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diff --git a/src/main/java/org/apache/commons/math3/ode/MultistepIntegrator.java b/src/main/java/org/apache/commons/math3/ode/MultistepIntegrator.java new file mode 100644 index 0000000..fd76124 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/ode/MultistepIntegrator.java @@ -0,0 +1,496 @@ +/* + * 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; + +import org.apache.commons.math3.exception.DimensionMismatchException; +import org.apache.commons.math3.exception.MathIllegalStateException; +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.exception.util.LocalizedFormats; +import org.apache.commons.math3.linear.Array2DRowRealMatrix; +import org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator; +import org.apache.commons.math3.ode.nonstiff.DormandPrince853Integrator; +import org.apache.commons.math3.ode.sampling.StepHandler; +import org.apache.commons.math3.ode.sampling.StepInterpolator; +import org.apache.commons.math3.util.FastMath; + +/** + * This class is the base class for multistep integrators for Ordinary Differential Equations. + * + * <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>Rather than storing several previous steps separately, this 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> + * + * (we omit the k index in the notation for clarity) + * + * <p>Multistep integrators with Nordsieck representation are highly sensitive to large step changes + * because when the step is multiplied by factor a, the k<sup>th</sup> component of the Nordsieck + * vector is multiplied by a<sup>k</sup> and the last components are the least accurate ones. The + * default max growth factor is therefore set to a quite low value: 2<sup>1/order</sup>. + * + * @see org.apache.commons.math3.ode.nonstiff.AdamsBashforthIntegrator + * @see org.apache.commons.math3.ode.nonstiff.AdamsMoultonIntegrator + * @since 2.0 + */ +public abstract class MultistepIntegrator extends AdaptiveStepsizeIntegrator { + + /** First scaled derivative (h y'). */ + protected double[] scaled; + + /** + * Nordsieck matrix of the higher scaled derivatives. + * + * <p>(h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ..., h<sup>k</sup>/k! y<sup>(k)</sup>) + */ + protected Array2DRowRealMatrix nordsieck; + + /** Starter integrator. */ + private FirstOrderIntegrator starter; + + /** Number of steps of the multistep method (excluding the one being computed). */ + private final int nSteps; + + /** Stepsize control exponent. */ + private double exp; + + /** Safety factor for stepsize control. */ + private double safety; + + /** Minimal reduction factor for stepsize control. */ + private double minReduction; + + /** Maximal growth factor for stepsize control. */ + private double maxGrowth; + + /** + * Build a multistep integrator with the given stepsize bounds. + * + * <p>The default starter integrator is set to the {@link DormandPrince853Integrator + * Dormand-Prince 8(5,3)} integrator with some defaults settings. + * + * <p>The default max growth factor is set to a quite low value: 2<sup>1/order</sup>. + * + * @param name name of the method + * @param nSteps number of steps of the multistep method (excluding the one being computed) + * @param order order of the method + * @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 NumberIsTooSmallException if number of steps is smaller than 2 + */ + protected MultistepIntegrator( + final String name, + final int nSteps, + final int order, + final double minStep, + final double maxStep, + final double scalAbsoluteTolerance, + final double scalRelativeTolerance) + throws NumberIsTooSmallException { + + super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance); + + if (nSteps < 2) { + throw new NumberIsTooSmallException( + LocalizedFormats.INTEGRATION_METHOD_NEEDS_AT_LEAST_TWO_PREVIOUS_POINTS, + nSteps, + 2, + true); + } + + starter = + new DormandPrince853Integrator( + minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance); + this.nSteps = nSteps; + + exp = -1.0 / order; + + // set the default values of the algorithm control parameters + setSafety(0.9); + setMinReduction(0.2); + setMaxGrowth(FastMath.pow(2.0, -exp)); + } + + /** + * Build a multistep integrator with the given stepsize bounds. + * + * <p>The default starter integrator is set to the {@link DormandPrince853Integrator + * Dormand-Prince 8(5,3)} integrator with some defaults settings. + * + * <p>The default max growth factor is set to a quite low value: 2<sup>1/order</sup>. + * + * @param name name of the method + * @param nSteps number of steps of the multistep method (excluding the one being computed) + * @param order order of the method + * @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 + */ + protected MultistepIntegrator( + final String name, + final int nSteps, + final int order, + final double minStep, + final double maxStep, + final double[] vecAbsoluteTolerance, + final double[] vecRelativeTolerance) { + super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance); + starter = + new DormandPrince853Integrator( + minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance); + this.nSteps = nSteps; + + exp = -1.0 / order; + + // set the default values of the algorithm control parameters + setSafety(0.9); + setMinReduction(0.2); + setMaxGrowth(FastMath.pow(2.0, -exp)); + } + + /** + * Get the starter integrator. + * + * @return starter integrator + */ + public ODEIntegrator getStarterIntegrator() { + return starter; + } + + /** + * Set the starter integrator. + * + * <p>The various step and event handlers for this starter integrator will be managed + * automatically by the multi-step integrator. Any user configuration for these elements will be + * cleared before use. + * + * @param starterIntegrator starter integrator + */ + public void setStarterIntegrator(FirstOrderIntegrator starterIntegrator) { + this.starter = starterIntegrator; + } + + /** + * Start the integration. + * + * <p>This method computes one step using the underlying starter integrator, and initializes the + * Nordsieck vector at step start. The starter integrator purpose is only to establish initial + * conditions, it does not really change time by itself. The top level multistep integrator + * remains in charge of handling time propagation and events handling as it will starts its own + * computation right from the beginning. In a sense, the starter integrator can be seen as a + * dummy one and so it will never trigger any user event nor call any user step handler. + * + * @param t0 initial time + * @param y0 initial value of the state vector at t0 + * @param t target time for the integration (can be set to a value smaller than <code>t0</code> + * for backward integration) + * @exception DimensionMismatchException if arrays dimension do not match equations settings + * @exception NumberIsTooSmallException if integration step is too small + * @exception MaxCountExceededException if the number of functions evaluations is exceeded + * @exception NoBracketingException if the location of an event cannot be bracketed + */ + protected void start(final double t0, final double[] y0, final double t) + throws DimensionMismatchException, + NumberIsTooSmallException, + MaxCountExceededException, + NoBracketingException { + + // make sure NO user event nor user step handler is triggered, + // this is the task of the top level integrator, not the task + // of the starter integrator + starter.clearEventHandlers(); + starter.clearStepHandlers(); + + // set up one specific step handler to extract initial Nordsieck vector + starter.addStepHandler(new NordsieckInitializer((nSteps + 3) / 2, y0.length)); + + // start integration, expecting a InitializationCompletedMarkerException + try { + + if (starter instanceof AbstractIntegrator) { + ((AbstractIntegrator) starter).integrate(getExpandable(), t); + } else { + starter.integrate( + new FirstOrderDifferentialEquations() { + + /** {@inheritDoc} */ + public int getDimension() { + return getExpandable().getTotalDimension(); + } + + /** {@inheritDoc} */ + public void computeDerivatives(double t, double[] y, double[] yDot) { + getExpandable().computeDerivatives(t, y, yDot); + } + }, + t0, + y0, + t, + new double[y0.length]); + } + + // we should not reach this step + throw new MathIllegalStateException(LocalizedFormats.MULTISTEP_STARTER_STOPPED_EARLY); + + } catch (InitializationCompletedMarkerException icme) { // NOPMD + // this is the expected nominal interruption of the start integrator + + // count the evaluations used by the starter + getCounter().increment(starter.getEvaluations()); + } + + // remove the specific step handler + starter.clearStepHandlers(); + } + + /** + * Initialize the high order scaled derivatives at step start. + * + * @param h step size to use for scaling + * @param t first steps times + * @param y first steps states + * @param yDot first steps derivatives + * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>, h<sup>3</sup>/6 + * y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>) + */ + protected abstract Array2DRowRealMatrix initializeHighOrderDerivatives( + final double h, final double[] t, final double[][] y, final double[][] yDot); + + /** + * Get the minimal reduction factor for stepsize control. + * + * @return minimal reduction factor + */ + public double getMinReduction() { + return minReduction; + } + + /** + * Set the minimal reduction factor for stepsize control. + * + * @param minReduction minimal reduction factor + */ + public void setMinReduction(final double minReduction) { + this.minReduction = minReduction; + } + + /** + * Get the maximal growth factor for stepsize control. + * + * @return maximal growth factor + */ + public double getMaxGrowth() { + return maxGrowth; + } + + /** + * Set the maximal growth factor for stepsize control. + * + * @param maxGrowth maximal growth factor + */ + public void setMaxGrowth(final double maxGrowth) { + this.maxGrowth = maxGrowth; + } + + /** + * Get the safety factor for stepsize control. + * + * @return