Field-based version of Luther method for solving ODE.
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Luc Maisonobe
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math4.ode.nonstiff;
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import org.apache.commons.math4.Field;
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import org.apache.commons.math4.RealFieldElement;
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import org.apache.commons.math4.util.FastMath;
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/**
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* This class implements the Luther sixth order Runge-Kutta
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* integrator for Ordinary Differential Equations.
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* <p>
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* This method is described in H. A. Luther 1968 paper <a
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* href="http://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99876-1/S0025-5718-68-99876-1.pdf">
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* An explicit Sixth-Order Runge-Kutta Formula</a>.
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* </p>
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* <p>This method is an explicit Runge-Kutta method, its Butcher-array
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* is the following one :
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* <pre>
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* 0 | 0 0 0 0 0 0
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* 1 | 1 0 0 0 0 0
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* 1/2 | 3/8 1/8 0 0 0 0
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* 2/3 | 8/27 2/27 8/27 0 0 0
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* (7-q)/14 | ( -21 + 9q)/392 ( -56 + 8q)/392 ( 336 - 48q)/392 ( -63 + 3q)/392 0 0
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* (7+q)/14 | (-1155 - 255q)/1960 ( -280 - 40q)/1960 ( 0 - 320q)/1960 ( 63 + 363q)/1960 ( 2352 + 392q)/1960 0
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* 1 | ( 330 + 105q)/180 ( 120 + 0q)/180 ( -200 + 280q)/180 ( 126 - 189q)/180 ( -686 - 126q)/180 ( 490 - 70q)/180
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* |--------------------------------------------------------------------------------------------------------------------------------------------------
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* | 1/20 0 16/45 0 49/180 49/180 1/20
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* </pre>
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* where q = √21</p>
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*
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* @see EulerFieldIntegrator
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* @see ClassicalRungeKuttaFieldIntegrator
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* @see GillFieldIntegrator
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* @see MidpointFieldIntegrator
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* @see ThreeEighthesFieldIntegrator
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* @param <T> the type of the field elements
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* @since 3.6
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*/
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public class LutherFieldIntegrator<T extends RealFieldElement<T>>
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extends RungeKuttaFieldIntegrator<T> {
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/** Square root. */
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private static final double Q = FastMath.sqrt(21);
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/** Time steps Butcher array. */
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private static final double[] STATIC_C = {
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1.0, 1.0 / 2.0, 2.0 / 3.0, (7.0 - Q) / 14.0, (7.0 + Q) / 14.0, 1.0
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};
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/** Internal weights Butcher array. */
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private static final double[][] STATIC_A = {
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{ 1.0 },
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{ 3.0 / 8.0, 1.0 / 8.0 },
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{ 8.0 / 27.0, 2.0 / 27.0, 8.0 / 27.0 },
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{ ( -21.0 + 9.0 * Q) / 392.0, ( -56.0 + 8.0 * Q) / 392.0, ( 336.0 - 48.0 * Q) / 392.0, (-63.0 + 3.0 * Q) / 392.0 },
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{ (-1155.0 - 255.0 * Q) / 1960.0, (-280.0 - 40.0 * Q) / 1960.0, ( 0.0 - 320.0 * Q) / 1960.0, ( 63.0 + 363.0 * Q) / 1960.0, (2352.0 + 392.0 * Q) / 1960.0 },
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{ ( 330.0 + 105.0 * Q) / 180.0, ( 120.0 + 0.0 * Q) / 180.0, (-200.0 + 280.0 * Q) / 180.0, (126.0 - 189.0 * Q) / 180.0, (-686.0 - 126.0 * Q) / 180.0, (490.0 - 70.0 * Q) / 180.0 }
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};
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/** Propagation weights Butcher array. */
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private static final double[] STATIC_B = {
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1.0 / 20.0, 0, 16.0 / 45.0, 0, 49.0 / 180.0, 49.0 / 180.0, 1.0 / 20.0
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};
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/** Simple constructor.
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* Build a fourth-order Luther integrator with the given step.
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* @param field field to which the time and state vector elements belong
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* @param step integration step
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*/
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public LutherFieldIntegrator(final Field<T> field, final T step) {
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super(field, "Luther", STATIC_C, STATIC_A, STATIC_B, new LutherFieldStepInterpolator<T>(), step);
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}
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}
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math4.ode.nonstiff;
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import org.apache.commons.math4.RealFieldElement;
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import org.apache.commons.math4.ode.FieldEquationsMapper;
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import org.apache.commons.math4.ode.FieldODEStateAndDerivative;
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import org.apache.commons.math4.util.FastMath;
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import org.apache.commons.math4.util.MathArrays;
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/**
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* This class represents an interpolator over the last step during an
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* ODE integration for the 6th order Luther integrator.
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*
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* <p>This interpolator computes dense output inside the last
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* step computed. The interpolation equation is consistent with the
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* integration scheme.</p>
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*
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* @see LutherFieldIntegrator
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* @param <T> the type of the field elements
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* @since 3.6
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*/
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class LutherFieldStepInterpolator<T extends RealFieldElement<T>>
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extends RungeKuttaFieldStepInterpolator<T> {
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/** Square root. */
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private static final double Q = FastMath.sqrt(21);
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/** Simple constructor.
