AdamsMoultonIntegrator.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 java.util.Arrays;

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


  34. /**
  35.  * This class implements implicit Adams-Moulton integrators for Ordinary
  36.  * Differential Equations.
  37.  *
  38.  * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
  39.  * multistep ODE solvers. This implementation is a variation of the classical
  40.  * one: it uses adaptive stepsize to implement error control, whereas
  41.  * classical implementations are fixed step size. The value of state vector
  42.  * at step n+1 is a simple combination of the value at step n and of the
  43.  * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
  44.  * compute y<sub>n+1</sub>, another method must be used to compute a first
  45.  * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
  46.  * a final estimate of y<sub>n+1</sub> using the following formulas. Depending
  47.  * on the number k of previous steps one wants to use for computing the next
  48.  * value, different formulas are available for the final estimate:</p>
  49.  * <ul>
  50.  *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
  51.  *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
  52.  *   <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>
  53.  *   <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>
  54.  *   <li>...</li>
  55.  * </ul>
  56.  *
  57.  * <p>A k-steps Adams-Moulton method is of order k+1.</p>
  58.  *
  59.  * <p><b>Implementation details</b></p>
  60.  *
  61.  * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
  62.  * <div style="white-space: pre"><code>
  63.  * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
  64.  * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
  65.  * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
  66.  * ...
  67.  * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
  68.  * </code></div>
  69.  *
  70.  * <p>The definitions above use the classical representation with several previous first
  71.  * derivatives. Lets define
  72.  * <div style="white-space: pre"><code>
  73.  *   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>
  74.  * </code></div>
  75.  * (we omit the k index in the notation for clarity). With these definitions,
  76.  * Adams-Moulton methods can be written:
  77.  * <ul>
  78.  *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
  79.  *   <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>
  80.  *   <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>
  81.  *   <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>
  82.  *   <li>...</li>
  83.  * </ul>
  84.  *
  85.  * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
  86.  * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
  87.  * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
  88.  * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
  89.  * <div style="white-space: pre"><code>
  90.  * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
  91.  * </code></div>
  92.  * (here again we omit the k index in the notation for clarity)
  93.  *
  94.  * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
  95.  * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
  96.  * for degree k polynomials.
  97.  * <div style="white-space: pre"><code>
  98.  * 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)
  99.  * </code></div>
  100.  * The previous formula can be used with several values for i to compute the transform between
  101.  * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
  102.  * and q<sub>n</sub> resulting from the Taylor series formulas above is:
  103.  * <div style="white-space: pre"><code>
  104.  * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
  105.  * </code></div>
  106.  * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
  107.  * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
  108.  * the column number starting from 1:
  109.  * <pre>
  110.  *        [  -2   3   -4    5  ... ]
  111.  *        [  -4  12  -32   80  ... ]
  112.  *   P =  [  -6  27 -108  405  ... ]
  113.  *        [  -8  48 -256 1280  ... ]
  114.  *        [          ...           ]
  115.  * </pre>
  116.  *
  117.  * <p>Using the Nordsieck vector has several advantages:
  118.  * <ul>
  119.  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
  120.  *   Taylor series formulas,</li>
  121.  *   <li>it simplifies step changes that occur when discrete events that truncate
  122.  *   the step are triggered,</li>
  123.  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
  124.  * </ul>
  125.  *
  126.  * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
  127.  * n as follows:
  128.  * <ul>
  129.  *   <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
  130.  *   <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
  131.  *   <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>
  132.  * </ul>
  133.  * where A is a rows shifting matrix (the lower left part is an identity matrix):
  134.  * <pre>
  135.  *        [ 0 0   ...  0 0 | 0 ]
  136.  *        [ ---------------+---]
  137.  *        [ 1 0   ...  0 0 | 0 ]
  138.  *    A = [ 0 1   ...  0 0 | 0 ]
  139.  *        [       ...      | 0 ]
  140.  *        [ 0 0   ...  1 0 | 0 ]
  141.  *        [ 0 0   ...  0 1 | 0 ]
  142.  * </pre>
  143.  * From this predicted vector, the corrected vector is computed as follows:
  144.  * <ul>
  145.  *   <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... &plusmn;1 ] r<sub>n+1</sub></li>
  146.  *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
  147.  *   <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>
  148.  * </ul>
  149.  * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
  150.  * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
  151.  * represent the corrected states.
  152.  *
  153.  * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
  154.  * they only depend on k and therefore are precomputed once for all.</p>
  155.  *
  156.  * @since 2.0
  157.  */
  158. public class AdamsMoultonIntegrator extends AdamsIntegrator {

  160.     /** Integrator method name. */
  161.     private static final String METHOD_NAME = "Adams-Moulton";

