/*
 * 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.math3.ode.nonstiff;

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


This class implements the 8(5,3) Dormand-Prince integrator for Ordinary
Differential Equations.
This integrator is an embedded Runge-Kutta integrator
of order 8(5,3) 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 12 functions evaluations per step for integration and 4
evaluations for interpolation. However, since the first
interpolation evaluation is the same as the first integration
evaluation of the next step, we have included it in the integrator
rather than in the interpolator and specified the method was an
fsal. Hence, despite we have 13 stages here, the cost is
really 12 evaluations per step even if no interpolation is done,
and the overcost of interpolation is only 3 evaluations.
This method is based on an 8(6) method by Dormand and Prince
(i.e. order 8 for the integration and order 6 for error estimation)
modified by Hairer and Wanner to use a 5th order error estimator
with 3rd order correction. This modification was introduced because
the original method failed in some cases (wrong steps can be
accepted when step size is too large, for example in the
Brusselator problem) and also had severe difficulties when
applied to problems with discontinuities. This modification is
explained in the second edition of the first volume (Nonstiff
Problems) of the reference book by Hairer, Norsett and Wanner:
Solving Ordinary Differential Equations (Springer-Verlag,
ISBN 3-540-56670-8).
Since:
1.2/**
 * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary
 * Differential Equations.
 *
 * <p>This integrator is an embedded Runge-Kutta integrator
 * of order 8(5,3) 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 12 functions evaluations per step for integration and 4
 * evaluations for interpolation. However, since the first
 * interpolation evaluation is the same as the first integration
 * evaluation of the next step, we have included it in the integrator
 * rather than in the interpolator and specified the method was an
 * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is
 * really 12 evaluations per step even if no interpolation is done,
 * and the overcost of interpolation is only 3 evaluations.</p>
 *
 * <p>This method is based on an 8(6) method by Dormand and Prince
 * (i.e. order 8 for the integration and order 6 for error estimation)
 * modified by Hairer and Wanner to use a 5th order error estimator
 * with 3rd order correction. This modification was introduced because
 * the original method failed in some cases (wrong steps can be
 * accepted when step size is too large, for example in the
 * Brusselator problem) and also had <i>severe difficulties when
 * applied to problems with discontinuities</i>. This modification is
 * explained in the second edition of the first volume (Nonstiff
 * Problems) of the reference book by Hairer, Norsett and Wanner:
 * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag,
 * ISBN 3-540-56670-8).</p>
 *
 * @since 1.2
 */

public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {

  
Integrator method name. /** Integrator method name. */
  private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";

  
Time steps Butcher array. /** Time steps Butcher array. */
  private static final double[] STATIC_C = {
    (12.0 - 2.0 * FastMath.sqrt(6.0)) / 135.0, (6.0 - FastMath.sqrt(6.0)) / 45.0, (6.0 - FastMath.sqrt(6.0)) / 30.0,
    (6.0 + FastMath.sqrt(6.0)) / 30.0, 1.0/3.0, 1.0/4.0, 4.0/13.0, 127.0/195.0, 3.0/5.0,
    6.0/7.0, 1.0, 1.0
  };

  
Internal weights Butcher array. /** Internal weights Butcher array. */
  private static final double[][] STATIC_A = {

    // k2
    {(12.0 - 2.0 * FastMath.sqrt(6.0)) / 135.0},

    // k3
    {(6.0 - FastMath.sqrt(6.0)) / 180.0, (6.0 - FastMath.sqrt(6.0)) / 60.0},

    // k4
    {(6.0 - FastMath.sqrt(6.0)) / 120.0, 0.0, (6.0 - FastMath.sqrt(6.0)) / 40.0},

    // k5
    {(462.0 + 107.0 * FastMath.sqrt(6.0)) / 3000.0, 0.0,
     (-402.0 - 197.0 * FastMath.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * FastMath.sqrt(6.0)) / 375.0},