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* This constructor builds an instance that is not usable yet, the
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* {@link
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* org.apache.commons.math4.ode.sampling.AbstractFieldStepInterpolator#reinitialize}
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* method should be called before using the instance in order to
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* initialize the internal arrays. This constructor is used only
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* in order to delay the initialization in some cases. The {@link
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* RungeKuttaFieldIntegrator} class uses the prototyping design pattern
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* to create the step interpolators by cloning an uninitialized model
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* and later initializing the copy.
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*/
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LutherFieldStepInterpolator() {
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}
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/** Copy constructor.
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* @param interpolator interpolator to copy from. The copy is a deep
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* copy: its arrays are separated from the original arrays of the
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* instance
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*/
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LutherFieldStepInterpolator(final LutherFieldStepInterpolator<T> interpolator) {
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super(interpolator);
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}
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/** {@inheritDoc} */
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@Override
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protected LutherFieldStepInterpolator<T> doCopy() {
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return new LutherFieldStepInterpolator<T>(this);
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}
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/** {@inheritDoc} */
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@Override
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protected FieldODEStateAndDerivative<T> computeInterpolatedStateAndDerivatives(final FieldEquationsMapper<T> mapper,
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final T time, final T theta,
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final T oneMinusThetaH) {
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// the coefficients below have been computed by solving the
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// order conditions from a theorem from Butcher (1963), using
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// the method explained in Folkmar Bornemann paper "Runge-Kutta
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// Methods, Trees, and Maple", Center of Mathematical Sciences, Munich
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// University of Technology, February 9, 2001
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//<http://wwwzenger.informatik.tu-muenchen.de/selcuk/sjam012101.html>
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// the method is implemented in the rkcheck tool
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// <https://www.spaceroots.org/software/rkcheck/index.html>.
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// Running it for order 5 gives the following order conditions
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// for an interpolator:
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// order 1 conditions
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// \sum_{i=1}^{i=s}\left(b_{i} \right) =1
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// order 2 conditions
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// \sum_{i=1}^{i=s}\left(b_{i} c_{i}\right) = \frac{\theta}{2}
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// order 3 conditions
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// \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}
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// \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{2}\right) = \frac{\theta^{2}}{3}
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// order 4 conditions
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// \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}
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// \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}
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// \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}
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// \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{3}\right) = \frac{\theta^{3}}{4}
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// order 5 conditions
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// \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}
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// \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}
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// \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}
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// \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}
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// \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}
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// \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}
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// \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}
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// \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}
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// \sum_{i=1}^{i=s}\left(b_{i} c_{i}^{4}\right) = \frac{\theta^{4}}{5}
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// The a_{j,k} and c_{k} are given by the integrator Butcher arrays. What remains to solve
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// are the b_i for the interpolator. They are found by solving the above equations.
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// For a given interpolator, some equations are redundant, so in our case when we select
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// all equations from order 1 to 4, we still don't have enough independent equations
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// to solve from b_1 to b_7. We need to also select one equation from order 5. Here,
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// we selected the last equation. It appears this choice implied at least the last 3 equations
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// are fulfilled, but some of the former ones are not, so the resulting interpolator is order 5.
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// At the end, we get the b_i as polynomials in theta.
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final T coeffDot1 = theta.multiply(theta.multiply(theta.multiply(theta.multiply( 21 ).add( -47 )).add( 36 )).add( -54 / 5.0)).add(1);
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// not really needed as it is zero: final T coeffDot2 = theta.getField().getZero();
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final T coeffDot3 = theta.multiply(theta.multiply(theta.multiply(theta.multiply( 112 ).add(-608 / 3.0)).add( 320 / 3.0 )).add(-208 / 15.0));
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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));
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final T coeffDot5 = theta.multiply(theta.multiply(theta.multiply(theta.multiply(( -49 - 49 * Q) / 5.0).add((392 + 287 * Q) / 15.0)).add((-637 - 357 * Q) / 30.0)).add((833 + 343 * Q) / 150.0));
