/*
* 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.math.ode.nonstiff;
import org.apache.commons.math.ode.DerivativeException;
import org.apache.commons.math.ode.FirstOrderDifferentialEquations;
import org.apache.commons.math.ode.IntegratorException;
import org.apache.commons.math.ode.events.EventHandler;
import org.apache.commons.math.ode.sampling.AbstractStepInterpolator;
import org.apache.commons.math.ode.sampling.DummyStepInterpolator;
import org.apache.commons.math.ode.sampling.StepHandler;
/**
* This class implements a Gragg-Bulirsch-Stoer integrator for
* Ordinary Differential Equations.
*
* <p>The Gragg-Bulirsch-Stoer algorithm is one of the most efficient
* ones currently available for smooth problems. It uses Richardson
* extrapolation to estimate what would be the solution if the step
* size could be decreased down to zero.</p>
*
* <p>
* This method changes both the step size and the order during
* integration, in order to minimize computation cost. It is
* particularly well suited when a very high precision is needed. The
* limit where this method becomes more efficient than high-order
* embedded Runge-Kutta methods like {@link DormandPrince853Integrator
* Dormand-Prince 8(5,3)} depends on the problem. Results given in the
* Hairer, Norsett and Wanner book show for example that this limit
* occurs for accuracy around 1e-6 when integrating Saltzam-Lorenz
* equations (the authors note this problem is <i>extremely sensitive
* to the errors in the first integration steps</i>), and around 1e-11
* for a two dimensional celestial mechanics problems with seven
* bodies (pleiades problem, involving quasi-collisions for which
* <i>automatic step size control is essential</i>).
* </p>
*
* <p>
* This implementation is basically a reimplementation in Java of the
* <a
* href="http://www.unige.ch/math/folks/hairer/prog/nonstiff/odex.f">odex</a>
* fortran code by E. Hairer and G. Wanner. The redistribution policy
* for this code is available <a
* href="http://www.unige.ch/~hairer/prog/licence.txt">here</a>, for
* convenience, it is reproduced below.</p>
* </p>
*
* <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
* <tr><td>Copyright (c) 2004, Ernst Hairer</td></tr>
*
* <tr><td>Redistribution and use in source and binary forms, with or
* without modification, are permitted provided that the following
* conditions are met:
* <ul>
* <li>Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.</li>
* <li>Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.</li>
* </ul></td></tr>
*
* <tr><td><strong>THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
* CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING,
* BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR
* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
* EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
* PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
* PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
* LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
* NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
* SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.</strong></td></tr>
* </table>
*
* @author E. Hairer and G. Wanner (fortran version)
* @version $Revision: 786881 $ $Date: 2009-06-20 14:53:08 -0400 (Sat, 20 Jun 2009) $
* @since 1.2
*/
public class GraggBulirschStoerIntegrator extends AdaptiveStepsizeIntegrator {
/** Integrator method name. */
private static final String METHOD_NAME = "Gragg-Bulirsch-Stoer";
/** Simple constructor.
* Build a Gragg-Bulirsch-Stoer integrator with the given step
* bounds. All tuning parameters are set to their default
* values. The default step handler does nothing.
* @param minStep minimal step (must be positive even for backward
* integration), the last step can be smaller than this
* @param maxStep maximal step (must be positive even for backward
* integration)
* @param scalAbsoluteTolerance allowed absolute error
* @param scalRelativeTolerance allowed relative error
*/
public GraggBulirschStoerIntegrator(final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance) {
super(METHOD_NAME, minStep, maxStep,
scalAbsoluteTolerance, scalRelativeTolerance);
denseOutput = requiresDenseOutput() || (! eventsHandlersManager.isEmpty());
setStabilityCheck(true, -1, -1, -1);
setStepsizeControl(-1, -1, -1, -1);
setOrderControl(-1, -1, -1);
setInterpolationControl(true, -1);
}
/** Simple constructor.
* Build a Gragg-Bulirsch-Stoer integrator with the given step
* bounds. All tuning parameters are set to their default
* values. The default step handler does nothing.
