AdamsBashforthIntegrator.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */

  18. package org.apache.commons.math4.legacy.ode.nonstiff;

  20. import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
  21. import org.apache.commons.math4.legacy.exception.MaxCountExceededException;
  22. import org.apache.commons.math4.legacy.exception.NoBracketingException;
  23. import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
  24. import org.apache.commons.math4.legacy.linear.Array2DRowRealMatrix;
  25. import org.apache.commons.math4.legacy.linear.RealMatrix;
  26. import org.apache.commons.math4.legacy.ode.EquationsMapper;
  27. import org.apache.commons.math4.legacy.ode.ExpandableStatefulODE;
  28. import org.apache.commons.math4.legacy.ode.sampling.NordsieckStepInterpolator;
  29. import org.apache.commons.math4.core.jdkmath.JdkMath;


  32. /**
  33.  * This class implements explicit Adams-Bashforth integrators for Ordinary
  34.  * Differential Equations.
  35.  *
  36.  * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
  37.  * multistep ODE solvers. This implementation is a variation of the classical
  38.  * one: it uses adaptive stepsize to implement error control, whereas
  39.  * classical implementations are fixed step size. The value of state vector
  40.  * at step n+1 is a simple combination of the value at step n and of the
  41.  * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
  42.  * steps one wants to use for computing the next value, different formulas
  43.  * are available:</p>
  44.  * <ul>
  45.  *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
  46.  *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
  47.  *   <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>
  48.  *   <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>
  49.  *   <li>...</li>
  50.  * </ul>
  51.  *
  52.  * <p>A k-steps Adams-Bashforth method is of order k.</p>
  53.  *
  54.  * <p><b>Implementation details</b></p>
  55.  *
  56.  * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
  57.  * <div style="white-space: pre"><code>
  58.  * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
  59.  * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
  60.  * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
  61.  * ...
  62.  * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
  63.  * </code></div>
  64.  *
  65.  * <p>The definitions above use the classical representation with several previous first
  66.  * derivatives. Lets define
  67.  * <div style="white-space: pre"><code>
  68.  *   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>
  69.  * </code></div>
  70.  * (we omit the k index in the notation for clarity). With these definitions,
  71.  * Adams-Bashforth methods can be written:
  72.  * <ul>
  73.  *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
  74.  *   <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>
  75.  *   <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>
  76.  *   <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>
  77.  *   <li>...</li>
  78.  * </ul>
  79.  *
  80.  * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
  81.  * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
  82.  * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
  83.  * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
  84.  * <div style="white-space: pre"><code>
  85.  * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
  86.  * </code></div>
  87.  * (here again we omit the k index in the notation for clarity)
  88.  *
  89.  * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
  90.  * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
  91.  * for degree k polynomials.
  92.  * <div style="white-space: pre"><code>
  93.  * 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)
  94.  * </code></div>
  95.  * The previous formula can be used with several values for i to compute the transform between
  96.  * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
  97.  * and q<sub>n</sub> resulting from the Taylor series formulas above is:
  98.  * <div style="white-space: pre"><code>
  99.  * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
  100.  * </code></div>
  101.  * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
  102.  * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
  103.  * the column number starting from 1:
  104.  * <pre>
  105.  *        [  -2   3   -4    5  ... ]
  106.  *        [  -4  12  -32   80  ... ]
  107.  *   P =  [  -6  27 -108  405  ... ]
  108.  *        [  -8  48 -256 1280  ... ]
  109.  *        [          ...           ]
  110.  * </pre>
  111.  *
  112.  * <p>Using the Nordsieck vector has several advantages:
  113.  * <ul>
  114.  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
  115.  *   Taylor series formulas,</li>
  116.  *   <li>it simplifies step changes that occur when discrete events that truncate
  117.  *   the step are triggered,</li>
  118.  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
  119.  * </ul>
  120.  *
  121.  * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
  122.  * <ul>
  123.  *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
  124.  *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
  125.  *   <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>
  126.  * </ul>
  127.  * where A is a rows shifting matrix (the lower left part is an identity matrix):
  128.  * <pre>
  129.  *        [ 0 0   ...  0 0 | 0 ]
  130.  *        [ ---------------+---]
  131.  *        [ 1 0   ...  0 0 | 0 ]
  132.  *    A = [ 0 1   ...  0 0 | 0 ]
  133.  *        [       ...      | 0 ]
  134.  *        [ 0 0   ...  1 0 | 0 ]
  135.  *        [ 0 0   ...  0 1 | 0 ]
  136.  * </pre>
  137.  *
  138.  * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
  139.  * they only depend on k and therefore are precomputed once for all.</p>
  140.  *
  141.  * @since 2.0
  142.  */
  143. public class AdamsBashforthIntegrator extends AdamsIntegrator {