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final T coeffDot6 = theta.multiply(theta.multiply(theta.multiply(theta.multiply(( -49 + 49 * Q) / 5.0).add((392 - 287 * Q) / 15.0)).add((-637 + 357 * Q) / 30.0)).add((833 - 343 * Q) / 150.0));
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final T coeffDot7 = theta.multiply(theta.multiply(theta.multiply( 3 ).add( -3 ).add( 3 / 5.0)));
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final T[] interpolatedState = MathArrays.buildArray(theta.getField(), previousState.length);
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final T[] interpolatedDerivatives = MathArrays.buildArray(theta.getField(), previousState.length);
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if ((previousState != null) && (theta.getReal() <= 0.5)) {
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final T coeff1 = theta.multiply(theta.multiply(theta.multiply(theta.multiply( 21 / 5.0).add( -47 / 4.0)).add( 12 )).add( -27 / 5.0)).add(1);
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// not really needed as it is zero: final T coeff2 = theta.getField().getZero();
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final T coeff3 = theta.multiply(theta.multiply(theta.multiply(theta.multiply( 112 / 5.0).add(-152 / 3.0)).add( 320 / 9.0 )).add(-104 / 15.0));
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final T coeff4 = theta.multiply(theta.multiply(theta.multiply(theta.multiply(-567 / 25.0).add( 243 / 5.0)).add( -162 / 5.0 )).add( 162 / 25.0));
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final T coeff5 = theta.multiply(theta.multiply(theta.multiply(theta.multiply((-49 - 49 * Q) / 25.0).add((392 + 287 * Q) / 60.0)).add((-637 - 357 * Q) / 90.0)).add((833 + 343 * Q) / 300.0));
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final T coeff6 = theta.multiply(theta.multiply(theta.multiply(theta.multiply((-49 + 49 * Q) / 25.0).add((392 - 287 * Q) / 60.0)).add((-637 + 357 * Q) / 90.0)).add((833 - 343 * Q) / 300.0));
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final T coeff7 = theta.multiply(theta.multiply(theta.multiply( 3 / 4.0 ).add( -1 )).add( 3 / 10.0));
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for (int i = 0; i < interpolatedState.length; ++i) {
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final T yDot1 = yDotK[0][i];
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// not really needed as associated coefficients are zero: final T yDot2 = yDotK[1][i];
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final T yDot3 = yDotK[2][i];
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final T yDot4 = yDotK[3][i];
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final T yDot5 = yDotK[4][i];
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final T yDot6 = yDotK[5][i];
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final T yDot7 = yDotK[6][i];
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interpolatedState[i] = previousState[i].
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add(theta.multiply(h).
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multiply( coeff1.multiply(yDot1).
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// not really needed as it is zero: add(coeff2.multiply(yDot2)).
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add(coeff3.multiply(yDot3)).
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add(coeff4.multiply(yDot4)).
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add(coeff5.multiply(yDot5)).
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add(coeff6.multiply(yDot6)).
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add(coeff7.multiply(yDot7))));
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interpolatedDerivatives[i] = coeffDot1.multiply(yDot1).
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// not really needed as it is zero: add(coeffDot2.multiply(yDot2)).
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add(coeffDot3.multiply(yDot3)).
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add(coeffDot4.multiply(yDot4)).
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add(coeffDot5.multiply(yDot5)).
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add(coeffDot6.multiply(yDot6)).
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add(coeffDot7.multiply(yDot7));
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}
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} else {
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final T coeff1 = 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);
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// not really needed as it is zero: final T coeff2 = theta.getField().getZero();
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final T coeff3 = 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);
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final T coeff4 = theta.multiply(theta.multiply(theta.multiply(theta.multiply( 567 / 25.0).add( -648 / 25.0)).add( 162 / 25.0)));
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final T coeff5 = theta.multiply(theta.multiply(theta.multiply(theta.multiply(( 49 + 49 * Q) / 25.0).add((-1372 - 847 * Q) / 300.0)).add((2254 + 1029 * Q) / 900.0)).add( -49 / 180.0)).add(-49 / 180.0);
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final T coeff6 = theta.multiply(theta.multiply(theta.multiply(theta.multiply(( 49 - 49 * Q) / 25.0).add((-1372 + 847 * Q) / 300.0)).add((2254 - 1029 * Q) / 900.0)).add( -49 / 180.0)).add(-49 / 180.0);
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final T coeff7 = theta.multiply(theta.multiply(theta.multiply( -3 / 4.0 ).add( 1 / 4.0)).add( -1 / 20.0)).add( -1 / 20.0);
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for (int i = 0; i < interpolatedState.length; ++i) {
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final T yDot1 = yDotK[0][i];
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// not really needed as associated coefficients are zero: final T yDot2 = yDotK[1][i];
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final T yDot3 = yDotK[2][i];
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final T yDot4 = yDotK[3][i];
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final T yDot5 = yDotK[4][i];
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final T yDot6 = yDotK[5][i];
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final T yDot7 = yDotK[6][i];
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interpolatedState[i] = currentState[i].
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add(oneMinusThetaH.
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multiply( coeff1.multiply(yDot1).
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// not really needed as it is zero: add(coeff2.multiply(yDot2)).
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add(coeff3.multiply(yDot3)).
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add(coeff4.multiply(yDot4)).
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add(coeff5.multiply(yDot5)).
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add(coeff6.multiply(yDot6)).
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add(coeff7.multiply(yDot7))));
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interpolatedDerivatives[i] = coeffDot1.multiply(yDot1).
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// not really needed as it is zero: add(coeffDot2.multiply(yDot2)).
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add(coeffDot3.multiply(yDot3)).
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add(coeffDot4.multiply(yDot4)).
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add(coeffDot5.multiply(yDot5)).
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add(coeffDot6.multiply(yDot6)).
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add(coeffDot7.multiply(yDot7));
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}
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}
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return new FieldODEStateAndDerivative<T>(time, interpolatedState, yDotK[0]);
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}
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}
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