* @param minStep minimal step (must be positive even for backward
* integration), the last step can be smaller than this
* @param maxStep maximal step (must be positive even for backward
* integration)
* @param vecAbsoluteTolerance allowed absolute error
* @param vecRelativeTolerance allowed relative error
*/
public GraggBulirschStoerIntegrator(final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance) {
super(METHOD_NAME, minStep, maxStep,
vecAbsoluteTolerance, vecRelativeTolerance);
denseOutput = requiresDenseOutput() || (! eventsHandlersManager.isEmpty());
setStabilityCheck(true, -1, -1, -1);
setStepsizeControl(-1, -1, -1, -1);
setOrderControl(-1, -1, -1);
setInterpolationControl(true, -1);
}
/** Set the stability check controls.
* <p>The stability check is performed on the first few iterations of
* the extrapolation scheme. If this test fails, the step is rejected
* and the stepsize is reduced.</p>
* <p>By default, the test is performed, at most during two
* iterations at each step, and at most once for each of these
* iterations. The default stepsize reduction factor is 0.5.</p>
* @param performTest if true, stability check will be performed,
if false, the check will be skipped
* @param maxIter maximal number of iterations for which checks are
* performed (the number of iterations is reset to default if negative
* or null)
* @param maxChecks maximal number of checks for each iteration
* (the number of checks is reset to default if negative or null)
* @param stabilityReduction stepsize reduction factor in case of
* failure (the factor is reset to default if lower than 0.0001 or
* greater than 0.9999)
*/
public void setStabilityCheck(final boolean performTest,
final int maxIter, final int maxChecks,
final double stabilityReduction) {
this.performTest = performTest;
this.maxIter = (maxIter <= 0) ? 2 : maxIter;
this.maxChecks = (maxChecks <= 0) ? 1 : maxChecks;
if ((stabilityReduction < 0.0001) || (stabilityReduction > 0.9999)) {
this.stabilityReduction = 0.5;
} else {
this.stabilityReduction = stabilityReduction;
}
}
/** Set the step size control factors.
* <p>The new step size hNew is computed from the old one h by:
* <pre>
* hNew = h * stepControl2 / (err/stepControl1)^(1/(2k+1))
* </pre>
* where err is the scaled error and k the iteration number of the
* extrapolation scheme (counting from 0). The default values are
* 0.65 for stepControl1 and 0.94 for stepControl2.</p>
* <p>The step size is subject to the restriction:
* <pre>
* stepControl3^(1/(2k+1))/stepControl4 <= hNew/h <= 1/stepControl3^(1/(2k+1))
* </pre>
* The default values are 0.02 for stepControl3 and 4.0 for
* stepControl4.</p>
* @param stepControl1 first stepsize control factor (the factor is
* reset to default if lower than 0.0001 or greater than 0.9999)
* @param stepControl2 second stepsize control factor (the factor
* is reset to default if lower than 0.0001 or greater than 0.9999)
* @param stepControl3 third stepsize control factor (the factor is
* reset to default if lower than 0.0001 or greater than 0.9999)
* @param stepControl4 fourth stepsize control factor (the factor
* is reset to default if lower than 1.0001 or greater than 999.9)
*/
public void setStepsizeControl(final double stepControl1, final double stepControl2,
final double stepControl3, final double stepControl4) {
if ((stepControl1 < 0.0001) || (stepControl1 > 0.9999)) {
this.stepControl1 = 0.65;
} else {
this.stepControl1 = stepControl1;
}
if ((stepControl2 < 0.0001) || (stepControl2 > 0.9999)) {
this.stepControl2 = 0.94;
} else {
this.stepControl2 = stepControl2;
}
if ((stepControl3 < 0.0001) || (stepControl3 > 0.9999)) {
this.stepControl3 = 0.02;
} else {
this.stepControl3 = stepControl3;
}
if ((stepControl4 < 1.0001) || (stepControl4 > 999.9)) {
this.stepControl4 = 4.0;
} else {
this.stepControl4 = stepControl4;
}
}
/** Set the order control parameters.