safety factor + */ + public double getSafety() { + return safety; + } + + /** + * Set the safety factor for stepsize control. + * + * @param safety safety factor + */ + public void setSafety(final double safety) { + this.safety = safety; + } + + /** + * Get the number of steps of the multistep method (excluding the one being computed). + * + * @return number of steps of the multistep method (excluding the one being computed) + */ + public int getNSteps() { + return nSteps; + } + + /** + * Compute step grow/shrink factor according to normalized error. + * + * @param error normalized error of the current step + * @return grow/shrink factor for next step + */ + protected double computeStepGrowShrinkFactor(final double error) { + return FastMath.min( + maxGrowth, FastMath.max(minReduction, safety * FastMath.pow(error, exp))); + } + + /** + * Transformer used to convert the first step to Nordsieck representation. + * + * @deprecated as of 3.6 this unused interface is deprecated + */ + @Deprecated + public interface NordsieckTransformer { + /** + * Initialize the high order scaled derivatives at step start. + * + * @param h step size to use for scaling + * @param t first steps times + * @param y first steps states + * @param yDot first steps derivatives + * @return Nordieck vector at first step (h<sup>2</sup>/2 y''<sub>n</sub>, h<sup>3</sup>/6 + * y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>) + */ + Array2DRowRealMatrix initializeHighOrderDerivatives( + final double h, final double[] t, final double[][] y, final double[][] yDot); + } + + /** Specialized step handler storing the first step. */ + private class NordsieckInitializer implements StepHandler { + + /** Steps counter. */ + private int count; + + /** First steps times. */ + private final double[] t; + + /** First steps states. */ + private final double[][] y; + + /** First steps derivatives. */ + private final double[][] yDot; + + /** + * Simple constructor. + * + * @param nbStartPoints number of start points (including the initial point) + * @param n problem dimension + */ + NordsieckInitializer(final int nbStartPoints, final int n) { + this.count = 0; + this.t = new double[nbStartPoints]; + this.y = new double[nbStartPoints][n]; + this.yDot = new double[nbStartPoints][n]; + } + + /** {@inheritDoc} */ + public void handleStep(StepInterpolator interpolator, boolean isLast) + throws MaxCountExceededException { + + final double prev = interpolator.getPreviousTime(); + final double curr = interpolator.getCurrentTime(); + + if (count == 0) { + // first step, we need to store also the point at the beginning of the step + interpolator.setInterpolatedTime(prev); + t[0] = prev; + final ExpandableStatefulODE expandable = getExpandable(); + final EquationsMapper primary = expandable.getPrimaryMapper(); + primary.insertEquationData(interpolator.getInterpolatedState(), y[count]); + primary.insertEquationData(interpolator.getInterpolatedDerivatives(), yDot[count]); + int index = 0; + for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { + secondary.insertEquationData( + interpolator.getInterpolatedSecondaryState(index), y[count]); + secondary.insertEquationData( + interpolator.getInterpolatedSecondaryDerivatives(index), yDot[count]); + ++index; + } + } + + // store the point at the end of the step + ++count; + interpolator.setInterpolatedTime(curr); + t[count] = curr; + + final ExpandableStatefulODE expandable = getExpandable(); + final EquationsMapper primary = expandable.getPrimaryMapper(); + primary.insertEquationData(interpolator.getInterpolatedState(), y[count]); + primary.insertEquationData(interpolator.getInterpolatedDerivatives(), yDot[count]); + int index = 0; + for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { + secondary.insertEquationData( + interpolator.getInterpolatedSecondaryState(index), y[count]); + secondary.insertEquationData( + interpolator.getInterpolatedSecondaryDerivatives(index), yDot[count]); + ++index; + } + + if (count == t.length - 1) { + + // this was the last point we needed, we can compute the derivatives + stepStart = t[0]; + stepSize = (t[t.length - 1] - t[0]) / (t.length - 1); + + // first scaled derivative + scaled = yDot[0].clone(); + for (int j = 0; j < scaled.length; ++j) { + scaled[j] *= stepSize; + } + + // higher order derivatives + nordsieck = initializeHighOrderDerivatives(stepSize, t, y, yDot); + + // stop the integrator now that all needed steps have been handled + throw new InitializationCompletedMarkerException(); + } + } + + /** {@inheritDoc} */ + public void init(double t0, double[] y0, double time) { + // nothing to do + } + } + + /** Marker exception used ONLY to stop the starter integrator after first step. */ + private static class InitializationCompletedMarkerException extends RuntimeException { + + /** Serializable version identifier. */ + private static final long serialVersionUID = -1914085471038046418L; + + /** Simple constructor. */ + InitializationCompletedMarkerException() { + super((Throwable) null); + } + } +} |