AdamsBashforthIntegrator.java

  1. /*
  2.  * Licensed to the Hipparchus project 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 Hipparchus project 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.  *      https://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.hipparchus.ode.nonstiff;

  20. import org.hipparchus.exception.MathIllegalArgumentException;
  21. import org.hipparchus.linear.Array2DRowRealMatrix;
  22. import org.hipparchus.linear.RealMatrix;
  23. import org.hipparchus.ode.EquationsMapper;
  24. import org.hipparchus.ode.ODEStateAndDerivative;
  25. import org.hipparchus.ode.nonstiff.interpolators.AdamsStateInterpolator;
  26. import org.hipparchus.util.FastMath;


  29. /**
  30.  * This class implements explicit Adams-Bashforth integrators for Ordinary
  31.  * Differential Equations.
  32.  *
  33.  * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
  34.  * multistep ODE solvers. This implementation is a variation of the classical
  35.  * one: it uses adaptive stepsize to implement error control, whereas
  36.  * classical implementations are fixed step size. The value of state vector
  37.  * at step n+1 is a simple combination of the value at step n and of the
  38.  * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous
  39.  * steps one wants to use for computing the next value, different formulas
  40.  * are available:</p>
  41.  * <ul>
  42.  *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
  43.  *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
  44.  *   <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>
  45.  *   <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>
  46.  *   <li>...</li>
  47.  * </ul>
  48.  *
  49.  * <p>A k-steps Adams-Bashforth method is of order k.</p>
  50.  *
  51.  * <p> There must be sufficient time for the {@link #setStarterIntegrator(org.hipparchus.ode.ODEIntegrator)
  52.  * starter integrator} to take several steps between the the last reset event, and the end
  53.  * of integration, otherwise an exception may be thrown during integration. The user can
  54.  * adjust the end date of integration, or the step size of the starter integrator to
  55.  * ensure a sufficient number of steps can be completed before the end of integration.
  56.  * </p>
  57.  *
  58.  * <p><strong>Implementation details</strong></p>
  59.  *
  60.  * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
  61.  * \[
  62.  *   \left\{\begin{align}
  63.  *   s_1(n) &amp;= h y'_n \text{ for first derivative}\\
  64.  *   s_2(n) &amp;= \frac{h^2}{2} y_n'' \text{ for second derivative}\\
  65.  *   s_3(n) &amp;= \frac{h^3}{6} y_n''' \text{ for third derivative}\\
  66.  *   &amp;\cdots\\
  67.  *   s_k(n) &amp;= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative}
  68.  *   \end{align}\right.
  69.  * \]</p>
  70.  *
  71.  * <p>The definitions above use the classical representation with several previous first
  72.  * derivatives. Lets define
  73.  * \[
  74.  *   q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T
  75.  * \]
  76.  * (we omit the k index in the notation for clarity). With these definitions,
  77.  * Adams-Bashforth methods can be written:</p>
  78.  * \[
  79.  *   \left\{\begin{align}
  80.  *   k = 1: &amp; y_{n+1} = y_n +               s_1(n) \\
  81.  *   k = 2: &amp; y_{n+1} = y_n + \frac{3}{2}   s_1(n) + [ \frac{-1}{2} ] q_n \\
  82.  *   k = 3: &amp; y_{n+1} = y_n + \frac{23}{12} s_1(n) + [ \frac{-16}{12} \frac{5}{12} ] q_n \\
  83.  *   k = 4: &amp; y_{n+1} = y_n + \frac{55}{24} s_1(n) + [ \frac{-59}{24} \frac{37}{24} \frac{-9}{24} ] q_n \\
  84.  *          &amp; \cdots
  85.  *   \end{align}\right.
  86.  * \]
  87.  *
  88.  * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
  89.  * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with
  90.  * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
  91.  * and r<sub>n</sub>) where r<sub>n</sub> is defined as:
  92.  * \[
  93.  * r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T
  94.  * \]
  95.  * (here again we omit the k index in the notation for clarity)
  96.  * </p>
  97.  *
  98.  * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
  99.  * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
  100.  * for degree k polynomials.
  101.  * \[
  102.  * s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)
  103.  * \]
  104.  * The previous formula can be used with several values for i to compute the transform between
  105.  * classical representation and Nordsieck vector. The transform between r<sub>n</sub>
  106.  * and q<sub>n</sub> resulting from the Taylor series formulas above is:
  107.  * \[
  108.  * q_n = s_1(n) u + P r_n
  109.  * \]
  110.  * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
  111.  * with the \((j+1) (-i)^j\) terms with i being the row number starting from 1 and j being
  112.  * the column number starting from 1:
  113.  * \[
  114.  *   P=\begin{bmatrix}
  115.  *   -2  &amp;  3 &amp;   -4 &amp;    5 &amp; \ldots \\
  116.  *   -4  &amp; 12 &amp;  -32 &amp;   80 &amp; \ldots \\
  117.  *   -6  &amp; 27 &amp; -108 &amp;  405 &amp; \ldots \\
  118.  *   -8  &amp; 48 &amp; -256 &amp; 1280 &amp; \ldots \\
  119.  *       &amp;    &amp;  \ldots\\
  120.  *    \end{bmatrix}
  121.  * \]
  122.  * </p>
  123.  *
  124.  * <p>Using the Nordsieck vector has several advantages:</p>
  125.  * <ul>
  126.  *   <li>it greatly simplifies step interpolation as the interpolator mainly applies
  127.  *   Taylor series formulas,</li>
  128.  *   <li>it simplifies step changes that occur when discrete events that truncate
  129.  *   the step are triggered,</li>
  130.  *   <li>it allows to extend the methods in order to support adaptive stepsize.</li>
  131.  * </ul>
  132.  *
  133.  * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
  134.  * <ul>
  135.  *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
  136.  *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
  137.  *   <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>
  138.  * </ul>
  139.  * <p>where A is a rows shifting matrix (the lower left part is an identity matrix):</p>
  140.  * <pre>
  141.  *        [ 0 0   ...  0 0 | 0 ]
  142.  *        [ ---------------+---]
  143.  *        [ 1 0   ...  0 0 | 0 ]
  144.  *    A = [ 0 1   ...  0 0 | 0 ]
  145.  *        [       ...      | 0 ]
  146.  *        [ 0 0   ...  1 0 | 0 ]
  147.  *        [ 0 0   ...  0 1 | 0 ]
  148.  * </pre>
  149.  *
  150.  * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
  151.  * they only depend on k and therefore are precomputed once for all.</p>
  152.  *
  153.  */
  154. public class AdamsBashforthIntegrator extends AdamsIntegrator {

