/*
 * 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.
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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;

This class represents an interpolator over the last step during an
ODE integration for the 6th order Luther integrator.
This interpolator computes dense output inside the last
step computed. The interpolation equation is consistent with the
integration scheme.
Type parameters:
<T> – the type of the field elements
See Also:
LutherFieldIntegrator
Since:
3.6/**
 * This class represents an interpolator over the last step during an
 * ODE integration for the 6th order Luther integrator.
 *
 * <p>This interpolator computes dense output inside the last
 * step computed. The interpolation equation is consistent with the
 * integration scheme.</p>
 *
 * @see LutherFieldIntegrator
 * @param <T> the type of the field elements
 * @since 3.6
 */

class LutherFieldStepInterpolator<T extends RealFieldElement<T>>
    extends RungeKuttaFieldStepInterpolator<T> {

    
-49 - 49 q. /** -49 - 49 q. */
    private final T c5a;

    
392 + 287 q. /** 392 + 287 q. */
    private final T c5b;

    
-637 - 357 q. /** -637 - 357 q. */
    private final T c5c;

    
833 + 343 q. /** 833 + 343 q. */
    private final T c5d;

    
-49 + 49 q. /** -49 + 49 q. */
    private final T c6a;

    
-392 - 287 q. /** -392 - 287 q. */
    private final T c6b;

    
-637 + 357 q. /** -637 + 357 q. */
    private final T c6c;

    
833 - 343 q. /** 833 - 343 q. */
    private final T c6d;

    
49 + 49 q. /** 49 + 49 q. */
    private final T d5a;

    
-1372 - 847 q. /** -1372 - 847 q. */
    private final T d5b;

    
2254 + 1029 q /** 2254 + 1029 q */
    private final T d5c;

    
49 - 49 q. /** 49 - 49 q. */
    private final T d6a;

    
-1372 + 847 q. /** -1372 + 847 q. */
    private final T d6b;

    
2254 - 1029 q /** 2254 - 1029 q */
    private final T d6c;

    
Simple constructor.
Params:
field – field to which the time and state vector elements belong
forward – integration direction indicator
yDotK – slopes at the intermediate points
globalPreviousState – start of the global step
globalCurrentState – end of the global step
softPreviousState – start of the restricted step
softCurrentState – end of the restricted step
mapper – equations mapper for the all equations/** Simple constructor.
     * @param field field to which the time and state vector elements belong
     * @param forward integration direction indicator
     * @param yDotK slopes at the intermediate points
     * @param globalPreviousState start of the global step
     * @param globalCurrentState end of the global step
     * @param softPreviousState start of the restricted step
     * @param softCurrentState end of the restricted step
     * @param mapper equations mapper for the all equations
     */
    LutherFieldStepInterpolator(final Field<T> field, final boolean forward,
                                final T[][] yDotK,
                                final FieldODEStateAndDerivative<T> globalPreviousState,
                                final FieldODEStateAndDerivative<T> globalCurrentState,
                                final FieldODEStateAndDerivative<T> softPreviousState,
                                final FieldODEStateAndDerivative<T> softCurrentState,
                                final FieldEquationsMapper<T> mapper) {
        super(field, forward, yDotK,
              globalPreviousState, globalCurrentState, softPreviousState, softCurrentState,
              mapper);
        final T q = field.getZero().add(21).sqrt();
        c5a = q.multiply(  -49).add(  -49);
        c5b = q.multiply(  287).add(  392);
        c5c = q.multiply( -357).add( -637);
        c5d = q.multiply(  343).add(  833);
        c6a = q.multiply(   49).add(  -49);
        c6b = q.multiply( -287).add(  392);
        c6c = q.multiply(  357).add( -637);
        c6d = q.multiply( -343).add(  833);
        d5a = q.multiply(   49).add(   49);
        d5b = q.multiply( -847).add(-1372);
        d5c = q.multiply( 1029).add( 2254);
        d6a = q.multiply(  -49).add(   49);
        d6b = q.multiply(  847).add(-1372);
        d6c = q.multiply(-1029).add( 2254);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    protected LutherFieldStepInterpolator<T> create(final Field<T> newField, final boolean newForward, final T[][] newYDotK,
                                                    final FieldODEStateAndDerivative<T> newGlobalPreviousState,
                                                    final FieldODEStateAndDerivative<T> newGlobalCurrentState,
                                                    final FieldODEStateAndDerivative<T> newSoftPreviousState,
                                                    final FieldODEStateAndDerivative<T> newSoftCurrentState,
                                                    final FieldEquationsMapper<T> newMapper) {
        return new LutherFieldStepInterpolator<T>(newField, newForward, newYDotK,
                                                  newGlobalPreviousState, newGlobalCurrentState,
                                                  newSoftPreviousState, newSoftCurrentState,
                                                  newMapper);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @SuppressWarnings("unchecked")
    @Override
    protected FieldODEStateAndDerivative<T> computeInterpolatedStateAndDerivatives(final FieldEquationsMapper<T> mapper,
                                                                                   final T time, final T theta,
                                                                                   final T thetaH, final T oneMinusThetaH) {

