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 * 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
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package org.apache.commons.math3.ode.nonstiff;

import org.apache.commons.math3.Field;
import org.apache.commons.math3.RealFieldElement;
import org.apache.commons.math3.ode.FieldEquationsMapper;
import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;


This class implements the 5(4) Dormand-Prince 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. However, since this
is an fsal, the last evaluation of one step is the same as
the first evaluation of the next step and hence can be avoided. So
the cost is really 6 functions evaluations per step.
This method has been published (whithout the continuous output
that was added by Shampine in 1986) in the following article :
 A family of embedded Runge-Kutta formulae
 J. R. Dormand and P. J. Prince
 Journal of Computational and Applied Mathematics
 volume 6, no 1, 1980, pp. 19-26

Type parameters:
<T> – the type of the field elements
Since:
3.6/**
 * This class implements the 5(4) Dormand-Prince 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. However, since this
 * is an <i>fsal</i>, the last evaluation of one step is the same as
 * the first evaluation of the next step and hence can be avoided. So
 * the cost is really 6 functions evaluations per step.</p>
 *
 * <p>This method has been published (whithout the continuous output
 * that was added by Shampine in 1986) in the following article :
 * <pre>
 *  A family of embedded Runge-Kutta formulae
 *  J. R. Dormand and P. J. Prince
 *  Journal of Computational and Applied Mathematics
 *  volume 6, no 1, 1980, pp. 19-26
 * </pre></p>
 *
 * @param <T> the type of the field elements
 * @since 3.6
 */

public class DormandPrince54FieldIntegrator<T extends RealFieldElement<T>>
    extends EmbeddedRungeKuttaFieldIntegrator<T> {

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

    
Error array, element 1. /** Error array, element 1. */
    private final T e1;

    // element 2 is zero, so it is neither stored nor used

    
Error array, element 3. /** Error array, element 3. */
    private final T e3;

    
Error array, element 4. /** Error array, element 4. */
    private final T e4;

    
Error array, element 5. /** Error array, element 5. */
    private final T e5;

    
Error array, element 6. /** Error array, element 6. */
    private final T e6;

    
Error array, element 7. /** Error array, element 7. */
    private final T e7;

    
Simple constructor.
Build a fifth order Dormand-Prince integrator with the given step bounds
Params:
field – field to which the time and state vector elements belong
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 Dormand-Prince integrator with the given step bounds
     * @param field field to which the time and state vector elements belong
     * @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 DormandPrince54FieldIntegrator(final Field<T> field,
                                          final double minStep, final double maxStep,
                                          final double scalAbsoluteTolerance,
                                          final double scalRelativeTolerance) {
        super(field, METHOD_NAME, 6,
              minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
        e1 = fraction(    71,  57600);
        e3 = fraction(   -71,  16695);
        e4 = fraction(    71,   1920);
        e5 = fraction(-17253, 339200);
        e6 = fraction(    22,    525);
        e7 = fraction(    -1,     40);
    }

    
Simple constructor.
Build a fifth order Dormand-Prince integrator with the given step bounds
Params:
field – field to which the time and state vector elements belong
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 Dormand-Prince integrator with the given step bounds
     * @param field field to which the time and state vector elements belong
     * @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 DormandPrince54FieldIntegrator(final Field<T> field,
                                          final double minStep, final double maxStep,
                                          final double[] vecAbsoluteTolerance,
                                          final double[] vecRelativeTolerance) {
        super(field, METHOD_NAME, 6,
              minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
        e1 = fraction(    71,  57600);
        e3 = fraction(   -71,  16695);
        e4 = fraction(    71,   1920);
        e5 = fraction(-17253, 339200);
        e6 = fraction(    22,    525);
        e7 = fraction(    -1,     40);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public T[] getC() {
        final T[] c = MathArrays.buildArray(getField(), 6);
        c[0] = fraction(1,  5);
        c[1] = fraction(3, 10);
        c[2] = fraction(4,  5);
        c[3] = fraction(8,  9);
        c[4] = getField().getOne();
        c[5] = getField().getOne();
        return c;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public T[][] getA() {
        final T[][] a = MathArrays.buildArray(getField(), 6, -1);
        for (int i = 0; i < a.length; ++i) {
            a[i] = MathArrays.buildArray(getField(), i + 1);
        }
        a[0][0] = fraction(     1,     5);
        a[1][0] = fraction(     3,    40);
        a[1][1] = fraction(     9,    40);
        a[2][0] = fraction(    44,    45);
        a[2][1] = fraction(   -56,    15);
        a[2][2] = fraction(    32,     9);
        a[3][0] = fraction( 19372,  6561);
        a[3][1] = fraction(-25360,  2187);
        a[3][2] = fraction( 64448,  6561);
        a[3][3] = fraction(  -212,   729);
        a[4][0] = fraction(  9017,  3168);
        a[4][1] = fraction(  -355,    33);
        a[4][2] = fraction( 46732,  5247);
        a[4][3] = fraction(    49,   176);
        a[4][4] = fraction( -5103, 18656);
        a[5][0] = fraction(    35,   384);
        a[5][1] = getField().getZero();
        a[5][2] = fraction(   500,  1113);
        a[5][3] = fraction(   125,   192);
        a[5][4] = fraction( -2187,  6784);
        a[5][5] = fraction(    11,    84);
        return a;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public T[] getB() {
        final T[] b = MathArrays.buildArray(getField(), 7);
        b[0] = fraction(   35,   384);
        b[1] = getField().getZero();
        b[2] = fraction(  500, 1113);
        b[3] = fraction(  125,  192);
        b[4] = fraction(-2187, 6784);
        b[5] = fraction(   11,   84);
        b[6] = getField().getZero();
        return b;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    protected DormandPrince54FieldStepInterpolator<T>
        createInterpolator(final boolean forward, T[][] yDotK,
                           final FieldODEStateAndDerivative<T> globalPreviousState,
                           final FieldODEStateAndDerivative<T> globalCurrentState, final FieldEquationsMapper<T> mapper) {
        return new DormandPrince54FieldStepInterpolator<T>(getField(), forward, yDotK,
                                                           globalPreviousState, globalCurrentState,
                                                           globalPreviousState, globalCurrentState,
                                                           mapper);
    }

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

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

        T error = getField().getZero();

        for (int j = 0; j < mainSetDimension; ++j) {
            final T errSum =     yDotK[0][j].multiply(e1).
                             add(yDotK[2][j].multiply(e3)).
                             add(yDotK[3][j].multiply(e4)).
                             add(yDotK[4][j].multiply(e5)).
                             add(yDotK[5][j].multiply(e6)).
                             add(yDotK[6][j].multiply(e7));

            final T yScale = MathUtils.max(y0[j].abs(), y1[j].abs());
            final T tol    = (vecAbsoluteTolerance == null) ?
                             yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) :
                             yScale.multiply(vecRelativeTolerance[j]).add(vecAbsoluteTolerance[j]);
            final T ratio  = h.multiply(errSum).divide(tol);
            error = error.add(ratio.multiply(ratio));

        }

        return error.divide(mainSetDimension).sqrt();

    }

}