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diff --git a/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthIntegrator.java b/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthIntegrator.java new file mode 100644 index 0000000..af4a435 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthIntegrator.java @@ -0,0 +1,362 @@ +/* + * 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.linear.RealMatrix; +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 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>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 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> + * + * @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 = FastMath.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() % 2 == 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 FastMath.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(); + + } + +} |