* <p>The Gragg-Bulirsch-Stoer method changes both the step size and
* the order during integration, in order to minimize computation
* cost. Each extrapolation step increases the order by 2, so the
* maximal order that will be used is always even, it is twice the
* maximal number of columns in the extrapolation table.</p>
* <pre>
* order is decreased if w(k-1) <= w(k) * orderControl1
* order is increased if w(k) <= w(k-1) * orderControl2
* </pre>
* <p>where w is the table of work per unit step for each order
* (number of function calls divided by the step length), and k is
* the current order.</p>
* <p>The default maximal order after construction is 18 (i.e. the
* maximal number of columns is 9). The default values are 0.8 for
* orderControl1 and 0.9 for orderControl2.</p>
* @param maxOrder maximal order in the extrapolation table (the
* maximal order is reset to default if order <= 6 or odd)
* @param orderControl1 first order control factor (the factor is
* reset to default if lower than 0.0001 or greater than 0.9999)
* @param orderControl2 second order control factor (the factor
* is reset to default if lower than 0.0001 or greater than 0.9999)
*/
public void setOrderControl(final int maxOrder,
final double orderControl1, final double orderControl2) {
if ((maxOrder <= 6) || (maxOrder % 2 != 0)) {
this.maxOrder = 18;
}
if ((orderControl1 < 0.0001) || (orderControl1 > 0.9999)) {
this.orderControl1 = 0.8;
} else {
this.orderControl1 = orderControl1;
}
if ((orderControl2 < 0.0001) || (orderControl2 > 0.9999)) {
this.orderControl2 = 0.9;
} else {
this.orderControl2 = orderControl2;
}
// reinitialize the arrays
initializeArrays();
}
/** {@inheritDoc} */
@Override
public void addStepHandler (final StepHandler handler) {
super.addStepHandler(handler);
denseOutput = requiresDenseOutput() || (! eventsHandlersManager.isEmpty());
// reinitialize the arrays
initializeArrays();
}
/** {@inheritDoc} */
@Override
public void addEventHandler(final EventHandler function,
final double maxCheckInterval,
final double convergence,
final int maxIterationCount) {
super.addEventHandler(function, maxCheckInterval, convergence, maxIterationCount);
denseOutput = requiresDenseOutput() || (! eventsHandlersManager.isEmpty());
// reinitialize the arrays
initializeArrays();
}
/** Initialize the integrator internal arrays. */
private void initializeArrays() {
final int size = maxOrder / 2;
if ((sequence == null) || (sequence.length != size)) {
// all arrays should be reallocated with the right size
sequence = new int[size];
costPerStep = new int[size];
coeff = new double[size][];
costPerTimeUnit = new double[size];
optimalStep = new double[size];
}
if (denseOutput) {
// step size sequence: 2, 6, 10, 14, ...
for (int k = 0; k < size; ++k) {
sequence[k] = 4 * k + 2;
}
} else {
// step size sequence: 2, 4, 6, 8, ...
for (int k = 0; k < size; ++k) {
sequence[k] = 2 * (k + 1);
}
}
// initialize the order selection cost array
// (number of function calls for each column of the extrapolation table)
costPerStep[0] = sequence[0] + 1;
for (int k = 1; k < size; ++k) {
costPerStep[k] = costPerStep[k-1] + sequence[k];
}
// initialize the extrapolation tables
for (int k = 0; k < size; ++k) {
coeff[k] = (k > 0) ? new double[k] : null;
for (int l = 0; l < k; ++l) {
final double ratio = ((double) sequence[k]) / sequence[k-l-1];
coeff[k][l] = 1.0 / (ratio * ratio - 1.0);
}
}
}
/** Set the interpolation order control parameter.
* The interpolation order for dense output is 2k - mudif + 1. The
* default value for mudif is 4 and the interpolation error is used
* in stepsize control by default.
* @param useInterpolationError if true, interpolation error is used
* for stepsize control
* @param mudif interpolation order control parameter (the parameter
* is reset to default if <= 0 or >= 7)
*/
public void setInterpolationControl(final boolean useInterpolationError,
final int mudif) {
this.useInterpolationError = useInterpolationError;
if ((mudif <= 0) || (mudif >= 7)) {
this.mudif = 4;
} else {
this.mudif = mudif;
}
}
/** Update scaling array.
* @param y1 first state vector to use for scaling
* @param y2 second state vector to use for scaling
* @param scale scaling array to update
*/
private void rescale(final double[] y1, final double[] y2, final double[] scale) {
if (vecAbsoluteTolerance == null) {
for (int i = 0; i < scale.length; ++i) {
final double yi = Math.max(Math.abs(y1[i]), Math.abs(y2[i]));
scale[i] = scalAbsoluteTolerance + scalRelativeTolerance * yi;
}
} else {
for (int i = 0; i < scale.length; ++i) {
final double yi = Math.max(Math.abs(y1[i]), Math.abs(y2[i]));
scale[i] = vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yi;
}
}
}
/** Perform integration over one step using substeps of a modified
* midpoint method.