    // k6
    {1.0 / 27.0, 0.0, 0.0, (16.0 + FastMath.sqrt(6.0)) / 108.0, (16.0 - FastMath.sqrt(6.0)) / 108.0},

    // k7
    {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * FastMath.sqrt(6.0)) / 1024.0,
     (118.0 - 23.0 * FastMath.sqrt(6.0)) / 1024.0, -9.0 / 512.0},

    // k8
    {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * FastMath.sqrt(6.0)) / 371293.0,
     (51544.0 - 4784.0 * FastMath.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},

    // k9
    {58656157643.0 / 93983540625.0, 0.0, 0.0,
     (-1324889724104.0 - 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0,
     (-1324889724104.0 + 318801444819.0 * FastMath.sqrt(6.0)) / 626556937500.0,
     96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0,
     -165125654.0 / 3796875.0},

    // k10
    {8909899.0 / 18653125.0, 0.0, 0.0,
     (-4521408.0 - 1137963.0 * FastMath.sqrt(6.0)) / 2937500.0,
     (-4521408.0 + 1137963.0 * FastMath.sqrt(6.0)) / 2937500.0,
     96663078.0 / 4553125.0, 2107245056.0 / 137915625.0,
     -4913652016.0 / 147609375.0, -78894270.0 / 3880452869.0},

    // k11
    {-20401265806.0 / 21769653311.0, 0.0, 0.0,
     (354216.0 + 94326.0 * FastMath.sqrt(6.0)) / 112847.0,
     (354216.0 - 94326.0 * FastMath.sqrt(6.0)) / 112847.0,
     -43306765128.0 / 5313852383.0, -20866708358144.0 / 1126708119789.0,
     14886003438020.0 / 654632330667.0, 35290686222309375.0 / 14152473387134411.0,
     -1477884375.0 / 485066827.0},

    // k12
    {39815761.0 / 17514443.0, 0.0, 0.0,
     (-3457480.0 - 960905.0 * FastMath.sqrt(6.0)) / 551636.0,
     (-3457480.0 + 960905.0 * FastMath.sqrt(6.0)) / 551636.0,
     -844554132.0 / 47026969.0, 8444996352.0 / 302158619.0,
     -2509602342.0 / 877790785.0, -28388795297996250.0 / 3199510091356783.0,
     226716250.0 / 18341897.0, 1371316744.0 / 2131383595.0},

    // k13 should be for interpolation only, but since it is the same
    // stage as the first evaluation of the next step, we perform it
    // here at no cost by specifying this is an fsal method
    {104257.0/1920240.0, 0.0, 0.0, 0.0, 0.0, 3399327.0/763840.0,
     66578432.0/35198415.0, -1674902723.0/288716400.0,
     54980371265625.0/176692375811392.0, -734375.0/4826304.0,
     171414593.0/851261400.0, 137909.0/3084480.0}

  };

  
Propagation weights Butcher array. /** Propagation weights Butcher array. */
  private static final double[] STATIC_B = {
      104257.0/1920240.0,
      0.0,
      0.0,
      0.0,
      0.0,
      3399327.0/763840.0,
      66578432.0/35198415.0,
      -1674902723.0/288716400.0,
      54980371265625.0/176692375811392.0,
      -734375.0/4826304.0,
      171414593.0/851261400.0,
      137909.0/3084480.0,
      0.0
  };

  
First error weights array, element 1. /** First error weights array, element 1. */
  private static final double E1_01 =         116092271.0 / 8848465920.0;

  // elements 2 to 5 are zero, so they are neither stored nor used

  
First error weights array, element 6. /** First error weights array, element 6. */
  private static final double E1_06 =          -1871647.0 / 1527680.0;

  
First error weights array, element 7. /** First error weights array, element 7. */
  private static final double E1_07 =         -69799717.0 / 140793660.0;

  
First error weights array, element 8. /** First error weights array, element 8. */
  private static final double E1_08 =     1230164450203.0 / 739113984000.0;

