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diff --git a/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.java b/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.java new file mode 100644 index 0000000..69d2469 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.java @@ -0,0 +1,421 @@ +/* + * 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 java.util.Arrays; + +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.linear.RealMatrixPreservingVisitor; +import org.apache.commons.math3.ode.EquationsMapper; +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 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<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-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>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</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+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> + * + * <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> + * + * @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 (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 AdamsMoultonIntegrator(final int nSteps, + final double minStep, final double maxStep, + final double scalAbsoluteTolerance, + final double scalRelativeTolerance) + throws NumberIsTooSmallException { + 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 (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 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 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]; + final double[] yTmp = new double[y.length]; + final double[] predictedScaled = new double[y.length]; + Array2DRowRealMatrix nordsieckTmp = null; + + // set up two interpolators 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); + + 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); + final ExpandableStatefulODE expandable = getExpandable(); + final EquationsMapper primary = expandable.getPrimaryMapper(); + primary.insertEquationData(interpolator.getInterpolatedState(), yTmp); + int index = 0; + for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { + secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), yTmp); + ++index; + } + + // 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, y.length); + 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); + + // dispatch results + equations.setTime(stepStart); + equations.setCompleteState(y); + + resetInternalState(); + + } + + /** 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) + */ + 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; // (corrected-predicted)/tol + error += ratio * ratio; + } + } + + return FastMath.sqrt(error / mainSetDimension); + + } + } + +} |