        // the coefficients below have been computed by solving the
        // order conditions from a theorem from Butcher (1963), using
        // the method explained in Folkmar Bornemann paper "Runge-Kutta
        // Methods, Trees, and Maple", Center of Mathematical Sciences, Munich
        // University of Technology, February 9, 2001
        //<http://wwwzenger.informatik.tu-muenchen.de/selcuk/sjam012101.html>

        // the method is implemented in the rkcheck tool
        // <https://www.spaceroots.org/software/rkcheck/index.html>.
        // Running it for order 5 gives the following order conditions
        // for an interpolator:
        // order 1 conditions
        // \sum_{i=1}^{i=s}\left(b_{i} \right) =1
        // order 2 conditions
        // \sum_{i=1}^{i=s}\left(b_{i} c_{i}\right) = \frac{\theta}{2}
        // order 3 conditions
        // \sum_{i=2}^{i=s}\left(b_{i} \sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)}\right) = \frac{\theta^{2}}{6}
        // \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{2}\right) = \frac{\theta^{2}}{3}
        // order 4 conditions
        // \sum_{i=3}^{i=s}\left(b_{i} \sum_{j=2}^{j=i-1}{\left(a_{i,j} \sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k} \right)} \right)}\right) = \frac{\theta^{3}}{24}
        // \sum_{i=2}^{i=s}\left(b_{i} \sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j}^{2} \right)}\right) = \frac{\theta^{3}}{12}
        // \sum_{i=2}^{i=s}\left(b_{i} c_{i}\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)}\right) = \frac{\theta^{3}}{8}
        // \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{3}\right) = \frac{\theta^{3}}{4}
        // order 5 conditions
        // \sum_{i=4}^{i=s}\left(b_{i} \sum_{j=3}^{j=i-1}{\left(a_{i,j} \sum_{k=2}^{k=j-1}{\left(a_{j,k} \sum_{l=1}^{l=k-1}{\left(a_{k,l} c_{l} \right)} \right)} \right)}\right) = \frac{\theta^{4}}{120}
        // \sum_{i=3}^{i=s}\left(b_{i} \sum_{j=2}^{j=i-1}{\left(a_{i,j} \sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k}^{2} \right)} \right)}\right) = \frac{\theta^{4}}{60}
        // \sum_{i=3}^{i=s}\left(b_{i} \sum_{j=2}^{j=i-1}{\left(a_{i,j} c_{j}\sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k} \right)} \right)}\right) = \frac{\theta^{4}}{40}
        // \sum_{i=2}^{i=s}\left(b_{i} \sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j}^{3} \right)}\right) = \frac{\theta^{4}}{20}
        // \sum_{i=3}^{i=s}\left(b_{i} c_{i}\sum_{j=2}^{j=i-1}{\left(a_{i,j} \sum_{k=1}^{k=j-1}{\left(a_{j,k} c_{k} \right)} \right)}\right) = \frac{\theta^{4}}{30}
        // \sum_{i=2}^{i=s}\left(b_{i} c_{i}\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j}^{2} \right)}\right) = \frac{\theta^{4}}{15}
        // \sum_{i=2}^{i=s}\left(b_{i} \left(\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)} \right)^{2}\right) = \frac{\theta^{4}}{20}
        // \sum_{i=2}^{i=s}\left(b_{i} c_{i}^{2}\sum_{j=1}^{j=i-1}{\left(a_{i,j} c_{j} \right)}\right) = \frac{\theta^{4}}{10}
        // \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{4}\right) = \frac{\theta^{4}}{5}

        // The a_{j,k} and c_{k} are given by the integrator Butcher arrays. What remains to solve
        // are the b_i for the interpolator. They are found by solving the above equations.
        // For a given interpolator, some equations are redundant, so in our case when we select
        // all equations from order 1 to 4, we still don't have enough independent equations
        // to solve from b_1 to b_7. We need to also select one equation from order 5. Here,
        // we selected the last equation. It appears this choice implied at least the last 3 equations
        // are fulfilled, but some of the former ones are not, so the resulting interpolator is order 5.
        // At the end, we get the b_i as polynomials in theta.

