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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
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 * Unless required by applicable law or agreed to in writing, software
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package org.apache.commons.math3.ode.nonstiff;

import org.apache.commons.math3.ode.sampling.StepInterpolator;
import org.apache.commons.math3.util.FastMath;

/**
 * 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.

* * @see LutherIntegrator * @since 3.3 */ class LutherStepInterpolator extends RungeKuttaStepInterpolator { /** Serializable version identifier */ private static final long serialVersionUID = 20140416L; /** Square root. */ private static final double Q = FastMath.sqrt(21); /** Simple constructor. * This constructor builds an instance that is not usable yet, the * {@link * org.apache.commons.math3.ode.sampling.AbstractStepInterpolator#reinitialize} * method should be called before using the instance in order to * initialize the internal arrays. This constructor is used only * in order to delay the initialization in some cases. The {@link * RungeKuttaIntegrator} class uses the prototyping design pattern * to create the step interpolators by cloning an uninitialized model * and later initializing the copy. */ // CHECKSTYLE: stop RedundantModifier // the public modifier here is needed for serialization public LutherStepInterpolator() { } // CHECKSTYLE: resume RedundantModifier /** Copy constructor. * @param interpolator interpolator to copy from. The copy is a deep * copy: its arrays are separated from the original arrays of the * instance */ LutherStepInterpolator(final LutherStepInterpolator interpolator) { super(interpolator); } /** {@inheritDoc} */ @Override protected StepInterpolator doCopy() { return new LutherStepInterpolator(this); } /** {@inheritDoc} */ @Override protected void computeInterpolatedStateAndDerivatives(final double theta, final double 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 // // the method is implemented in the rkcheck tool // . // 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 double coeffDot1 = 1 + theta * ( -54 / 5.0 + theta * ( 36 + theta * ( -47 + theta * 21))); final double coeffDot2 = 0; final double coeffDot3 = theta * (-208 / 15.0 + theta * ( 320 / 3.0 + theta * (-608 / 3.0 + theta * 112))); final double coeffDot4 = theta * ( 324 / 25.0 + theta * ( -486 / 5.0 + theta * ( 972 / 5.0 + theta * -567 / 5.0))); final double coeffDot5 = theta * ((833 + 343 * Q) / 150.0 + theta * ((-637 - 357 * Q) / 30.0 + theta * ((392 + 287 * Q) / 15.0 + theta * (-49 - 49 * Q) / 5.0))); final double coeffDot6 = theta * ((833 - 343 * Q) / 150.0 + theta * ((-637 + 357 * Q) / 30.0 + theta * ((392 - 287 * Q) / 15.0 + theta * (-49 + 49 * Q) / 5.0))); final double coeffDot7 = theta * ( 3 / 5.0 + theta * ( -3 + theta * 3)); if ((previousState != null) && (theta <= 0.5)) { final double coeff1 = 1 + theta * ( -27 / 5.0 + theta * ( 12 + theta * ( -47 / 4.0 + theta * 21 / 5.0))); final double coeff2 = 0; final double coeff3 = theta * (-104 / 15.0 + theta * ( 320 / 9.0 + theta * (-152 / 3.0 + theta * 112 / 5.0))); final double coeff4 = theta * ( 162 / 25.0 + theta * ( -162 / 5.0 + theta * ( 243 / 5.0 + theta * -567 / 25.0))); final double coeff5 = theta * ((833 + 343 * Q) / 300.0 + theta * ((-637 - 357 * Q) / 90.0 + theta * ((392 + 287 * Q) / 60.0 + theta * (-49 - 49 * Q) / 25.0))); final double coeff6 = theta * ((833 - 343 * Q) / 300.0 + theta * ((-637 + 357 * Q) / 90.0 + theta * ((392 - 287 * Q) / 60.0 + theta * (-49 + 49 * Q) / 25.0))); final double coeff7 = theta * ( 3 / 10.0 + theta * ( -1 + theta * ( 3 / 4.0))); for (int i = 0; i < interpolatedState.length; ++i) { final double yDot1 = yDotK[0][i]; final double yDot2 = yDotK[1][i]; final double yDot3 = yDotK[2][i]; final double yDot4 = yDotK[3][i]; final double yDot5 = yDotK[4][i]; final double yDot6 = yDotK[5][i]; final double yDot7 = yDotK[6][i]; interpolatedState[i] = previousState[i] + theta * h * (coeff1 * yDot1 + coeff2 * yDot2 + coeff3 * yDot3 + coeff4 * yDot4 + coeff5 * yDot5 + coeff6 * yDot6 + coeff7 * yDot7); interpolatedDerivatives[i] = coeffDot1 * yDot1 + coeffDot2 * yDot2 + coeffDot3 * yDot3 + coeffDot4 * yDot4 + coeffDot5 * yDot5 + coeffDot6 * yDot6 + coeffDot7 * yDot7; } } else { final double coeff1 = -1 / 20.0 + theta * ( 19 / 20.0 + theta * ( -89 / 20.0 + theta * ( 151 / 20.0 + theta * -21 / 5.0))); final double coeff2 = 0; final double coeff3 = -16 / 45.0 + theta * ( -16 / 45.0 + theta * ( -328 / 45.0 + theta * ( 424 / 15.0 + theta * -112 / 5.0))); final double coeff4 = theta * ( theta * ( 162 / 25.0 + theta * ( -648 / 25.0 + theta * 567 / 25.0))); final double coeff5 = -49 / 180.0 + theta * ( -49 / 180.0 + theta * ((2254 + 1029 * Q) / 900.0 + theta * ((-1372 - 847 * Q) / 300.0 + theta * ( 49 + 49 * Q) / 25.0))); final double coeff6 = -49 / 180.0 + theta * ( -49 / 180.0 + theta * ((2254 - 1029 * Q) / 900.0 + theta * ((-1372 + 847 * Q) / 300.0 + theta * ( 49 - 49 * Q) / 25.0))); final double coeff7 = -1 / 20.0 + theta * ( -1 / 20.0 + theta * ( 1 / 4.0 + theta * ( -3 / 4.0))); for (int i = 0; i < interpolatedState.length; ++i) { final double yDot1 = yDotK[0][i]; final double yDot2 = yDotK[1][i]; final double yDot3 = yDotK[2][i]; final double yDot4 = yDotK[3][i]; final double yDot5 = yDotK[4][i]; final double yDot6 = yDotK[5][i]; final double yDot7 = yDotK[6][i]; interpolatedState[i] = currentState[i] + oneMinusThetaH * (coeff1 * yDot1 + coeff2 * yDot2 + coeff3 * yDot3 + coeff4 * yDot4 + coeff5 * yDot5 + coeff6 * yDot6 + coeff7 * yDot7); interpolatedDerivatives[i] = coeffDot1 * yDot1 + coeffDot2 * yDot2 + coeffDot3 * yDot3 + coeffDot4 * yDot4 + coeffDot5 * yDot5 + coeffDot6 * yDot6 + coeffDot7 * yDot7; } } } }




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