yDotK[1]);
firstTime = false;
}
stepSize = hNew;
// step adjustment near bounds
if ((forward && (stepStart + stepSize > t))
|| ((! forward) && (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;
}
equations.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) {
// Switching functions handling
interpolator.storeTime(stepStart + stepSize);
if (switchesHandler.evaluateStep(interpolator)) {
// reject the step to match exactly the next switch time
hNew = switchesHandler.getEventTime() - stepStart;
} else {
// accept the step
loop = false;
}
} else {
// reject the step and attempt to reduce error by stepsize control
double factor = Math.min(maxGrowth,
Math.max(minReduction,
safety * Math.pow(error, exp)));
hNew = filterStep(stepSize * factor, false);
}
}
// the step has been accepted
stepStart += stepSize;
System.arraycopy(yTmp, 0, y, 0, y0.length);
switchesHandler.stepAccepted(stepStart, y);
if (switchesHandler.stop()) {
lastStep = true;
} else {
lastStep = forward ? (stepStart >= t) : (stepStart <= t);
}
// provide the step data to the step handler
interpolator.storeTime(stepStart);
handler.handleStep(interpolator, lastStep);
if (fsal) {
// save the last evaluation for the next step
System.arraycopy(yDotK[stages - 1], 0, yDotK[0], 0, y0.length);
}
if (switchesHandler.reset(stepStart, y) && ! lastStep) {
// some switching function has triggered changes that
// invalidate the derivatives, we need to recompute them
equations.computeDerivatives(stepStart, y, yDotK[0]);
}
if (! lastStep) {
// stepsize control for next step
double factor = Math.min(maxGrowth,
Math.max(minReduction,
safety * Math.pow(error, exp)));
double scaledH = stepSize * factor;
double nextT = stepStart + scaledH;
boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
hNew = filterStep(scaledH, nextIsLast);
}
} while (! lastStep);
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(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(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);
/** Indicator for fsal methods. */
private boolean fsal;
/** Time steps from Butcher array (without the first zero). */
private double[] c;
/** Internal weights from Butcher array (without the first empty row). */
private double[][] a;
/** External weights for the high order method from Butcher array. */
private double[] b;
/** Prototype of the step interpolator. */
private RungeKuttaStepInterpolator prototype;
/** Stepsize control exponent. */
private 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;
}
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