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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.math.ode.nonstiff;
import java.util.Arrays;
import org.apache.commons.math.linear.Array2DRowRealMatrix;
import org.apache.commons.math.linear.MatrixVisitorException;
import org.apache.commons.math.linear.RealMatrixPreservingVisitor;
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 implicit Adams-Moulton integrators for Ordinary
* Differential Equations.
*
* <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
* 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+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
* compute y<sub>n+1</sub>,another method must be used to compute a first
* estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
* a final estimate of y<sub>n+1</sub> using the following formulas. Depending
* on the number k of previous steps one wants to use for computing the next
* value, different formulas are available for the final estimate:</p>
* <ul>
* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li>
* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li>
* <li>...</li>
* </ul>
*
* <p>A k-steps Adams-Moulton method is of order k+1.</p>
*
* <h3>Implementation details</h3>
*
* <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
* <pre>
* s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
* s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
* s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
* ...
* s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative
* </pre></p>
*
* <p>The definitions above use the classical representation with several previous first
* derivatives. Lets define
* <pre>
* q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup>
* </pre>
* (we omit the k index in the notation for clarity). With these definitions,
* Adams-Moulton methods can be written:
* <ul>
* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li>
* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li>
* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li>
* <li>...</li>
* </ul></p>
*
* <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
* s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
* higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
* and r<sub>n</sub>) where r<sub>n</sub> is defined as:
* <pre>
* r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
* </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, s<sub>1</sub>(n-i) can be
* computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
* for degree k polynomials.
* <pre>
* s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(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</sub>
* and q<sub>n</sub> resulting from the Taylor series formulas above is:
* <pre>
* q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
* </pre>
* where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
* with the j (-i)<sup>j-1</sup> terms:
* <pre>
* [ -2 3 -4 5 ... ]
* [ -4 12 -32 80 ... ]
* P = [ -6 27 -108 405 ... ]
* [ -8 48 -256 1280 ... ]
* [ ... ]
* </pre></p>
*
* <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.</li>
* </ul></p>
*
* <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
* n as follows:
* <ul>
* <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
* <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
* <li>R<sub>n+1</sub> = (s<sub>1</sub>(n) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li>
* </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>
* From this predicted vector, the corrected vector is computed as follows:
* <ul>
* <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li>
* <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
* <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
* </ul>
* where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
* predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
* represent the corrected states.</p>
*
* <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
* they only depend on k and therefore are precomputed once for all.</p>
*
* @version $Revision: 789159 $ $Date: 2009-06-28 17:56:20 -0400 (Sun, 28 Jun 2009) $
* @since 2.0
*/
public class AdamsMoultonIntegrator extends AdamsIntegrator {
/**
* Build an Adams-Moulton integrator with the given order and error 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 AdamsMoultonIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws IllegalArgumentException {
super("Adams-Moulton", nSteps, nSteps + 1, minStep, maxStep,
scalAbsoluteTolerance, scalRelativeTolerance);
}
/**
* Build an Adams-Moulton integrator with the given order and error 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 AdamsMoultonIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance)
throws IllegalArgumentException {
super("Adams-Moulton", nSteps, nSteps + 1, 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[y0.length];
final double[] yTmp = new double[y0.length];
// set up two interpolators 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);
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;
// predict a first estimate of the state at step end (P in the PECE sequence)
final double stepEnd = stepStart + stepSize;
interpolator.setInterpolatedTime(stepEnd);
System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);
// evaluate a first estimate of the derivative (first E in the PECE sequence)
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);
// apply correction (C in the PECE sequence)
error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp));
if (error <= 1.0) {
// evaluate a final estimate of the derivative (second E in the PECE sequence)
computeDerivatives(stepEnd, yTmp, yDot);
// update Nordsieck vector
final double[] correctedScaled = new double[y0.length];
for (int j = 0; j < y0.length; ++j) {
correctedScaled[j] = stepSize * yDot[j];
}
updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp);
// discrete events handling
interpolatorTmp.reinitialize(stepEnd, stepSize, correctedScaled, 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)) {
// rejecting the step would lead to a too small next step, we accept it
loop = false;
} else {
// reject the step to match exactly the next switch time
hNew = dt;
interpolator.rescale(hNew);
}
} else {
// accept the step
scaled = correctedScaled;
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;
}
/** Corrector for current state in Adams-Moulton method.
* <p>
* This visitor implements the Taylor series formula:
* <pre>
* Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub>
* </pre>
* </p>
*/
private class Corrector implements RealMatrixPreservingVisitor {
/** Previous state. */
private final double[] previous;
/** Current scaled first derivative. */
private final double[] scaled;
/** Current state before correction. */
private final double[] before;
/** Current state after correction. */
private final double[] after;
/** Simple constructor.
* @param previous previous state
* @param scaled current scaled first derivative
* @param state state to correct (will be overwritten after visit)
*/
public Corrector(final double[] previous, final double[] scaled, final double[] state) {
this.previous = previous;
this.scaled = scaled;
this.after = state;
this.before = state.clone();
}
/** {@inheritDoc} */
public void start(int rows, int columns,
int startRow, int endRow, int startColumn, int endColumn) {
Arrays.fill(after, 0.0);
}
/** {@inheritDoc} */
public void visit(int row, int column, double value)
throws MatrixVisitorException {
if ((row & 0x1) == 0) {
after[column] -= value;
} else {
after[column] += value;
}
}
/**
* End visiting te Nordsieck vector.
* <p>The correction is used to control stepsize. So its amplitude is
* considered to be an error, which must be normalized according to
* error control settings. If the normalized value is greater than 1,
* the correction was too large and the step must be rejected.</p>
* @return the normalized correction, if greater than 1, the step
* must be rejected
*/
public double end() {
double error = 0;
for (int i = 0; i < after.length; ++i) {
after[i] += previous[i] + scaled[i];
final double yScale = Math.max(Math.abs(previous[i]), Math.abs(after[i]));
final double tol = (vecAbsoluteTolerance == null) ?
(scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
(vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
final double ratio = (after[i] - before[i]) / tol;
error += ratio * ratio;
}
return Math.sqrt(error / after.length);
}
}
}