* @param t0 initial time
* @param y0 initial value of the state vector at t0
* @param step global step
* @param k iteration number (from 0 to sequence.length - 1)
* @param scale scaling array
* @param f placeholder where to put the state vector derivatives at each substep
* (element 0 already contains initial derivative)
* @param yMiddle placeholder where to put the state vector at the middle of the step
* @param yEnd placeholder where to put the state vector at the end
* @param yTmp placeholder for one state vector
* @return true if computation was done properly,
* false if stability check failed before end of computation
* @throws DerivativeException this exception is propagated to the caller if the
* underlying user function triggers one
*/
private boolean tryStep(final double t0, final double[] y0, final double step, final int k,
final double[] scale, final double[][] f,
final double[] yMiddle, final double[] yEnd,
final double[] yTmp)
throws DerivativeException {
final int n = sequence[k];
final double subStep = step / n;
final double subStep2 = 2 * subStep;
// first substep
double t = t0 + subStep;
for (int i = 0; i < y0.length; ++i) {
yTmp[i] = y0[i];
yEnd[i] = y0[i] + subStep * f[0][i];
}
computeDerivatives(t, yEnd, f[1]);
// other substeps
for (int j = 1; j < n; ++j) {
if (2 * j == n) {
// save the point at the middle of the step
System.arraycopy(yEnd, 0, yMiddle, 0, y0.length);
}
t += subStep;
for (int i = 0; i < y0.length; ++i) {
final double middle = yEnd[i];
yEnd[i] = yTmp[i] + subStep2 * f[j][i];
yTmp[i] = middle;
}
computeDerivatives(t, yEnd, f[j+1]);
// stability check
if (performTest && (j <= maxChecks) && (k < maxIter)) {
double initialNorm = 0.0;
for (int l = 0; l < y0.length; ++l) {
final double ratio = f[0][l] / scale[l];
initialNorm += ratio * ratio;
}
double deltaNorm = 0.0;
for (int l = 0; l < y0.length; ++l) {
final double ratio = (f[j+1][l] - f[0][l]) / scale[l];
deltaNorm += ratio * ratio;
}
if (deltaNorm > 4 * Math.max(1.0e-15, initialNorm)) {
return false;
}
}
}
// correction of the last substep (at t0 + step)
for (int i = 0; i < y0.length; ++i) {
yEnd[i] = 0.5 * (yTmp[i] + yEnd[i] + subStep * f[n][i]);
}
return true;
}
/** Extrapolate a vector.
* @param offset offset to use in the coefficients table
* @param k index of the last updated point
* @param diag working diagonal of the Aitken-Neville's
* triangle, without the last element
* @param last last element
*/
private void extrapolate(final int offset, final int k,
final double[][] diag, final double[] last) {
// update the diagonal
for (int j = 1; j < k; ++j) {
for (int i = 0; i < last.length; ++i) {
// Aitken-Neville's recursive formula
diag[k-j-1][i] = diag[k-j][i] +
coeff[k+offset][j-1] * (diag[k-j][i] - diag[k-j-1][i]);
}
}
// update the last element
for (int i = 0; i < last.length; ++i) {
// Aitken-Neville's recursive formula
last[i] = diag[0][i] + coeff[k+offset][k-1] * (diag[0][i] - last[i]);
}
}
/** {@inheritDoc} */
@Override
public double integrate(final FirstOrderDifferentialEquations equations,
final double t0, final double[] y0, final double t, final double[] y)
throws DerivativeException, IntegratorException {
sanityChecks(equations, t0, y0, t, y);
setEquations(equations);
resetEvaluations();
final boolean forward = (t > t0);
// create some internal working arrays
final double[] yDot0 = new double[y0.length];
final double[] y1 = new double[y0.length];
