ode/nonstiff/AdamsBashforthIntegrator.java

/*
 * 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) + &sum;<sub>j&gt;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)&times;(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();

    }

}