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Commons Math example source code file (EmbeddedRungeKuttaIntegrator.java)

This example Commons Math source code file (EmbeddedRungeKuttaIntegrator.java) is included in the DevDaily.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

Java - Commons Math tags/keywords

abstractstepinterpolator, combinedeventsmanager, derivativeexception, embeddedrungekuttaintegrator, embeddedrungekuttaintegrator, firstorderdifferentialequations, integratorexception, override, rungekuttastepinterpolator, rungekuttastepinterpolator, stephandler, stephandler, string, string

The Commons Math EmbeddedRungeKuttaIntegrator.java source code

/*
 * 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.math.ode.nonstiff;

import org.apache.commons.math.ode.DerivativeException;
import org.apache.commons.math.ode.FirstOrderDifferentialEquations;
import org.apache.commons.math.ode.IntegratorException;
import org.apache.commons.math.ode.events.CombinedEventsManager;
import org.apache.commons.math.ode.sampling.AbstractStepInterpolator;
import org.apache.commons.math.ode.sampling.DummyStepInterpolator;
import org.apache.commons.math.ode.sampling.StepHandler;

/**
 * This class implements the common part of all embedded Runge-Kutta
 * integrators for Ordinary Differential Equations.
 *
 * <p>These methods are embedded explicit Runge-Kutta methods with two
 * sets of coefficients allowing to estimate the error, their Butcher
 * arrays are as follows :
 * <pre>
 *    0  |
 *   c2  | a21
 *   c3  | a31  a32
 *   ... |        ...
 *   cs  | as1  as2  ...  ass-1
 *       |--------------------------
 *       |  b1   b2  ...   bs-1  bs
 *       |  b'1  b'2 ...   b's-1 b's
 * </pre>
 * </p>
 *
 * <p>In fact, we rather use the array defined by ej = bj - b'j to
 * compute directly the error rather than computing two estimates and
 * then comparing them.</p>
 *
 * <p>Some methods are qualified as fsal (first same as last)
 * methods. This means the last evaluation of the derivatives in one
 * step is the same as the first in the next step. Then, this
 * evaluation can be reused from one step to the next one and the cost
 * of such a method is really s-1 evaluations despite the method still
 * has s stages. This behaviour is true only for successful steps, if
 * the step is rejected after the error estimation phase, no
 * evaluation is saved. For an <i>fsal method, we have cs = 1 and
 * asi = bi for all i.</p>
 *
 * @version $Revision: 927202 $ $Date: 2010-03-24 18:11:51 -0400 (Wed, 24 Mar 2010) $
 * @since 1.2
 */

public abstract class EmbeddedRungeKuttaIntegrator
  extends AdaptiveStepsizeIntegrator {

    /** Indicator for <i>fsal methods. */
    private final boolean fsal;

    /** Time steps from Butcher array (without the first zero). */
    private final double[] c;

    /** Internal weights from Butcher array (without the first empty row). */
    private final double[][] a;

    /** External weights for the high order method from Butcher array. */
    private final double[] b;

    /** Prototype of the step interpolator. */
    private final RungeKuttaStepInterpolator prototype;

    /** Stepsize control exponent. */
    private final 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 Runge-Kutta integrator with the given Butcher array.
   * @param name name of the method
   * @param fsal indicate that the method is an <i>fsal
   * @param c time steps from Butcher array (without the first zero)
   * @param a internal weights from Butcher array (without the first empty row)
   * @param b propagation weights for the high order method from Butcher array
   * @param prototype prototype of the step interpolator to use
   * @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
   */
  protected EmbeddedRungeKuttaIntegrator(final String name, final boolean fsal,
                                         final double[] c, final double[][] a, final double[] b,
                                         final RungeKuttaStepInterpolator prototype,
                                         final double minStep, final double maxStep,
                                         final double scalAbsoluteTolerance,
                                         final double scalRelativeTolerance) {

    super(name, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);

    this.fsal      = fsal;
    this.c         = c;
    this.a         = a;
    this.b         = b;
    this.prototype = prototype;

    exp = -1.0 / getOrder();

    // set the default values of the algorithm control parameters
    setSafety(0.9);
    setMinReduction(0.2);
    setMaxGrowth(10.0);

  }

  /** Build a Runge-Kutta integrator with the given Butcher array.
   * @param name name of the method
   * @param fsal indicate that the method is an <i>fsal
   * @param c time steps from Butcher array (without the first zero)
   * @param a internal weights from Butcher array (without the first empty row)
   * @param b propagation weights for the high order method from Butcher array
   * @param prototype prototype of the step interpolator to use
   * @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 EmbeddedRungeKuttaIntegrator(final String name, final boolean fsal,
                                         final double[] c, final double[][] a, final double[] b,
                                         final RungeKuttaStepInterpolator prototype,
                                         final double   minStep, final double maxStep,
                                         final double[] vecAbsoluteTolerance,
                                         final double[] vecRelativeTolerance) {

    super(name, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);

    this.fsal      = fsal;
    this.c         = c;
    this.a         = a;
    this.b         = b;
    this.prototype = prototype;

    exp = -1.0 / getOrder();

    // set the default values of the algorithm control parameters
    setSafety(0.9);
    setMinReduction(0.2);
    setMaxGrowth(10.0);

  }

  /** Get the order of the method.
   * @return order of the method
   */
  public abstract int getOrder();

  /** 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;
  }

  /** {@inheritDoc} */
  @Override
  public double integrate(final FirstOrderDifferentialEquations equations,
                          final double t0, final double[] y0,
                          final double t, final double[] y)
  throws DerivativeException, IntegratorException {

    sanityChecks(equations, t0, y0, t, y);
    setEquations(equations);
    resetEvaluations();
    final boolean forward = t > t0;

