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With inspiration from other libraries
/*
* 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.util.FastMath;
/**
* This class implements the 5(4) Dormand-Prince integrator for Ordinary
* Differential Equations.
* This integrator is an embedded Runge-Kutta integrator
* of order 5(4) used in local extrapolation mode (i.e. the solution
* is computed using the high order formula) with stepsize control
* (and automatic step initialization) and continuous output. This
* method uses 7 functions evaluations per step. However, since this
* is an fsal, the last evaluation of one step is the same as
* the first evaluation of the next step and hence can be avoided. So
* the cost is really 6 functions evaluations per step.
*
* This method has been published (whithout the continuous output
* that was added by Shampine in 1986) in the following article :
*
* A family of embedded Runge-Kutta formulae
* J. R. Dormand and P. J. Prince
* Journal of Computational and Applied Mathematics
* volume 6, no 1, 1980, pp. 19-26
*
*
* @since 1.2
*/
public class DormandPrince54Integrator extends EmbeddedRungeKuttaIntegrator {
/** Integrator method name. */
private static final String METHOD_NAME = "Dormand-Prince 5(4)";
/** Time steps Butcher array. */
private static final double[] STATIC_C = {
1.0/5.0, 3.0/10.0, 4.0/5.0, 8.0/9.0, 1.0, 1.0
};
/** Internal weights Butcher array. */
private static final double[][] STATIC_A = {
{1.0/5.0},
{3.0/40.0, 9.0/40.0},
{44.0/45.0, -56.0/15.0, 32.0/9.0},
{19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0, -212.0/729.0},
{9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0},
{35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0}
};
/** Propagation weights Butcher array. */
private static final double[] STATIC_B = {
35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0, 0.0
};
/** Error array, element 1. */
private static final double E1 = 71.0 / 57600.0;
// element 2 is zero, so it is neither stored nor used
/** Error array, element 3. */
private static final double E3 = -71.0 / 16695.0;
/** Error array, element 4. */
private static final double E4 = 71.0 / 1920.0;
/** Error array, element 5. */
private static final double E5 = -17253.0 / 339200.0;
/** Error array, element 6. */
private static final double E6 = 22.0 / 525.0;
/** Error array, element 7. */
private static final double E7 = -1.0 / 40.0;
/** Simple constructor.
* Build a fifth order Dormand-Prince integrator with the given step bounds
* @param minStep minimal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param maxStep maximal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param scalAbsoluteTolerance allowed absolute error
* @param scalRelativeTolerance allowed relative error
*/
public DormandPrince54Integrator(final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance) {
super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
}
/** Simple constructor.
* Build a fifth order Dormand-Prince integrator with the given step bounds
* @param minStep minimal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param maxStep maximal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param vecAbsoluteTolerance allowed absolute error
* @param vecRelativeTolerance allowed relative error
*/
public DormandPrince54Integrator(final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance) {
super(METHOD_NAME, true, STATIC_C, STATIC_A, STATIC_B, new DormandPrince54StepInterpolator(),
minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
}
/** {@inheritDoc} */
@Override
public int getOrder() {
return 5;
}
/** {@inheritDoc} */
@Override
protected double estimateError(final double[][] yDotK,
final double[] y0, final double[] y1,
final double h) {
double error = 0;
for (int j = 0; j < mainSetDimension; ++j) {
final double errSum = E1 * yDotK[0][j] + E3 * yDotK[2][j] +
E4 * yDotK[3][j] + E5 * yDotK[4][j] +
E6 * yDotK[5][j] + E7 * yDotK[6][j];
final double yScale = FastMath.max(FastMath.abs(y0[j]), FastMath.abs(y1[j]));
final double tol = (vecAbsoluteTolerance == null) ?
(scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
(vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
final double ratio = h * errSum / tol;
error += ratio * ratio;
}
return FastMath.sqrt(error / mainSetDimension);
}
}