Commons Math example source code file (AdamsBashforthIntegrator.java)
The Commons Math AdamsBashforthIntegrator.java source code
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* http://www.apache.org/licenses/LICENSE-2.0
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package org.apache.commons.math.ode.nonstiff;
import org.apache.commons.math.linear.Array2DRowRealMatrix;
import org.apache.commons.math.ode.DerivativeException;
import org.apache.commons.math.ode.FirstOrderDifferentialEquations;
import org.apache.commons.math.ode.IntegratorException;
import org.apache.commons.math.ode.events.CombinedEventsManager;
import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator;
import org.apache.commons.math.ode.sampling.StepHandler;
/**
* This class implements explicit Adams-Bashforth integrators for Ordinary
* Differential Equations.
*
* <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
* multistep ODE solvers. This implementation is a variation of the classical
* one: it uses adaptive stepsize to implement error control, whereas
* classical implementations are fixed step size. The value of state vector
* at step n+1 is a simple combination of the value at step n and of the
* derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
* steps one wants to use for computing the next value, different formulas
* are available:</p>
* <ul>
* <li>k = 1: yn+1 = yn + h y'n
* <li>k = 2: yn+1 = yn + h (3y'n-y'n-1)/2
* <li>k = 3: yn+1 = yn + h (23y'n-16y'n-1+5y'n-2)/12
* <li>k = 4: yn+1 = yn + h (55y'n-59y'n-1+37y'n-2-9y'n-3)/24
* <li>...
* </ul>
*
* <p>A k-steps Adams-Bashforth method is of order k.
*
* <h3>Implementation details
*
* <p>We define scaled derivatives si(n) at step n as:
* <pre>
* s<sub>1(n) = h y'n for first derivative
* s<sub>2(n) = h2/2 y''n for second derivative
* s<sub>3(n) = h3/6 y'''n for third derivative
* ...
* s<sub>k(n) = hk/k! y(k)n for kth derivative
* </pre>
*
* <p>The definitions above use the classical representation with several previous first
* derivatives. Lets define
* <pre>
* q<sub>n = [ s1(n-1) s1(n-2) ... s1(n-(k-1)) ]T
* </pre>
* (we omit the k index in the notation for clarity). With these definitions,
* Adams-Bashforth methods can be written:
* <ul>
* <li>k = 1: yn+1 = yn + s1(n)
* <li>k = 2: yn+1 = yn + 3/2 s1(n) + [ -1/2 ] qn
* <li>k = 3: yn+1 = yn + 23/12 s1(n) + [ -16/12 5/12 ] qn
* <li>k = 4: yn+1 = yn + 55/24 s1(n) + [ -59/24 37/24 -9/24 ] qn
* <li>...
* </ul>
*
* <p>Instead of using the classical representation with first derivatives only (yn,
* s<sub>1(n) and qn), our implementation uses the Nordsieck vector with
* higher degrees scaled derivatives all taken at the same step (y<sub>n, s1(n)
* and r<sub>n) where rn is defined as:
* <pre>
* r<sub>n = [ s2(n), s3(n) ... sk(n) ]T
* </pre>
* (here again we omit the k index in the notation for clarity)
* </p>
*
* <p>Taylor series formulas show that for any index offset i, s1(n-i) can be
* computed from s<sub>1(n), s2(n) ... sk(n), the formula being exact
* for degree k polynomials.
* <pre>
* s<sub>1(n-i) = s1(n) + ∑j j (-i)j-1 sj(n)
* </pre>
* The previous formula can be used with several values for i to compute the transform between
* classical representation and Nordsieck vector. The transform between r<sub>n
* and q<sub>n resulting from the Taylor series formulas above is:
* <pre>
* q<sub>n = s1(n) u + P rn
* </pre>
* where u is the [ 1 1 ... 1 ]<sup>T vector and P is the (k-1)×(k-1) matrix built
* with the j (-i)<sup>j-1 terms:
* <pre>
* [ -2 3 -4 5 ... ]
* [ -4 12 -32 80 ... ]
* P = [ -6 27 -108 405 ... ]
* [ -8 48 -256 1280 ... ]
* [ ... ]
* </pre>
*
* <p>Using the Nordsieck vector has several advantages:
* <ul>
* <li>it greatly simplifies step interpolation as the interpolator mainly applies
* Taylor series formulas,</li>
* <li>it simplifies step changes that occur when discrete events that truncate
* the step are triggered,</li>
* <li>it allows to extend the methods in order to support adaptive stepsize.
* </ul>
*
* <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
* <ul>
* <li>yn+1 = yn + s1(n) + uT rn
* <li>s1(n+1) = h f(tn+1, yn+1)
* <li>rn+1 = (s1(n) - s1(n+1)) P-1 u + P-1 A P rn
* </ul>
* where A is a rows shifting matrix (the lower left part is an identity matrix):
* <pre>
* [ 0 0 ... 0 0 | 0 ]
* [ ---------------+---]
* [ 1 0 ... 0 0 | 0 ]
* A = [ 0 1 ... 0 0 | 0 ]
* [ ... | 0 ]
* [ 0 0 ... 1 0 | 0 ]
* [ 0 0 ... 0 1 | 0 ]
* </pre>
*
* <p>The P-1u vector and the P-1 A P matrix do not depend on the state,
* they only depend on k and therefore are precomputed once for all.</p>
*
* @version $Revision: 927202 $ $Date: 2010-03-24 18:11:51 -0400 (Wed, 24 Mar 2010) $
* @since 2.0
*/
public class AdamsBashforthIntegrator extends AdamsIntegrator {
/**
* Build an Adams-Bashforth integrator with the given order and step control parameters.
