LutherStateInterpolator.java
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* this work for additional information regarding copyright ownership.
* The Hipparchus project licenses this file to You under the Apache License, Version 2.0
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* https://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.hipparchus.ode.nonstiff;
import org.hipparchus.ode.EquationsMapper;
import org.hipparchus.ode.ODEStateAndDerivative;
import org.hipparchus.util.FastMath;
/**
* 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 LutherIntegrator
*/
class LutherStateInterpolator extends RungeKuttaStateInterpolator {
/** Serializable version identifier */
private static final long serialVersionUID = 20160328;
/** Square root. */
private static final double Q = FastMath.sqrt(21);
/** Simple constructor.
* @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
*/
LutherStateInterpolator(final boolean forward,
final double[][] yDotK,
final ODEStateAndDerivative globalPreviousState,
final ODEStateAndDerivative globalCurrentState,
final ODEStateAndDerivative softPreviousState,
final ODEStateAndDerivative softCurrentState,
final EquationsMapper mapper) {
super(forward, yDotK,
globalPreviousState, globalCurrentState, softPreviousState, softCurrentState,
mapper);
}
/** {@inheritDoc} */
@Override
protected LutherStateInterpolator create(final boolean newForward, final double[][] newYDotK,
final ODEStateAndDerivative newGlobalPreviousState,
final ODEStateAndDerivative newGlobalCurrentState,
final ODEStateAndDerivative newSoftPreviousState,
final ODEStateAndDerivative newSoftCurrentState,
final EquationsMapper newMapper) {
return new LutherStateInterpolator(newForward, newYDotK,
newGlobalPreviousState, newGlobalCurrentState,
newSoftPreviousState, newSoftCurrentState,
newMapper);
}
/** {@inheritDoc} */
@Override
protected ODEStateAndDerivative computeInterpolatedStateAndDerivatives(final EquationsMapper mapper,
final double time, final double theta,
final double thetaH, 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
//<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 double[] interpolatedState;
final double[] interpolatedDerivatives;
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 (getGlobalPreviousState() != 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)));
interpolatedState = previousStateLinearCombination(thetaH * coeff1, thetaH * coeff2,
thetaH * coeff3, thetaH * coeff4,
thetaH * coeff5, thetaH * coeff6,
thetaH * coeff7);
interpolatedDerivatives = derivativeLinearCombination(coeffDot1, coeffDot2, coeffDot3, coeffDot4, coeffDot5, coeffDot6, coeffDot7);
} 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)));
interpolatedState = currentStateLinearCombination(oneMinusThetaH * coeff1, oneMinusThetaH * coeff2,
oneMinusThetaH * coeff3, oneMinusThetaH * coeff4,
oneMinusThetaH * coeff5, oneMinusThetaH * coeff6,
oneMinusThetaH * coeff7);
interpolatedDerivatives = derivativeLinearCombination(coeffDot1, coeffDot2, coeffDot3, coeffDot4, coeffDot5, coeffDot6, coeffDot7);
}
return mapper.mapStateAndDerivative(time, interpolatedState, interpolatedDerivatives);
}
}