| /* |
| * 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.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.sampling.NordsieckStepInterpolator; |
| import org.apache.commons.math.ode.sampling.StepHandler; |
| import org.apache.commons.math.util.FastMath; |
| |
| |
| /** |
| * 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: 1073158 $ $Date: 2011-02-21 22:46:52 +0100 (lun. 21 févr. 2011) $ |
| * @since 2.0 |
| */ |
| public class AdamsMoultonIntegrator extends AdamsIntegrator { |
| |
| /** Integrator method name. */ |
| private static final String METHOD_NAME = "Adams-Moulton"; |
| |
| /** |
| * 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(METHOD_NAME, 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(METHOD_NAME, 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]; |
| final double[] predictedScaled = new double[y0.length]; |
| Array2DRowRealMatrix nordsieckTmp = null; |
| |
| // set up two interpolators sharing the integrator arrays |
| final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); |
| interpolator.reinitialize(y, forward); |
| |
| // set up integration control objects |
| for (StepHandler handler : stepHandlers) { |
| handler.reset(); |
| } |
| setStateInitialized(false); |
| |
| // 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); |
| |
| isLastStep = false; |
| do { |
| |
| double error = 10; |
| while (error >= 1.0) { |
| |
| 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 |
| for (int j = 0; j < y0.length; ++j) { |
| predictedScaled[j] = stepSize * yDot[j]; |
| } |
| 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) { |
| // 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); |
| } |
| } |
| |
| // evaluate a final estimate of the derivative (second E in the PECE sequence) |
| final double stepEnd = stepStart + stepSize; |
| 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 |
| System.arraycopy(yTmp, 0, y, 0, n); |
| interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp); |
| interpolator.storeTime(stepStart); |
| interpolator.shift(); |
| interpolator.storeTime(stepEnd); |
| stepStart = acceptStep(interpolator, y, yDot, t); |
| scaled = correctedScaled; |
| nordsieck = nordsieckTmp; |
| |
| 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); |
| |
| 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) { |
| if ((row & 0x1) == 0) { |
| after[column] -= value; |
| } else { |
| after[column] += value; |
| } |
| } |
| |
| /** |
| * End visiting the 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]; |
| if (i < mainSetDimension) { |
| final double yScale = FastMath.max(FastMath.abs(previous[i]), FastMath.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 FastMath.sqrt(error / mainSetDimension); |
| |
| } |
| } |
| |
| } |
