AdamsBashforthIntegrator.java
/*
* 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.
*/
package org.apache.commons.math4.legacy.ode.nonstiff;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.MaxCountExceededException;
import org.apache.commons.math4.legacy.exception.NoBracketingException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
import org.apache.commons.math4.legacy.linear.Array2DRowRealMatrix;
import org.apache.commons.math4.legacy.linear.RealMatrix;
import org.apache.commons.math4.legacy.ode.EquationsMapper;
import org.apache.commons.math4.legacy.ode.ExpandableStatefulODE;
import org.apache.commons.math4.legacy.ode.sampling.NordsieckStepInterpolator;
import org.apache.commons.math4.core.jdkmath.JdkMath;
/**
* 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: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li>
* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li>
* <li>...</li>
* </ul>
*
* <p>A k-steps Adams-Bashforth method is of order k.</p>
*
* <p><b>Implementation details</b></p>
*
* <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
* <div style="white-space: pre"><code>
* 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<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
* </code></div>
*
* <p>The definitions above use the classical representation with several previous first
* derivatives. Lets define
* <div style="white-space: pre"><code>
* 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>
* </code></div>
* (we omit the k index in the notation for clarity). With these definitions,
* Adams-Bashforth methods can be written:
* <ul>
* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n) + [ -1/2 ] q<sub>n</sub></li>
* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n) + [ -16/12 5/12 ] q<sub>n</sub></li>
* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n) + [ -59/24 37/24 -9/24 ] q<sub>n</sub></li>
* <li>...</li>
* </ul>
*
* <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
* s<sub>1</sub>(n) and q<sub>n</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:
* <div style="white-space: pre"><code>
* r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
* </code></div>
* (here again we omit the k index in the notation for clarity)
*
* <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.
* <div style="white-space: pre"><code>
* s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n)
* </code></div>
* 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:
* <div style="white-space: pre"><code>
* q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
* </code></div>
* where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
* with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
* the column number starting from 1:
* <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.</li>
* </ul>
*
* <p>The 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>
*
* <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>
*
* @since 2.0
*/
public class AdamsBashforthIntegrator extends AdamsIntegrator {
/** Integrator method name. */
private static final String METHOD_NAME = "Adams-Bashforth";
/**
* 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 (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
* @exception NumberIsTooSmallException if order is 1 or less
*/
public AdamsBashforthIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws NumberIsTooSmallException {
super(METHOD_NAME, 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 (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 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(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
vecAbsoluteTolerance, vecRelativeTolerance);
}
/** Estimate error.
* <p>
* Error is estimated by interpolating back to previous state using
* the state Taylor expansion and comparing to real previous state.
* </p>
* @param previousState state vector at step start
* @param predictedState predicted state vector at step end
* @param predictedScaled predicted value of the scaled derivatives at step end
* @param predictedNordsieck predicted value of the Nordsieck vector at step end
* @return estimated normalized local discretization error
*/
private double errorEstimation(final double[] previousState,
final double[] predictedState,
final double[] predictedScaled,
final RealMatrix predictedNordsieck) {
double error = 0;
for (int i = 0; i < mainSetDimension; ++i) {
final double yScale = JdkMath.abs(predictedState[i]);
final double tol = (vecAbsoluteTolerance == null) ?
(scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
(vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
// apply Taylor formula from high order to low order,
// for the sake of numerical accuracy
double variation = 0;
int sign = (predictedNordsieck.getRowDimension() & 1) == 0 ? -1 : 1;
for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) {
variation += sign * predictedNordsieck.getEntry(k, i);
sign = -sign;
}
variation -= predictedScaled[i];
final double ratio = (predictedState[i] - previousState[i] + variation) / tol;
error += ratio * ratio;
}
return JdkMath.sqrt(error / mainSetDimension);
}
/** {@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();
// initialize working arrays
final double[] y = equations.getCompleteState();
final double[] yDot = new double[y.length];
// set up an interpolator sharing the integrator arrays
final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
interpolator.reinitialize(y, forward,
equations.getPrimaryMapper(), equations.getSecondaryMappers());
// set up integration control objects
initIntegration(equations.getTime(), y, t);
// compute the initial Nordsieck vector using the configured starter integrator
start(equations.getTime(), y, t);
interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
interpolator.storeTime(stepStart);
// reuse the step that was chosen by the starter integrator
double hNew = stepSize;
interpolator.rescale(hNew);
// main integration loop
isLastStep = false;
do {
interpolator.shift();
final double[] predictedY = new double[y.length];
final double[] predictedScaled = new double[y.length];
Array2DRowRealMatrix predictedNordsieck = null;
double error = 10;
while (error >= 1.0) {
// predict a first estimate of the state at step end
final double stepEnd = stepStart + hNew;
interpolator.storeTime(stepEnd);
final ExpandableStatefulODE expandable = getExpandable();
final EquationsMapper primary = expandable.getPrimaryMapper();
primary.insertEquationData(interpolator.getInterpolatedState(), predictedY);
int index = 0;
for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), predictedY);
++index;
}
// evaluate the derivative
computeDerivatives(stepEnd, predictedY, yDot);
// predict Nordsieck vector at step end
for (int j = 0; j < predictedScaled.length; ++j) {
predictedScaled[j] = hNew * yDot[j];
}
predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
// evaluate error
error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck);
if (error >= 1.0) {
// reject the step and attempt to reduce error by stepsize control
final double factor = computeStepGrowShrinkFactor(error);
hNew = filterStep(hNew * factor, forward, false);
interpolator.rescale(hNew);
}
}
stepSize = hNew;
final double stepEnd = stepStart + stepSize;
interpolator.reinitialize(stepEnd, stepSize, predictedScaled, predictedNordsieck);
// discrete events handling
interpolator.storeTime(stepEnd);
System.arraycopy(predictedY, 0, y, 0, y.length);
stepStart = acceptStep(interpolator, y, yDot, t);
scaled = predictedScaled;
nordsieck = predictedNordsieck;
interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
if (!isLastStep) {
// prepare next step
interpolator.storeTime(stepStart);
if (resetOccurred) {
// 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);
}
// 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);
final double filteredNextT = stepStart + hNew;
final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
if (filteredNextIsLast) {
hNew = t - stepStart;
}
interpolator.rescale(hNew);
}
} while (!isLastStep);
// dispatch results
equations.setTime(stepStart);
equations.setCompleteState(y);
resetInternalState();
}
}