  156.     /** Name of integration scheme. */
  157.     public static final String METHOD_NAME = "Adams-Bashforth";

  159.     /**
  160.      * Build an Adams-Bashforth integrator with the given order and step control parameters.
  161.      * @param nSteps number of steps of the method excluding the one being computed
  162.      * @param minStep minimal step (sign is irrelevant, regardless of
  163.      * integration direction, forward or backward), the last step can
  164.      * be smaller than this
  165.      * @param maxStep maximal step (sign is irrelevant, regardless of
  166.      * integration direction, forward or backward), the last step can
  167.      * be smaller than this
  168.      * @param scalAbsoluteTolerance allowed absolute error
  169.      * @param scalRelativeTolerance allowed relative error
  170.      * @exception MathIllegalArgumentException if order is 1 or less
  171.      */
  172.     public AdamsBashforthIntegrator(final int nSteps,
  173.                                     final double minStep, final double maxStep,
  174.                                     final double scalAbsoluteTolerance,
  175.                                     final double scalRelativeTolerance)
  176.         throws MathIllegalArgumentException {
  177.         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
  178.               scalAbsoluteTolerance, scalRelativeTolerance);
  179.     }

  181.     /**
  182.      * Build an Adams-Bashforth integrator with the given order and step control parameters.
  183.      * @param nSteps number of steps of the method excluding the one being computed
  184.      * @param minStep minimal step (sign is irrelevant, regardless of
  185.      * integration direction, forward or backward), the last step can
  186.      * be smaller than this
  187.      * @param maxStep maximal step (sign is irrelevant, regardless of
  188.      * integration direction, forward or backward), the last step can
  189.      * be smaller than this
  190.      * @param vecAbsoluteTolerance allowed absolute error
  191.      * @param vecRelativeTolerance allowed relative error
  192.      * @exception IllegalArgumentException if order is 1 or less
  193.      */
  194.     public AdamsBashforthIntegrator(final int nSteps,
  195.                                     final double minStep, final double maxStep,
  196.                                     final double[] vecAbsoluteTolerance,
  197.                                     final double[] vecRelativeTolerance)
  198.         throws IllegalArgumentException {
  199.         super(METHOD_NAME, nSteps, nSteps, minStep, maxStep,
  200.               vecAbsoluteTolerance, vecRelativeTolerance);
  201.     }

  203.     /** {@inheritDoc} */
  204.     @Override
  205.     protected double errorEstimation(final double[] previousState, final double predictedTime,
  206.                                      final double[] predictedState,
  207.                                      final double[] predictedScaled,
  208.                                      final RealMatrix predictedNordsieck) {

  210.         final StepsizeHelper helper = getStepSizeHelper();
  211.         double error = 0;
  212.         for (int i = 0; i < helper.getMainSetDimension(); ++i) {
  213.             final double tol = helper.getTolerance(i, FastMath.abs(predictedState[i]));

  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() % 2 == 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;

  228.         }

  230.         return FastMath.sqrt(error / helper.getMainSetDimension());

  232.     }

  234.     /** {@inheritDoc} */
  235.     @Override
  236.     protected AdamsStateInterpolator finalizeStep(final double stepSize, final double[] predictedState,
  237.                                                   final double[] predictedScaled, final Array2DRowRealMatrix predictedNordsieck,
  238.                                                   final boolean isForward,
  239.                                                   final ODEStateAndDerivative globalPreviousState,
  240.                                                   final ODEStateAndDerivative globalCurrentState,
  241.                                                   final EquationsMapper equationsMapper) {
  242.         return new AdamsStateInterpolator(getStepSize(), globalCurrentState,
  243.                                           predictedScaled, predictedNordsieck, isForward,
  244.                                           getStepStart(), globalCurrentState, equationsMapper);
  245.     }

  247. }
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