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* The ASF licenses this file to You under the Apache License, Version 2.0
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* http://www.apache.org/licenses/LICENSE-2.0
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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 Also: - LutherIntegrator
Since: 3.3
/**
* 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
* @since 3.3
*/
class LutherStepInterpolator extends RungeKuttaStepInterpolator {
Serializable version identifier /** Serializable version identifier */
private static final long serialVersionUID = 20140416L;
Square root. /** Square root. */
private static final double Q = FastMath.sqrt(21);
Simple constructor. This constructor builds an instance that is not usable yet, the 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 RungeKuttaIntegrator class uses the prototyping design pattern to create the step interpolators by cloning an uninitialized model and later initializing the copy. /** 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.
Params: - interpolator – interpolator to copy from. The copy is a deep
copy: its arrays are separated from the original arrays of the
instance
/** 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} /** {@inheritDoc} */
@Override
protected StepInterpolator doCopy() {
return new LutherStepInterpolator(this);
}
{@inheritDoc} /** {@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
//<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 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;
}
}
}
}