  
First error weights array, element 9. /** First error weights array, element 9. */
  private static final double E1_09 = -1980813971228885.0 / 5654156025964544.0;

  
First error weights array, element 10. /** First error weights array, element 10. */
  private static final double E1_10 =         464500805.0 / 1389975552.0;

  
First error weights array, element 11. /** First error weights array, element 11. */
  private static final double E1_11 =     1606764981773.0 / 19613062656000.0;

  
First error weights array, element 12. /** First error weights array, element 12. */
  private static final double E1_12 =           -137909.0 / 6168960.0;


  
Second error weights array, element 1. /** Second error weights array, element 1. */
  private static final double E2_01 =           -364463.0 / 1920240.0;

  // elements 2 to 5 are zero, so they are neither stored nor used

  
Second error weights array, element 6. /** Second error weights array, element 6. */
  private static final double E2_06 =           3399327.0 / 763840.0;

  
Second error weights array, element 7. /** Second error weights array, element 7. */
  private static final double E2_07 =          66578432.0 / 35198415.0;

  
Second error weights array, element 8. /** Second error weights array, element 8. */
  private static final double E2_08 =       -1674902723.0 / 288716400.0;

  
Second error weights array, element 9. /** Second error weights array, element 9. */
  private static final double E2_09 =   -74684743568175.0 / 176692375811392.0;

  
Second error weights array, element 10. /** Second error weights array, element 10. */
  private static final double E2_10 =           -734375.0 / 4826304.0;

  
Second error weights array, element 11. /** Second error weights array, element 11. */
  private static final double E2_11 =         171414593.0 / 851261400.0;

  
Second error weights array, element 12. /** Second error weights array, element 12. */
  private static final double E2_12 =             69869.0 / 3084480.0;

  
Simple constructor.
Build an eighth order Dormand-Prince 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 an eighth order Dormand-Prince 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 DormandPrince853Integrator(final double minStep, final double maxStep,
                                    final double scalAbsoluteTolerance,
                                    final double scalRelativeTolerance) {
    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
          new DormandPrince853StepInterpolator(),
          minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
  }

  
Simple constructor.
Build an eighth order Dormand-Prince 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 an eighth order Dormand-Prince 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 DormandPrince853Integrator(final double minStep, final double maxStep,
                                    final double[] vecAbsoluteTolerance,
                                    final double[] vecRelativeTolerance) {
    super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B,
          new DormandPrince853StepInterpolator(),
          minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
  }

  
{@inheritDoc} /** {@inheritDoc} */
  @Override
  public int getOrder() {
    return 8;
  }

  
{@inheritDoc} /** {@inheritDoc} */
  @Override
  protected double estimateError(final double[][] yDotK,
                                 final double[] y0, final double[] y1,
                                 final double h) {
    double error1 = 0;
    double error2 = 0;

    for (int j = 0; j < mainSetDimension; ++j) {
      final double errSum1 = E1_01 * yDotK[0][j]  + E1_06 * yDotK[5][j] +
                             E1_07 * yDotK[6][j]  + E1_08 * yDotK[7][j] +
                             E1_09 * yDotK[8][j]  + E1_10 * yDotK[9][j] +
                             E1_11 * yDotK[10][j] + E1_12 * yDotK[11][j];
      final double errSum2 = E2_01 * yDotK[0][j]  + E2_06 * yDotK[5][j] +
                             E2_07 * yDotK[6][j]  + E2_08 * yDotK[7][j] +
                             E2_09 * yDotK[8][j]  + E2_10 * yDotK[9][j] +
                             E2_11 * yDotK[10][j] + E2_12 * yDotK[11][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 ratio1  = errSum1 / tol;
      error1        += ratio1 * ratio1;
      final double ratio2  = errSum2 / tol;
      error2        += ratio2 * ratio2;
    }

    double den = error1 + 0.01 * error2;
    if (den <= 0.0) {
      den = 1.0;
    }

    return FastMath.abs(h) * error1 / FastMath.sqrt(mainSetDimension * den);

  }

}