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 * 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
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 *      http://www.apache.org/licenses/LICENSE-2.0
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 * See the License for the specific language governing permissions and
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package org.apache.commons.math3.ode.nonstiff;

import org.apache.commons.math3.util.FastMath;


This class implements the 5(4) Higham and Hall integrator for
Ordinary Differential Equations.
This integrator is an embedded Runge-Kutta integrator
of order 5(4) used in local extrapolation mode (i.e. the solution
is computed using the high order formula) with stepsize control
(and automatic step initialization) and continuous output. This
method uses 7 functions evaluations per step.
Since:
1.2/**
 * This class implements the 5(4) Higham and Hall integrator for
 * Ordinary Differential Equations.
 *
 * <p>This integrator is an embedded Runge-Kutta integrator
 * of order 5(4) used in local extrapolation mode (i.e. the solution
 * is computed using the high order formula) with stepsize control
 * (and automatic step initialization) and continuous output. This
 * method uses 7 functions evaluations per step.</p>
 *
 * @since 1.2
 */

public class HighamHall54Integrator extends EmbeddedRungeKuttaIntegrator {

  
Integrator method name. /** Integrator method name. */
  private static final String METHOD_NAME = "Higham-Hall 5(4)";

  
Time steps Butcher array. /** Time steps Butcher array. */
  private static final double[] STATIC_C = {
    2.0/9.0, 1.0/3.0, 1.0/2.0, 3.0/5.0, 1.0, 1.0
  };

  
Internal weights Butcher array. /** Internal weights Butcher array. */
  private static final double[][] STATIC_A = {
    {2.0/9.0},
    {1.0/12.0, 1.0/4.0},
    {1.0/8.0, 0.0, 3.0/8.0},
    {91.0/500.0, -27.0/100.0, 78.0/125.0, 8.0/125.0},
    {-11.0/20.0, 27.0/20.0, 12.0/5.0, -36.0/5.0, 5.0},
    {1.0/12.0, 0.0, 27.0/32.0, -4.0/3.0, 125.0/96.0, 5.0/48.0}
  };

  
Propagation weights Butcher array. /** Propagation weights Butcher array. */
  private static final double[] STATIC_B = {
    1.0/12.0, 0.0, 27.0/32.0, -4.0/3.0, 125.0/96.0, 5.0/48.0, 0.0
  };

  
Error weights Butcher array. /** Error weights Butcher array. */
  private static final double[] STATIC_E = {
    -1.0/20.0, 0.0, 81.0/160.0, -6.0/5.0, 25.0/32.0, 1.0/16.0, -1.0/10.0
  };

  
Simple constructor.
Build a fifth order Higham and Hall integrator with the given step bounds
Params:
minStep – minimal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this
maxStep – maximal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this
scalAbsoluteTolerance – allowed absolute error
scalRelativeTolerance – allowed relative error/** Simple constructor.
   * Build a fifth order Higham and Hall integrator with the given step bounds
   * @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
   */
  public HighamHall54Integrator(final double minStep, final double maxStep,
                                final double scalAbsoluteTolerance,
                                final double scalRelativeTolerance) {
    super(METHOD_NAME, false, STATIC_C, STATIC_A, STATIC_B, new HighamHall54StepInterpolator(),
          minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
  }

  
Simple constructor.
Build a fifth order Higham and Hall integrator with the given step bounds
Params:
minStep – minimal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this
maxStep – maximal step (sign is irrelevant, regardless of
integration direction, forward or backward), the last step can
be smaller than this
vecAbsoluteTolerance – allowed absolute error
vecRelativeTolerance – allowed relative error/** Simple constructor.
   * Build a fifth order Higham and Hall integrator with the given step bounds
   * @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
   */
  public HighamHall54Integrator(final double minStep, final double maxStep,
                                final double[] vecAbsoluteTolerance,
                                final double[] vecRelativeTolerance) {
    super(METHOD_NAME, false, STATIC_C, STATIC_A, STATIC_B, new HighamHall54StepInterpolator(),
          minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
  }

  
{@inheritDoc} /** {@inheritDoc} */
  @Override
  public int getOrder() {
    return 5;
  }

  
{@inheritDoc} /** {@inheritDoc} */
  @Override
  protected double estimateError(final double[][] yDotK,
                                 final double[] y0, final double[] y1,
                                 final double h) {

    double error = 0;

    for (int j = 0; j < mainSetDimension; ++j) {
      double errSum = STATIC_E[0] * yDotK[0][j];
      for (int l = 1; l < STATIC_E.length; ++l) {
        errSum += STATIC_E[l] * yDotK[l][j];
      }

      final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j]));
      final double tol = (vecAbsoluteTolerance == null) ?
                         (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
                         (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
      final double ratio  = h * errSum / tol;
      error += ratio * ratio;

    }

    return FastMath.sqrt(error / mainSetDimension);

  }

}