        final T coeffDot1 =  theta.multiply(theta.multiply(theta.multiply(theta.multiply(   21        ).add( -47          )).add(   36         )).add( -54     /   5.0)).add(1);
        final T coeffDot2 =  time.getField().getZero();
        final T coeffDot3 =  theta.multiply(theta.multiply(theta.multiply(theta.multiply(  112        ).add(-608    /  3.0)).add(  320   / 3.0 )).add(-208    /  15.0));
        final T coeffDot4 =  theta.multiply(theta.multiply(theta.multiply(theta.multiply( -567  /  5.0).add( 972    /  5.0)).add( -486   / 5.0 )).add( 324    /  25.0));
        final T coeffDot5 =  theta.multiply(theta.multiply(theta.multiply(theta.multiply(c5a.divide(5)).add(c5b.divide(15))).add(c5c.divide(30))).add(c5d.divide(150)));
        final T coeffDot6 =  theta.multiply(theta.multiply(theta.multiply(theta.multiply(c6a.divide(5)).add(c6b.divide(15))).add(c6c.divide(30))).add(c6d.divide(150)));
        final T coeffDot7 =  theta.multiply(theta.multiply(theta.multiply(                                             3.0 ).add(   -3         )).add(   3   /   5.0));
        final T[] interpolatedState;
        final T[] interpolatedDerivatives;

        if (getGlobalPreviousState() != null && theta.getReal() <= 0.5) {

            final T s         = thetaH;
            final T coeff1    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(  21    /  5.0).add( -47    /  4.0)).add(   12         )).add( -27    /   5.0)).add(1));
            final T coeff2    = time.getField().getZero();
            final T coeff3    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply( 112    /  5.0).add(-152    /  3.0)).add(  320   / 9.0 )).add(-104    /  15.0)));
            final T coeff4    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(-567    / 25.0).add( 243    /  5.0)).add( -162   / 5.0 )).add( 162    /  25.0)));
            final T coeff5    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(c5a.divide(25)).add(c5b.divide(60))).add(c5c.divide(90))).add(c5d.divide(300))));
            final T coeff6    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(c6a.divide(25)).add(c6b.divide(60))).add(c6c.divide(90))).add(c6d.divide(300))));
            final T coeff7    = s.multiply(theta.multiply(theta.multiply(theta.multiply(                                      3    /  4.0 ).add(   -1         )).add(   3    /  10.0)));
            interpolatedState       = previousStateLinearCombination(coeff1, coeff2, coeff3, coeff4, coeff5, coeff6, coeff7);
            interpolatedDerivatives = derivativeLinearCombination(coeffDot1, coeffDot2, coeffDot3, coeffDot4, coeffDot5, coeffDot6, coeffDot7);
        } else {

            final T s         = oneMinusThetaH;
            final T coeff1    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply( -21   /   5.0).add(   151  /  20.0)).add( -89   /   20.0)).add(  19 /  20.0)).add(- 1 / 20.0));
            final T coeff2    = time.getField().getZero();
            final T coeff3    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(-112   /   5.0).add(   424  /  15.0)).add( -328  /   45.0)).add( -16 /  45.0)).add(-16 /  45.0));
            final T coeff4    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply( 567   /  25.0).add(  -648  /  25.0)).add(  162  /   25.0))));
            final T coeff5    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(d5a.divide(25)).add(d5b.divide(300))).add(d5c.divide(900))).add( -49 / 180.0)).add(-49 / 180.0));
            final T coeff6    = s.multiply(theta.multiply(theta.multiply(theta.multiply(theta.multiply(d6a.divide(25)).add(d6b.divide(300))).add(d6c.divide(900))).add( -49 / 180.0)).add(-49 / 180.0));
            final T coeff7    = s.multiply(               theta.multiply(theta.multiply(theta.multiply(                        -3  /   4.0 ).add(   1   /    4.0)).add(  -1 /  20.0)).add( -1 /  20.0));
            interpolatedState       = currentStateLinearCombination(coeff1, coeff2, coeff3, coeff4, coeff5, coeff6, coeff7);
            interpolatedDerivatives = derivativeLinearCombination(coeffDot1, coeffDot2, coeffDot3, coeffDot4, coeffDot5, coeffDot6, coeffDot7);
        }

        return new FieldODEStateAndDerivative<T>(time, interpolatedState, interpolatedDerivatives);

    }

}