Field-based version of classical Runge-Kutta 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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/**
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* This class implements the classical fourth order Runge-Kutta
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* integrator for Ordinary Differential Equations (it is the most
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* often used Runge-Kutta method).
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*
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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
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* 1/2 | 1/2 0 0 0
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* 1/2 | 0 1/2 0 0
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* 1 | 0 0 1 0
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* |--------------------
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* | 1/6 1/3 1/3 1/6
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* </pre>
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* </p>
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*
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* @see EulerFieldIntegrator
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* @see GillFieldIntegrator
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* @see MidpointFieldIntegrator
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* @see ThreeEighthesFieldIntegrator
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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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public class ClassicalRungeKuttaFieldIntegrator<T extends RealFieldElement<T>>
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extends RungeKuttaFieldIntegrator<T> {
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/** Time steps Butcher array. */
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private static final double[] STATIC_C = {
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1.0 / 2.0, 1.0 / 2.0, 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 / 2.0 },
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{ 0.0, 1.0 / 2.0 },
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{ 0.0, 0.0, 1.0 }
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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 / 6.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 6.0
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};
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/** Simple constructor.
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* Build a fourth-order Runge-Kutta 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 ClassicalRungeKuttaFieldIntegrator(final Field<T> field, final T step) {
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super(field, "classical Runge-Kutta", STATIC_C, STATIC_A, STATIC_B,
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new ClassicalRungeKuttaFieldStepInterpolator<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.MathArrays;
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/**
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* This class implements a step interpolator for the classical fourth
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* order Runge-Kutta integrator.
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*
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* <p>This interpolator allows to compute dense output inside the last
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* step computed. The interpolation equation is consistent with the
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* integration scheme :
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* <ul>
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* <li>Using reference point at step start:<br>
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* y(t<sub>n</sub> + θ h) = y (t<sub>n</sub>)
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* + θ (h/6) [ (6 - 9 θ + 4 θ<sup>2</sup>) y'<sub>1</sub>
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* + ( 6 θ - 4 θ<sup>2</sup>) (y'<sub>2</sub> + y'<sub>3</sub>)
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* + ( -3 θ + 4 θ<sup>2</sup>) y'<sub>4</sub>
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* ]
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* </li>
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* <li>Using reference point at step end:<br>
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* y(t<sub>n</sub> + θ h) = y (t<sub>n</sub> + h)
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* + (1 - θ) (h/6) [ (-4 θ^2 + 5 θ - 1) y'<sub>1</sub>
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* +(4 θ^2 - 2 θ - 2) (y'<sub>2</sub> + y'<sub>3</sub>)
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* -(4 θ^2 + θ + 1) y'<sub>4</sub>
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* ]
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* </li>
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* </ul>
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* </p>
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*
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* where θ belongs to [0 ; 1] and where y'<sub>1</sub> to y'<sub>4</sub> are the four
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* evaluations of the derivatives already computed during the
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* step.</p>
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*
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* @see ClassicalRungeKuttaFieldIntegrator
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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 ClassicalRungeKuttaFieldStepInterpolator<T extends RealFieldElement<T>>
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extends RungeKuttaFieldStepInterpolator<T> {
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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 RungeKuttaFieldStepInterpolator#reinitialize} method should be
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* called before using the instance in order to initialize the
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* internal arrays. This constructor is used only in order to delay
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* the initialization in some cases. The {@link RungeKuttaFieldIntegrator}
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* class uses the prototyping design pattern to create the step
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* interpolators by cloning an uninitialized model and latter initializing
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* the copy.
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*/
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ClassicalRungeKuttaFieldStepInterpolator() {
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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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ClassicalRungeKuttaFieldStepInterpolator(final ClassicalRungeKuttaFieldStepInterpolator<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 ClassicalRungeKuttaFieldStepInterpolator<T> doCopy() {
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return new ClassicalRungeKuttaFieldStepInterpolator<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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final T one = time.getField().getOne();
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final T oneMinusTheta = one.subtract(theta);
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final T oneMinus2Theta = one.subtract(theta.multiply(2));
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final T coeffDot1 = oneMinusTheta.multiply(oneMinus2Theta);
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final T coeffDot23 = theta.multiply(oneMinusTheta).multiply(2);
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final T coeffDot4 = theta.multiply(oneMinus2Theta).negate();
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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 fourTheta2 = theta.multiply(theta).multiply(4);
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final T s = theta.multiply(h).divide(6.0);
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final T coeff1 = s.multiply(fourTheta2.subtract(theta.multiply(9)).add(6));
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final T coeff23 = s.multiply(theta.multiply(6).subtract(fourTheta2));
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final T coeff4 = s.multiply(fourTheta2.subtract(theta.multiply(3)));
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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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final T yDot23 = yDotK[1][i].add(yDotK[2][i]);
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final T yDot4 = yDotK[3][i];
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interpolatedState[i] =
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previousState[i].add(coeff1.multiply(yDot1)).add(coeff23.multiply(yDot23)).add(coeff4.multiply(yDot4));
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interpolatedDerivatives[i] =
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coeffDot1.multiply(yDot1).add(coeffDot23.multiply(yDot23)).add(coeffDot4.multiply(yDot4));
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}
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} else {
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final T fourTheta = theta.multiply(4);
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final T s = oneMinusThetaH.divide(6);
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final T coeff1 = s.multiply(theta.multiply(fourTheta.negate().add(5)).subtract(1));
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final T coeff23 = s.multiply(theta.multiply(fourTheta.subtract(2)).subtract(2));
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final T coeff4 = s.multiply(theta.multiply(fourTheta.negate().subtract(1)).subtract(1));
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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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final T yDot23 = yDotK[1][i].add(yDotK[2][i]);
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final T yDot4 = yDotK[3][i];
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interpolatedState[i] =
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currentState[i].add(coeff1.multiply(yDot1)).add(coeff23.multiply(yDot23)).add(coeff4.multiply(yDot4));
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interpolatedDerivatives[i] =
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coeffDot1.multiply(yDot1).add(coeffDot23.multiply(yDot23)).add(coeffDot4.multiply(yDot4));
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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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