  145.     /** Integrator method name. */
  146.     private static final String METHOD_NAME = "Adams-Bashforth";

  148.     /**
  149.      * Build an Adams-Bashforth integrator with the given order and step control parameters.
  150.      * @param nSteps number of steps of the method excluding the one being computed
  151.      * @param minStep minimal step (sign is irrelevant, regardless of
  152.      * integration direction, forward or backward), the last step can
  153.      * be smaller than this
  154.      * @param maxStep maximal step (sign is irrelevant, regardless of
  155.      * integration direction, forward or backward), the last step can
  156.      * be smaller than this
  157.      * @param scalAbsoluteTolerance allowed absolute error
  158.      * @param scalRelativeTolerance allowed relative error
  159.      * @exception NumberIsTooSmallException if order is 1 or less
  160.      */
  161.     public AdamsBashforthIntegrator(final int nSteps,
  162.                                     final double minStep, final double maxStep,
  163.                                     final double scalAbsoluteTolerance,
  164.                                     final double scalRelativeTolerance)
  165.         throws NumberIsTooSmallException {
  166.         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
  167.               scalAbsoluteTolerance, scalRelativeTolerance);
  168.     }

  170.     /**
  171.      * Build an Adams-Bashforth integrator with the given order and step control parameters.
  172.      * @param nSteps number of steps of the method excluding the one being computed
  173.      * @param minStep minimal step (sign is irrelevant, regardless of
  174.      * integration direction, forward or backward), the last step can
  175.      * be smaller than this
  176.      * @param maxStep maximal step (sign is irrelevant, regardless of
  177.      * integration direction, forward or backward), the last step can
  178.      * be smaller than this
  179.      * @param vecAbsoluteTolerance allowed absolute error
  180.      * @param vecRelativeTolerance allowed relative error
  181.      * @exception IllegalArgumentException if order is 1 or less
  182.      */
  183.     public AdamsBashforthIntegrator(final int nSteps,
  184.                                     final double minStep, final double maxStep,
  185.                                     final double[] vecAbsoluteTolerance,
  186.                                     final double[] vecRelativeTolerance)
  187.         throws IllegalArgumentException {
  188.         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
  189.               vecAbsoluteTolerance, vecRelativeTolerance);
  190.     }

  192.     /** Estimate error.
  193.      * <p>
  194.      * Error is estimated by interpolating back to previous state using
  195.      * the state Taylor expansion and comparing to real previous state.
  196.      * </p>
  197.      * @param previousState state vector at step start
  198.      * @param predictedState predicted state vector at step end
  199.      * @param predictedScaled predicted value of the scaled derivatives at step end
  200.      * @param predictedNordsieck predicted value of the Nordsieck vector at step end
  201.      * @return estimated normalized local discretization error
  202.      */
  203.     private double errorEstimation(final double[] previousState,
  204.                                    final double[] predictedState,
  205.                                    final double[] predictedScaled,
  206.                                    final RealMatrix predictedNordsieck) {

  208.         double error = 0;
  209.         for (int i = 0; i < mainSetDimension; ++i) {
  210.             final double yScale = JdkMath.abs(predictedState[i]);
  211.             final double tol = (vecAbsoluteTolerance == null) ?
  212.                                (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
  213.                                (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);

  215.             // apply Taylor formula from high order to low order,
  216.             // for the sake of numerical accuracy
  217.             double variation = 0;
  218.             int sign = (predictedNordsieck.getRowDimension() & 1) == 0 ? -1 : 1;
  219.             for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) {
  220.                 variation += sign * predictedNordsieck.getEntry(k, i);
  221.                 sign       = -sign;
  222.             }
  223.             variation -= predictedScaled[i];

  225.             final double ratio  = (predictedState[i] - previousState[i] + variation) / tol;
  226.             error              += ratio * ratio;
  227.         }

  229.         return JdkMath.sqrt(error / mainSetDimension);
  230.     }

  232.     /** {@inheritDoc} */
  233.     @Override
  234.     public void integrate(final ExpandableStatefulODE equations, final double t)
  235.         throws NumberIsTooSmallException, DimensionMismatchException,
  236.                MaxCountExceededException, NoBracketingException {