  163.     /**
  164.      * Build an Adams-Moulton integrator with the given order and error control parameters.
  165.      * @param nSteps number of steps of the method excluding the one being computed
  166.      * @param minStep minimal step (sign is irrelevant, regardless of
  167.      * integration direction, forward or backward), the last step can
  168.      * be smaller than this
  169.      * @param maxStep maximal step (sign is irrelevant, regardless of
  170.      * integration direction, forward or backward), the last step can
  171.      * be smaller than this
  172.      * @param scalAbsoluteTolerance allowed absolute error
  173.      * @param scalRelativeTolerance allowed relative error
  174.      * @exception NumberIsTooSmallException if order is 1 or less
  175.      */
  176.     public AdamsMoultonIntegrator(final int nSteps,
  177.                                   final double minStep, final double maxStep,
  178.                                   final double scalAbsoluteTolerance,
  179.                                   final double scalRelativeTolerance)
  180.         throws NumberIsTooSmallException {
  181.         super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
  182.               scalAbsoluteTolerance, scalRelativeTolerance);
  183.     }

  185.     /**
  186.      * Build an Adams-Moulton integrator with the given order and error control parameters.
  187.      * @param nSteps number of steps of the method excluding the one being computed
  188.      * @param minStep minimal step (sign is irrelevant, regardless of
  189.      * integration direction, forward or backward), the last step can
  190.      * be smaller than this
  191.      * @param maxStep maximal step (sign is irrelevant, regardless of
  192.      * integration direction, forward or backward), the last step can
  193.      * be smaller than this
  194.      * @param vecAbsoluteTolerance allowed absolute error
  195.      * @param vecRelativeTolerance allowed relative error
  196.      * @exception IllegalArgumentException if order is 1 or less
  197.      */
  198.     public AdamsMoultonIntegrator(final int nSteps,
  199.                                   final double minStep, final double maxStep,
  200.                                   final double[] vecAbsoluteTolerance,
  201.                                   final double[] vecRelativeTolerance)
  202.         throws IllegalArgumentException {
  203.         super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
  204.               vecAbsoluteTolerance, vecRelativeTolerance);
  205.     }

  207.     /** {@inheritDoc} */
  208.     @Override
  209.     public void integrate(final ExpandableStatefulODE equations,final double t)
  210.         throws NumberIsTooSmallException, DimensionMismatchException,
  211.                MaxCountExceededException, NoBracketingException {

  213.         sanityChecks(equations, t);
  214.         setEquations(equations);
  215.         final boolean forward = t > equations.getTime();

  217.         // initialize working arrays
  218.         final double[] y0   = equations.getCompleteState();
  219.         final double[] y    = y0.clone();
  220.         final double[] yDot = new double[y.length];
  221.         final double[] yTmp = new double[y.length];
  222.         final double[] predictedScaled = new double[y.length];
  223.         Array2DRowRealMatrix nordsieckTmp = null;

  225.         // set up two interpolators sharing the integrator arrays
  226.         final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
  227.         interpolator.reinitialize(y, forward,
  228.                                   equations.getPrimaryMapper(), equations.getSecondaryMappers());

  230.         // set up integration control objects
  231.         initIntegration(equations.getTime(), y0, t);

  233.         // compute the initial Nordsieck vector using the configured starter integrator
  234.         start(equations.getTime(), y, t);
  235.         interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
  236.         interpolator.storeTime(stepStart);

  238.         double hNew = stepSize;
  239.         interpolator.rescale(hNew);

  241.         isLastStep = false;
  242.         do {

  244.             double error = 10;
  245.             while (error >= 1.0) {

  247.                 stepSize = hNew;

  249.                 // predict a first estimate of the state at step end (P in the PECE sequence)
  250.                 final double stepEnd = stepStart + stepSize;
  251.                 interpolator.setInterpolatedTime(stepEnd);
  252.                 final ExpandableStatefulODE expandable = getExpandable();
  253.                 final EquationsMapper primary = expandable.getPrimaryMapper();
  254.                 primary.insertEquationData(interpolator.getInterpolatedState(), yTmp);
  255.                 int index = 0;
  256.                 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) {
  257.                     secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), yTmp);
  258.                     ++index;
  259.                 }

  261.                 // evaluate a first estimate of the derivative (first E in the PECE sequence)
  262.                 computeDerivatives(stepEnd, yTmp, yDot);

  264.                 // update Nordsieck vector
  265.                 for (int j = 0; j < y0.length; ++j) {
  266.                     predictedScaled[j] = stepSize * yDot[j];
  267.                 }
  268.                 nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
  269.                 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);

  271.                 // apply correction (C in the PECE sequence)
  272.                 error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp));

  274.                 if (error >= 1.0) {
  275.                     // reject the step and attempt to reduce error by stepsize control
  276.                     final double factor = computeStepGrowShrinkFactor(error);
  277.                     hNew = filterStep(stepSize * factor, forward, false);
  278.                     interpolator.rescale(hNew);
  279.                 }
  280.             }

