Java tutorial
/* * 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.math3.ode.nonstiff; import org.apache.commons.math3.exception.DimensionMismatchException; import org.apache.commons.math3.exception.MaxCountExceededException; import org.apache.commons.math3.exception.NoBracketingException; import org.apache.commons.math3.exception.NumberIsTooSmallException; import org.apache.commons.math3.linear.Array2DRowRealMatrix; import org.apache.commons.math3.ode.ExpandableStatefulODE; import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator; import org.apache.commons.math3.util.FastMath; /** * 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> * * <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<sup>(k)</sup><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-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> * * <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: * <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 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> * * <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 $Id: AdamsBashforthIntegrator.java 1416643 2012-12-03 19:37:14Z tn $ * @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); } /** {@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[] y0 = equations.getCompleteState(); final double[] y = y0.clone(); 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(), y0, 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); final int lastRow = nordsieck.getRowDimension() - 1; // reuse the step that was chosen by the starter integrator double hNew = stepSize; interpolator.rescale(hNew); // main integration loop isLastStep = false; do { double error = 10; while (error >= 1.0) { stepSize = hNew; // evaluate error using the last term of the Taylor expansion error = 0; for (int i = 0; i < mainSetDimension; ++i) { final double yScale = FastMath.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 = FastMath.sqrt(error / mainSetDimension); if (error >= 1.0) { // 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); } } // predict a first estimate of the state at step end final double stepEnd = stepStart + stepSize; interpolator.shift(); interpolator.setInterpolatedTime(stepEnd); System.arraycopy(interpolator.getInterpolatedState(), 0, y, 0, y0.length); // evaluate the derivative computeDerivatives(stepEnd, y, 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); interpolator.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp); // discrete events handling interpolator.storeTime(stepEnd); stepStart = acceptStep(interpolator, y, yDot, t); scaled = predictedScaled; nordsieck = nordsieckTmp; 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(); } }