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Commons Math example source code file (AdamsMoultonIntegrator.java)
The Commons Math AdamsMoultonIntegrator.java source code/* * 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.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 is needed to * compute y<sub>n+1,another method must be used to compute a first * estimate of y<sub>n+1, then compute y'n+1, then compute * a final estimate of y<sub>n+1 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: yn+1 = yn + h y'n+1 * <li>k = 2: yn+1 = yn + h (y'n+1+y'n)/2 * <li>k = 3: yn+1 = yn + h (5y'n+1+8y'n-y'n-1)/12 * <li>k = 4: yn+1 = yn + h (9y'n+1+19y'n-5y'n-1+y'n-2)/24 * <li>... * </ul> * * <p>A k-steps Adams-Moulton method is of order k+1. * * <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-Moulton methods can be written: * <ul> * <li>k = 1: yn+1 = yn + s1(n+1) * <li>k = 2: yn+1 = yn + 1/2 s1(n+1) + [ 1/2 ] qn+1 * <li>k = 3: yn+1 = yn + 5/12 s1(n+1) + [ 8/12 -1/12 ] qn+1 * <li>k = 4: yn+1 = yn + 9/24 s1(n+1) + [ 19/24 -5/24 1/24 ] qn+1 * <li>... * </ul> * * <p>Instead of using the classical representation with first derivatives only (yn, * s<sub>1(n+1) and qn+1), 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 predicted 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> * From this predicted vector, the corrected vector is computed as follows: * <ul> * <li>yn+1 = yn + S1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1 * <li>s1(n+1) = h f(tn+1, yn+1) * <li>rn+1 = Rn+1 + (s1(n+1) - S1(n+1)) P-1 u * </ul> * where the upper case Y<sub>n+1, S1(n+1) and Rn+1 represent the * predicted states whereas the lower case y<sub>n+1, sn+1 and rn+1 * represent the corrected states.</p> * * <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 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)) { // 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 = 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 = yn + s1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1 * </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); } } } Other Commons Math examples (source code examples)Here is a short list of links related to this Commons Math AdamsMoultonIntegrator.java source code file: |
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