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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;

/**
 * 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 LutherFieldIntegrator * @param the type of the field elements * @since 3.6 */ class LutherFieldStepInterpolator> extends RungeKuttaFieldStepInterpolator { /** -49 - 49 q. */ private final T c5a; /** 392 + 287 q. */ private final T c5b; /** -637 - 357 q. */ private final T c5c; /** 833 + 343 q. */ private final T c5d; /** -49 + 49 q. */ private final T c6a; /** -392 - 287 q. */ private final T c6b; /** -637 + 357 q. */ private final T c6c; /** 833 - 343 q. */ private final T c6d; /** 49 + 49 q. */ private final T d5a; /** -1372 - 847 q. */ private final T d5b; /** 2254 + 1029 q */ private final T d5c; /** 49 - 49 q. */ private final T d6a; /** -1372 + 847 q. */ private final T d6b; /** 2254 - 1029 q */ private final T d6c; /** 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 field, final boolean forward, final T[][] yDotK, final FieldODEStateAndDerivative globalPreviousState, final FieldODEStateAndDerivative globalCurrentState, final FieldODEStateAndDerivative softPreviousState, final FieldODEStateAndDerivative softCurrentState, final FieldEquationsMapper 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} */ @Override protected LutherFieldStepInterpolator create(final Field newField, final boolean newForward, final T[][] newYDotK, final FieldODEStateAndDerivative newGlobalPreviousState, final FieldODEStateAndDerivative newGlobalCurrentState, final FieldODEStateAndDerivative newSoftPreviousState, final FieldODEStateAndDerivative newSoftCurrentState, final FieldEquationsMapper newMapper) { return new LutherFieldStepInterpolator(newField, newForward, newYDotK, newGlobalPreviousState, newGlobalCurrentState, newSoftPreviousState, newSoftCurrentState, newMapper); } /** {@inheritDoc} */ @SuppressWarnings("unchecked") @Override protected FieldODEStateAndDerivative computeInterpolatedStateAndDerivatives(final FieldEquationsMapper 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 // // 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 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(time, interpolatedState, interpolatedDerivatives); } }




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