final double[] yTmp = new double[y0.length];
final double[] yTmpDot = new double[y0.length];
final double[][] diagonal = new double[sequence.length-1][];
final double[][] y1Diag = new double[sequence.length-1][];
for (int k = 0; k < sequence.length-1; ++k) {
diagonal[k] = new double[y0.length];
y1Diag[k] = new double[y0.length];
}
final double[][][] fk = new double[sequence.length][][];
for (int k = 0; k < sequence.length; ++k) {
fk[k] = new double[sequence[k] + 1][];
// all substeps start at the same point, so share the first array
fk[k][0] = yDot0;
for (int l = 0; l < sequence[k]; ++l) {
fk[k][l+1] = new double[y0.length];
}
}
if (y != y0) {
System.arraycopy(y0, 0, y, 0, y0.length);
}
double[] yDot1 = null;
double[][] yMidDots = null;
if (denseOutput) {
yDot1 = new double[y0.length];
yMidDots = new double[1 + 2 * sequence.length][];
for (int j = 0; j < yMidDots.length; ++j) {
yMidDots[j] = new double[y0.length];
}
} else {
yMidDots = new double[1][];
yMidDots[0] = new double[y0.length];
}
// initial scaling
final double[] scale = new double[y0.length];
rescale(y, y, scale);
// initial order selection
final double tol =
(vecRelativeTolerance == null) ? scalRelativeTolerance : vecRelativeTolerance[0];
final double log10R = Math.log(Math.max(1.0e-10, tol)) / Math.log(10.0);
int targetIter = Math.max(1,
Math.min(sequence.length - 2,
(int) Math.floor(0.5 - 0.6 * log10R)));
// set up an interpolator sharing the integrator arrays
AbstractStepInterpolator interpolator = null;
if (denseOutput || (! eventsHandlersManager.isEmpty())) {
interpolator = new GraggBulirschStoerStepInterpolator(y, yDot0,
y1, yDot1,
yMidDots, forward);
} else {
interpolator = new DummyStepInterpolator(y, forward);
}
interpolator.storeTime(t0);
stepStart = t0;
double hNew = 0;
double maxError = Double.MAX_VALUE;
boolean previousRejected = false;
boolean firstTime = true;
boolean newStep = true;
boolean lastStep = false;
boolean firstStepAlreadyComputed = false;
for (StepHandler handler : stepHandlers) {
handler.reset();
}
costPerTimeUnit[0] = 0;
while (! lastStep) {
double error;
boolean reject = false;
if (newStep) {
interpolator.shift();
// first evaluation, at the beginning of the step
if (! firstStepAlreadyComputed) {
computeDerivatives(stepStart, y, yDot0);
}
if (firstTime) {
hNew = initializeStep(equations, forward,
2 * targetIter + 1, scale,
stepStart, y, yDot0, yTmp, yTmpDot);
if (! forward) {
hNew = -hNew;
}
}
newStep = false;
}
stepSize = hNew;
// step adjustment near bounds
if ((forward && (stepStart + stepSize > t)) ||
((! forward) && (stepStart + stepSize < t))) {
stepSize = t - stepStart;
}
final double nextT = stepStart + stepSize;
lastStep = forward ? (nextT >= t) : (nextT <= t);
// iterate over several substep sizes
int k = -1;
for (boolean loop = true; loop; ) {
++k;
// modified midpoint integration with the current substep
if ( ! tryStep(stepStart, y, stepSize, k, scale, fk[k],
(k == 0) ? yMidDots[0] : diagonal[k-1],
(k == 0) ? y1 : y1Diag[k-1],
yTmp)) {
// the stability check failed, we reduce the global step
hNew = Math.abs(filterStep(stepSize * stabilityReduction, forward, false));
reject = true;
loop = false;
} else {
// the substep was computed successfully
if (k > 0) {
// extrapolate the state at the end of the step
// using last iteration data
extrapolate(0, k, y1Diag, y1);
rescale(y, y1, scale);
// estimate the error at the end of the step.