  238.         sanityChecks(equations, t);
  239.         setEquations(equations);
  240.         final boolean forward = t > equations.getTime();

  242.         // initialize working arrays
  243.         final double[] y    = equations.getCompleteState();
  244.         final double[] yDot = new double[y.length];

  246.         // set up an interpolator sharing the integrator arrays
  247.         final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
  248.         interpolator.reinitialize(y, forward,
  249.                                   equations.getPrimaryMapper(), equations.getSecondaryMappers());

  251.         // set up integration control objects
  252.         initIntegration(equations.getTime(), y, t);

  254.         // compute the initial Nordsieck vector using the configured starter integrator
  255.         start(equations.getTime(), y, t);
  256.         interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
  257.         interpolator.storeTime(stepStart);

  259.         // reuse the step that was chosen by the starter integrator
  260.         double hNew = stepSize;
  261.         interpolator.rescale(hNew);

  263.         // main integration loop
  264.         isLastStep = false;
  265.         do {

  267.             interpolator.shift();
  268.             final double[] predictedY      = new double[y.length];
  269.             final double[] predictedScaled = new double[y.length];
  270.             Array2DRowRealMatrix predictedNordsieck = null;
  271.             double error = 10;
  272.             while (error >= 1.0) {

  274.                 // predict a first estimate of the state at step end
  275.                 final double stepEnd = stepStart + hNew;
  276.                 interpolator.storeTime(stepEnd);
  277.                 final ExpandableStatefulODE expandable = getExpandable();
  278.                 final EquationsMapper primary = expandable.getPrimaryMapper();
  279.                 primary.insertEquationData(interpolator.getInterpolatedState(), predictedY);
  280.                 int index = 0;
  281.                 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
  282.                     secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), predictedY);
  283.                     ++index;
  284.                 }

  286.                 // evaluate the derivative
  287.                 computeDerivatives(stepEnd, predictedY, yDot);

  289.                 // predict Nordsieck vector at step end
  290.                 for (int j = 0; j < predictedScaled.length; ++j) {
  291.                     predictedScaled[j] = hNew * yDot[j];
  292.                 }
  293.                 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
  294.                 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);

  296.                 // evaluate error
  297.                 error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck);

  299.                 if (error >= 1.0) {
  300.                     // reject the step and attempt to reduce error by stepsize control
  301.                     final double factor = computeStepGrowShrinkFactor(error);
  302.                     hNew = filterStep(hNew * factor, forward, false);
  303.                     interpolator.rescale(hNew);
  304.                 }
  305.             }

  307.             stepSize = hNew;
  308.             final double stepEnd = stepStart + stepSize;
  309.             interpolator.reinitialize(stepEnd, stepSize, predictedScaled, predictedNordsieck);

  311.             // discrete events handling
  312.             interpolator.storeTime(stepEnd);
  313.             System.arraycopy(predictedY, 0, y, 0, y.length);
  314.             stepStart = acceptStep(interpolator, y, yDot, t);
  315.             scaled    = predictedScaled;
  316.             nordsieck = predictedNordsieck;
  317.             interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);

  319.             if (!isLastStep) {

  321.                 // prepare next step
  322.                 interpolator.storeTime(stepStart);

  324.                 if (resetOccurred) {
  325.                     // some events handler has triggered changes that
  326.                     // invalidate the derivatives, we need to restart from scratch
  327.                     start(stepStart, y, t);
  328.                     interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
  329.                 }

  331.                 // stepsize control for next step
  332.                 final double  factor     = computeStepGrowShrinkFactor(error);
  333.                 final double  scaledH    = stepSize * factor;
  334.                 final double  nextT      = stepStart + scaledH;
  335.                 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
  336.                 hNew = filterStep(scaledH, forward, nextIsLast);

  338.                 final double  filteredNextT      = stepStart + hNew;
  339.                 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
  340.                 if (filteredNextIsLast) {
  341.                     hNew = t - stepStart;
  342.                 }

  344.                 interpolator.rescale(hNew);
  345.             }
  346.         } while (!isLastStep);

  348.         // dispatch results
  349.         equations.setTime(stepStart);
  350.         equations.setCompleteState(y);

  352.         resetInternalState();
  353.     }
  354. }
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