    // create some internal working arrays
    final int stages = c.length + 1;
    if (y != y0) {
      System.arraycopy(y0, 0, y, 0, y0.length);
    }
    final double[][] yDotK = new double[stages][y0.length];
    final double[] yTmp = new double[y0.length];

    // set up an interpolator sharing the integrator arrays
    AbstractStepInterpolator interpolator;
    if (requiresDenseOutput() || (! eventsHandlersManager.isEmpty())) {
      final RungeKuttaStepInterpolator rki = (RungeKuttaStepInterpolator) prototype.copy();
      rki.reinitialize(this, yTmp, yDotK, forward);
      interpolator = rki;
    } else {
      interpolator = new DummyStepInterpolator(yTmp, yDotK[stages - 1], forward);
    }
    interpolator.storeTime(t0);

    // set up integration control objects
    stepStart         = t0;
    double  hNew      = 0;
    boolean firstTime = true;
    for (StepHandler handler : stepHandlers) {
        handler.reset();
    }
    CombinedEventsManager manager = addEndTimeChecker(t0, t, eventsHandlersManager);
    boolean lastStep = false;

    // main integration loop
    while (!lastStep) {

      interpolator.shift();

      double error = 0;
      for (boolean loop = true; loop;) {

        if (firstTime || !fsal) {
          // first stage
          computeDerivatives(stepStart, y, yDotK[0]);
        }

        if (firstTime) {
          final double[] scale = new double[y0.length];
          if (vecAbsoluteTolerance == null) {
              for (int i = 0; i < scale.length; ++i) {
                scale[i] = scalAbsoluteTolerance + scalRelativeTolerance * Math.abs(y[i]);
              }
            } else {
              for (int i = 0; i < scale.length; ++i) {
                scale[i] = vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * Math.abs(y[i]);
              }
            }
          hNew = initializeStep(equations, forward, getOrder(), scale,
                                stepStart, y, yDotK[0], yTmp, yDotK[1]);
          firstTime = false;
        }

        stepSize = hNew;

        // next stages
        for (int k = 1; k < stages; ++k) {

          for (int j = 0; j < y0.length; ++j) {
            double sum = a[k-1][0] * yDotK[0][j];
            for (int l = 1; l < k; ++l) {
              sum += a[k-1][l] * yDotK[l][j];
            }
            yTmp[j] = y[j] + stepSize * sum;
          }

          computeDerivatives(stepStart + c[k-1] * stepSize, yTmp, yDotK[k]);

        }

        // estimate the state at the end of the step
        for (int j = 0; j < y0.length; ++j) {
          double sum    = b[0] * yDotK[0][j];
          for (int l = 1; l < stages; ++l) {
            sum    += b[l] * yDotK[l][j];
          }
          yTmp[j] = y[j] + stepSize * sum;
        }

        // estimate the error at the end of the step
        error = estimateError(yDotK, y, yTmp, stepSize);
        if (error <= 1.0) {

          // discrete events handling
          interpolator.storeTime(stepStart + stepSize);
          if (manager.evaluateStep(interpolator)) {
              final double dt = manager.getEventTime() - stepStart;
              if (Math.abs(dt) <= Math.ulp(stepStart)) {
                  // we cannot simply truncate the step, reject the current computation
                  // and let the loop compute another state with the truncated step.
                  // it is so small (much probably exactly 0 due to limited accuracy)
                  // that the code above would fail handling it.
                  // So we set up an artificial 0 size step by copying states
                  interpolator.storeTime(stepStart);
                  System.arraycopy(y, 0, yTmp, 0, y0.length);
                  hNew     = 0;
                  stepSize = 0;
                  loop     = false;
              } else {
                  // reject the step to match exactly the next switch time
                  hNew = dt;
              }
          } else {
            // accept the step
            loop = false;
          }

        } else {
          // reject the step and attempt to reduce error by stepsize control
          final double factor =
              Math.min(maxGrowth,
                       Math.max(minReduction, safety * Math.pow(error, exp)));
          hNew = filterStep(stepSize * factor, forward, false);
        }

      }

      // the step has been accepted
      final double nextStep = stepStart + stepSize;
      System.arraycopy(yTmp, 0, y, 0, y0.length);
      manager.stepAccepted(nextStep, y);
      lastStep = manager.stop();

      // provide the step data to the step handler
      interpolator.storeTime(nextStep);
      for (StepHandler handler : stepHandlers) {
          handler.handleStep(interpolator, lastStep);
      }
      stepStart = nextStep;

      if (fsal) {
        // save the last evaluation for the next step
        System.arraycopy(yDotK[stages - 1], 0, yDotK[0], 0, y0.length);
      }

      if (manager.reset(stepStart, y) && ! lastStep) {
        // some event handler has triggered changes that
        // invalidate the derivatives, we need to recompute them
        computeDerivatives(stepStart, y, yDotK[0]);
      }

      if (! lastStep) {
        // in some rare cases we may get here with stepSize = 0, for example
        // when an event occurs at integration start, reducing the first step
        // to zero; we have to reset the step to some safe non zero value
          stepSize = filterStep(stepSize, forward, true);

        // stepsize control for next step
        final double factor = Math.min(maxGrowth,
                                       Math.max(minReduction,
                                                safety * Math.pow(error, exp)));
        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 stopTime = stepStart;
    resetInternalState();
    return stopTime;

  }

  /** 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;
  }

  /** Compute the error ratio.
   * @param yDotK derivatives computed during the first stages
   * @param y0 estimate of the step at the start of the step
   * @param y1 estimate of the step at the end of the step
   * @param h  current step
   * @return error ratio, greater than 1 if step should be rejected
   */
  protected abstract double estimateError(double[][] yDotK,
                                          double[] y0, double[] y1,
                                          double h);

}

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