* @param nSteps number of steps of the method excluding the one being computed
* @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
* @exception IllegalArgumentException if order is 1 or less
*/
public AdamsBashforthIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws IllegalArgumentException {
super("Adams-Bashforth", nSteps, nSteps, minStep, maxStep,
scalAbsoluteTolerance, scalRelativeTolerance);
}
/**
* Build an Adams-Bashforth integrator with the given order and step control parameters.
* @param nSteps number of steps of the method excluding the one being computed
* @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
* @exception IllegalArgumentException if order is 1 or less
*/
public AdamsBashforthIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance)
throws IllegalArgumentException {
super("Adams-Bashforth", nSteps, nSteps, minStep, maxStep,
vecAbsoluteTolerance, vecRelativeTolerance);
}
/** {@inheritDoc} */
@Override
public double integrate(final FirstOrderDifferentialEquations equations,
final double t0, final double[] y0,
final double t, final double[] y)
throws DerivativeException, IntegratorException {
final int n = y0.length;
sanityChecks(equations, t0, y0, t, y);
setEquations(equations);
resetEvaluations();
final boolean forward = t > t0;
// initialize working arrays
if (y != y0) {
System.arraycopy(y0, 0, y, 0, n);
}
final double[] yDot = new double[n];
final double[] yTmp = new double[y0.length];
// set up an interpolator sharing the integrator arrays
final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
interpolator.reinitialize(y, forward);
final NordsieckStepInterpolator interpolatorTmp = new NordsieckStepInterpolator();
interpolatorTmp.reinitialize(yTmp, forward);
// set up integration control objects
for (StepHandler handler : stepHandlers) {
handler.reset();
}
CombinedEventsManager manager = addEndTimeChecker(t0, t, eventsHandlersManager);
// compute the initial Nordsieck vector using the configured starter integrator
start(t0, y, t);
interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
interpolator.storeTime(stepStart);
final int lastRow = nordsieck.getRowDimension() - 1;
// reuse the step that was chosen by the starter integrator
double hNew = stepSize;
interpolator.rescale(hNew);
boolean lastStep = false;
while (!lastStep) {
// shift all data
interpolator.shift();
double error = 0;
for (boolean loop = true; loop;) {
stepSize = hNew;
// evaluate error using the last term of the Taylor expansion
error = 0;
for (int i = 0; i < y0.length; ++i) {
final double yScale = Math.abs(y[i]);
final double tol = (vecAbsoluteTolerance == null) ?
(scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
(vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
final double ratio = nordsieck.getEntry(lastRow, i) / tol;
error += ratio * ratio;
}
error = Math.sqrt(error / y0.length);
if (error <= 1.0) {
// predict a first estimate of the state at step end
final double stepEnd = stepStart + stepSize;
interpolator.setInterpolatedTime(stepEnd);
System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);
// evaluate the derivative
computeDerivatives(stepEnd, yTmp, yDot);
// update Nordsieck vector
final double[] predictedScaled = new double[y0.length];
for (int j = 0; j < y0.length; ++j) {
predictedScaled[j] = stepSize * yDot[j];
}
final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);
// discrete events handling
interpolatorTmp.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp);
interpolatorTmp.storeTime(stepStart);
interpolatorTmp.shift();
interpolatorTmp.storeTime(stepEnd);
if (manager.evaluateStep(interpolatorTmp)) {
final double dt = manager.getEventTime() - stepStart;
if (Math.abs(dt) <= Math.ulp(stepStart)) {
// we cannot simply truncate the step, reject the current computation
// and let the loop compute another state with the truncated step.
// it is so small (much probably exactly 0 due to limited accuracy)
// that the code above would fail handling it.
// So we set up an artificial 0 size step by copying states
interpolator.storeTime(stepStart);
System.arraycopy(y, 0, yTmp, 0, y0.length);
hNew = 0;
stepSize = 0;
loop = false;
} else {
// reject the step to match exactly the next switch time
hNew = dt;
interpolator.rescale(hNew);
}
} else {
// accept the step
scaled = predictedScaled;
nordsieck = nordsieckTmp;
interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
loop = false;
}
} else {
// reject the step and attempt to reduce error by stepsize control
final double factor = computeStepGrowShrinkFactor(error);
hNew = filterStep(stepSize * factor, forward, false);
interpolator.rescale(hNew);
}
}
// the step has been accepted (may have been truncated)
final double nextStep = stepStart + stepSize;
System.arraycopy(yTmp, 0, y, 0, n);
interpolator.storeTime(nextStep);
manager.stepAccepted(nextStep, y);
lastStep = manager.stop();
// provide the step data to the step handler
for (StepHandler handler : stepHandlers) {
interpolator.setInterpolatedTime(nextStep);
handler.handleStep(interpolator, lastStep);
}
stepStart = nextStep;
if (!lastStep && manager.reset(stepStart, y)) {
// some events handler has triggered changes that
// invalidate the derivatives, we need to restart from scratch
start(stepStart, y, t);
interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
}
if (! lastStep) {
// in some rare cases we may get here with stepSize = 0, for example
// when an event occurs at integration start, reducing the first step
// to zero; we have to reset the step to some safe non zero value
stepSize = filterStep(stepSize, forward, true);
// stepsize control for next step
final double factor = computeStepGrowShrinkFactor(error);
final double scaledH = stepSize * factor;
final double nextT = stepStart + scaledH;
final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
hNew = filterStep(scaledH, forward, nextIsLast);
interpolator.rescale(hNew);
}
}
final double stopTime = stepStart;
stepStart = Double.NaN;
stepSize = Double.NaN;
return stopTime;
}
}
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