error = 0;
for (int j = 0; j < y0.length; ++j) {
final double e = Math.abs(y1[j] - y1Diag[0][j]) / scale[j];
error += e * e;
}
error = Math.sqrt(error / y0.length);
if ((error > 1.0e15) || ((k > 1) && (error > maxError))) {
// error is too big, we reduce the global step
hNew = Math.abs(filterStep(stepSize * stabilityReduction, forward, false));
reject = true;
loop = false;
} else {
maxError = Math.max(4 * error, 1.0);
// compute optimal stepsize for this order
final double exp = 1.0 / (2 * k + 1);
double fac = stepControl2 / Math.pow(error / stepControl1, exp);
final double pow = Math.pow(stepControl3, exp);
fac = Math.max(pow / stepControl4, Math.min(1 / pow, fac));
optimalStep[k] = Math.abs(filterStep(stepSize * fac, forward, true));
costPerTimeUnit[k] = costPerStep[k] / optimalStep[k];
// check convergence
switch (k - targetIter) {
case -1 :
if ((targetIter > 1) && ! previousRejected) {
// check if we can stop iterations now
if (error <= 1.0) {
// convergence have been reached just before targetIter
loop = false;
} else {
// estimate if there is a chance convergence will
// be reached on next iteration, using the
// asymptotic evolution of error
final double ratio = ((double) sequence [k] * sequence[k+1]) /
(sequence[0] * sequence[0]);
if (error > ratio * ratio) {
// we don't expect to converge on next iteration
// we reject the step immediately and reduce order
reject = true;
loop = false;
targetIter = k;
if ((targetIter > 1) &&
(costPerTimeUnit[targetIter-1] <
orderControl1 * costPerTimeUnit[targetIter])) {
--targetIter;
}
hNew = optimalStep[targetIter];
}
}
}
break;
case 0:
if (error <= 1.0) {
// convergence has been reached exactly at targetIter
loop = false;
} else {
// estimate if there is a chance convergence will
// be reached on next iteration, using the
// asymptotic evolution of error
final double ratio = ((double) sequence[k+1]) / sequence[0];
if (error > ratio * ratio) {
// we don't expect to converge on next iteration
// we reject the step immediately
reject = true;
loop = false;
if ((targetIter > 1) &&
(costPerTimeUnit[targetIter-1] <
orderControl1 * costPerTimeUnit[targetIter])) {
--targetIter;
}
hNew = optimalStep[targetIter];
}
}
break;
case 1 :
if (error > 1.0) {
reject = true;
if ((targetIter > 1) &&
(costPerTimeUnit[targetIter-1] <
orderControl1 * costPerTimeUnit[targetIter])) {
--targetIter;
}
hNew = optimalStep[targetIter];
}
loop = false;
break;
default :
if ((firstTime || lastStep) && (error <= 1.0)) {
loop = false;
}
break;
}
}
}
}
}
// dense output handling
double hInt = getMaxStep();
if (denseOutput && ! reject) {
// extrapolate state at middle point of the step
for (int j = 1; j <= k; ++j) {
extrapolate(0, j, diagonal, yMidDots[0]);
}
// derivative at end of step
computeDerivatives(stepStart + stepSize, y1, yDot1);
final int mu = 2 * k - mudif + 3;
for (int l = 0; l < mu; ++l) {
// derivative at middle point of the step
final int l2 = l / 2;
double factor = Math.pow(0.5 * sequence[l2], l);
int middleIndex = fk[l2].length / 2;
for (int i = 0; i < y0.length; ++i) {
yMidDots[l+1][i] = factor * fk[l2][middleIndex + l][i];
}
for (int j = 1; j <= k - l2; ++j) {
factor = Math.pow(0.5 * sequence[j + l2], l);
middleIndex = fk[l2+j].length / 2;
for (int i = 0; i < y0.length; ++i) {
diagonal[j-1][i] = factor * fk[l2+j][middleIndex+l][i];
}
extrapolate(l2, j, diagonal, yMidDots[l+1]);
}
for (int i = 0; i < y0.length; ++i) {
yMidDots[l+1][i] *= stepSize;
}
// compute centered differences to evaluate next derivatives
for (int j = (l + 1) / 2; j <= k; ++j) {
for (int m = fk[j].length - 1; m >= 2 * (l + 1); --m) {
for (int i = 0; i < y0.length; ++i) {
fk[j][m][i] -= fk[j][m-2][i];
}
}
}
}
if (mu >= 0) {
// estimate the dense output coefficients
final GraggBulirschStoerStepInterpolator gbsInterpolator
= (GraggBulirschStoerStepInterpolator) interpolator;
gbsInterpolator.computeCoefficients(mu, stepSize);
if (useInterpolationError) {
// use the interpolation error to limit stepsize
final double interpError = gbsInterpolator.estimateError(scale);