  282.             // evaluate a final estimate of the derivative (second E in the PECE sequence)
  283.             final double stepEnd = stepStart + stepSize;
  284.             computeDerivatives(stepEnd, yTmp, yDot);

  286.             // update Nordsieck vector
  287.             final double[] correctedScaled = new double[y0.length];
  288.             for (int j = 0; j < y0.length; ++j) {
  289.                 correctedScaled[j] = stepSize * yDot[j];
  290.             }
  291.             updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp);

  293.             // discrete events handling
  294.             System.arraycopy(yTmp, 0, y, 0, y.length);
  295.             interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp);
  296.             interpolator.storeTime(stepStart);
  297.             interpolator.shift();
  298.             interpolator.storeTime(stepEnd);
  299.             stepStart = acceptStep(interpolator, y, yDot, t);
  300.             scaled    = correctedScaled;
  301.             nordsieck = nordsieckTmp;

  303.             if (!isLastStep) {

  305.                 // prepare next step
  306.                 interpolator.storeTime(stepStart);

  308.                 if (resetOccurred) {
  309.                     // some events handler has triggered changes that
  310.                     // invalidate the derivatives, we need to restart from scratch
  311.                     start(stepStart, y, t);
  312.                     interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
  313.                 }

  315.                 // stepsize control for next step
  316.                 final double  factor     = computeStepGrowShrinkFactor(error);
  317.                 final double  scaledH    = stepSize * factor;
  318.                 final double  nextT      = stepStart + scaledH;
  319.                 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
  320.                 hNew = filterStep(scaledH, forward, nextIsLast);

  322.                 final double  filteredNextT      = stepStart + hNew;
  323.                 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t);
  324.                 if (filteredNextIsLast) {
  325.                     hNew = t - stepStart;
  326.                 }

  328.                 interpolator.rescale(hNew);
  329.             }
  330.         } while (!isLastStep);

  332.         // dispatch results
  333.         equations.setTime(stepStart);
  334.         equations.setCompleteState(y);

  336.         resetInternalState();
  337.     }

  339.     /** Corrector for current state in Adams-Moulton method.
  340.      * <p>
  341.      * This visitor implements the Taylor series formula:
  342.      * <pre>
  343.      * Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... &plusmn;1 ] r<sub>n+1</sub>
  344.      * </pre>
  345.      * </p>
  346.      */
  347.     private final class Corrector implements RealMatrixPreservingVisitor {

  349.         /** Previous state. */
  350.         private final double[] previous;

  352.         /** Current scaled first derivative. */
  353.         private final double[] scaled;

  355.         /** Current state before correction. */
  356.         private final double[] before;

  358.         /** Current state after correction. */
  359.         private final double[] after;

  361.         /** Simple constructor.
  362.          * @param previous previous state
  363.          * @param scaled current scaled first derivative
  364.          * @param state state to correct (will be overwritten after visit)
  365.          */
  366.         Corrector(final double[] previous, final double[] scaled, final double[] state) {
  367.             this.previous = previous;
  368.             this.scaled   = scaled;
  369.             this.after    = state;
  370.             this.before   = state.clone();
  371.         }

  373.         /** {@inheritDoc} */
  374.         @Override
  375.         public void start(int rows, int columns,
  376.                           int startRow, int endRow, int startColumn, int endColumn) {
  377.             Arrays.fill(after, 0.0);
  378.         }

  380.         /** {@inheritDoc} */
  381.         @Override
  382.         public void visit(int row, int column, double value) {
  383.             if ((row & 0x1) == 0) {
  384.                 after[column] -= value;
  385.             } else {
  386.                 after[column] += value;
  387.             }
  388.         }

  390.         /**
  391.          * End visiting the Nordsieck vector.
  392.          * <p>The correction is used to control stepsize. So its amplitude is
  393.          * considered to be an error, which must be normalized according to
  394.          * error control settings. If the normalized value is greater than 1,
  395.          * the correction was too large and the step must be rejected.</p>
  396.          * @return the normalized correction, if greater than 1, the step
  397.          * must be rejected
  398.          */
  399.         @Override
  400.         public double end() {

  402.             double error = 0;
  403.             for (int i = 0; i < after.length; ++i) {
  404.                 after[i] += previous[i] + scaled[i];
  405.                 if (i < mainSetDimension) {
  406.                     final double yScale = JdkMath.max(JdkMath.abs(previous[i]), JdkMath.abs(after[i]));
  407.                     final double tol    = (vecAbsoluteTolerance == null) ?
  408.                                           (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
  409.                                           (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
  410.                     final double ratio  = (after[i] - before[i]) / tol; // (corrected-predicted)/tol
  411.                     error += ratio * ratio;
  412.                 }
  413.             }

  415.             return JdkMath.sqrt(error / mainSetDimension);
  416.         }
  417.     }
  418. }
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