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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
* limitations under the License.
*/
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.
*
* <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. */
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<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} */
@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} */
@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);
}
}
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