hInt = Math.abs(stepSize / Math.max(Math.pow(interpError, 1.0 / (mu+4)),
0.01));
if (interpError > 10.0) {
hNew = hInt;
reject = true;
}
}
// Discrete events handling
if (!reject) {
interpolator.storeTime(stepStart + stepSize);
if (eventsHandlersManager.evaluateStep(interpolator)) {
final double dt = eventsHandlersManager.getEventTime() - stepStart;
if (Math.abs(dt) > Math.ulp(stepStart)) {
// reject the step to match exactly the next switch time
hNew = Math.abs(dt);
reject = true;
}
}
}
}
if (!reject) {
// we will reuse the slope for the beginning of next step
firstStepAlreadyComputed = true;
System.arraycopy(yDot1, 0, yDot0, 0, y0.length);
}
}
if (! reject) {
// store end of step state
final double nextStep = stepStart + stepSize;
System.arraycopy(y1, 0, y, 0, y0.length);
eventsHandlersManager.stepAccepted(nextStep, y);
if (eventsHandlersManager.stop()) {
lastStep = true;
}
// provide the step data to the step handler
interpolator.storeTime(nextStep);
for (StepHandler handler : stepHandlers) {
handler.handleStep(interpolator, lastStep);
}
stepStart = nextStep;
if (eventsHandlersManager.reset(stepStart, y) && ! lastStep) {
// some switching function has triggered changes that
// invalidate the derivatives, we need to recompute them
firstStepAlreadyComputed = false;
}
int optimalIter;
if (k == 1) {
optimalIter = 2;
if (previousRejected) {
optimalIter = 1;
}
} else if (k <= targetIter) {
optimalIter = k;
if (costPerTimeUnit[k-1] < orderControl1 * costPerTimeUnit[k]) {
optimalIter = k-1;
} else if (costPerTimeUnit[k] < orderControl2 * costPerTimeUnit[k-1]) {
optimalIter = Math.min(k+1, sequence.length - 2);
}
} else {
optimalIter = k - 1;
if ((k > 2) &&
(costPerTimeUnit[k-2] < orderControl1 * costPerTimeUnit[k-1])) {
optimalIter = k - 2;
}
if (costPerTimeUnit[k] < orderControl2 * costPerTimeUnit[optimalIter]) {
optimalIter = Math.min(k, sequence.length - 2);
}
}
if (previousRejected) {
// after a rejected step neither order nor stepsize
// should increase
targetIter = Math.min(optimalIter, k);
hNew = Math.min(Math.abs(stepSize), optimalStep[targetIter]);
} else {
// stepsize control
if (optimalIter <= k) {
hNew = optimalStep[optimalIter];
} else {
if ((k < targetIter) &&
(costPerTimeUnit[k] < orderControl2 * costPerTimeUnit[k-1])) {
hNew = filterStep(optimalStep[k] * costPerStep[optimalIter+1] / costPerStep[k],
forward, false);
} else {
hNew = filterStep(optimalStep[k] * costPerStep[optimalIter] / costPerStep[k],
forward, false);
}
}
targetIter = optimalIter;
}
newStep = true;
}
hNew = Math.min(hNew, hInt);
if (! forward) {
hNew = -hNew;
}
firstTime = false;
if (reject) {
lastStep = false;
previousRejected = true;
} else {
previousRejected = false;
}
}
return stepStart;
}
/** maximal order. */
private int maxOrder;
/** step size sequence. */
private int[] sequence;
/** overall cost of applying step reduction up to iteration k+1,
* in number of calls.
*/
private int[] costPerStep;
/** cost per unit step. */
private double[] costPerTimeUnit;
/** optimal steps for each order. */
private double[] optimalStep;
/** extrapolation coefficients. */
private double[][] coeff;
/** stability check enabling parameter. */
private boolean performTest;
/** maximal number of checks for each iteration. */
private int maxChecks;
/** maximal number of iterations for which checks are performed. */
private int maxIter;
/** stepsize reduction factor in case of stability check failure. */
private double stabilityReduction;
/** first stepsize control factor. */
private double stepControl1;
/** second stepsize control factor. */
private double stepControl2;
/** third stepsize control factor. */
private double stepControl3;
/** fourth stepsize control factor. */
private double stepControl4;
/** first order control factor. */
private double orderControl1;
/** second order control factor. */
private double orderControl2;
/** dense outpute required. */
private boolean denseOutput;
/** use interpolation error in stepsize control. */
private boolean useInterpolationError;
/** interpolation order control parameter. */
private int mudif;
}