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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.ode.AbstractFieldIntegrator;
import org.apache.commons.math3.ode.FieldEquationsMapper;
import org.apache.commons.math3.ode.FieldExpandableODE;
import org.apache.commons.math3.ode.FirstOrderFieldDifferentialEquations;
import org.apache.commons.math3.ode.FieldODEState;
import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
import org.apache.commons.math3.util.MathArrays;
/**
* This class implements the common part of all fixed step Runge-Kutta
* integrators for Ordinary Differential Equations.
*
* These methods are explicit Runge-Kutta methods, their Butcher
* arrays are as follows :
*
* 0 |
* c2 | a21
* c3 | a31 a32
* ... | ...
* cs | as1 as2 ... ass-1
* |--------------------------
* | b1 b2 ... bs-1 bs
*
*
*
* @see EulerFieldIntegrator
* @see ClassicalRungeKuttaFieldIntegrator
* @see GillFieldIntegrator
* @see MidpointFieldIntegrator
* @param the type of the field elements
* @since 3.6
*/
public abstract class RungeKuttaFieldIntegrator>
extends AbstractFieldIntegrator
implements FieldButcherArrayProvider {
/** Time steps from Butcher array (without the first zero). */
private final T[] c;
/** Internal weights from Butcher array (without the first empty row). */
private final T[][] a;
/** External weights for the high order method from Butcher array. */
private final T[] b;
/** Integration step. */
private final T step;
/** Simple constructor.
* Build a Runge-Kutta integrator with the given
* step. The default step handler does nothing.
* @param field field to which the time and state vector elements belong
* @param name name of the method
* @param step integration step
*/
protected RungeKuttaFieldIntegrator(final Field field, final String name, final T step) {
super(field, name);
this.c = getC();
this.a = getA();
this.b = getB();
this.step = step.abs();
}
/** Create a fraction.
* @param p numerator
* @param q denominator
* @return p/q computed in the instance field
*/
protected T fraction(final int p, final int q) {
return getField().getZero().add(p).divide(q);
}
/** Create an interpolator.
* @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 mapper equations mapper for the all equations
* @return external weights for the high order method from Butcher array
*/
protected abstract RungeKuttaFieldStepInterpolator createInterpolator(boolean forward, T[][] yDotK,
final FieldODEStateAndDerivative globalPreviousState,
final FieldODEStateAndDerivative globalCurrentState,
FieldEquationsMapper mapper);
/** {@inheritDoc} */
public FieldODEStateAndDerivative integrate(final FieldExpandableODE equations,
final FieldODEState initialState, final T finalTime)
throws NumberIsTooSmallException, DimensionMismatchException,
MaxCountExceededException, NoBracketingException {
sanityChecks(initialState, finalTime);
final T t0 = initialState.getTime();
final T[] y0 = equations.getMapper().mapState(initialState);
setStepStart(initIntegration(equations, t0, y0, finalTime));
final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;
// create some internal working arrays
final int stages = c.length + 1;
T[] y = y0;
final T[][] yDotK = MathArrays.buildArray(getField(), stages, -1);
final T[] yTmp = MathArrays.buildArray(getField(), y0.length);
// set up integration control objects
if (forward) {
if (getStepStart().getTime().add(step).subtract(finalTime).getReal() >= 0) {
setStepSize(finalTime.subtract(getStepStart().getTime()));
} else {
setStepSize(step);
}
} else {
if (getStepStart().getTime().subtract(step).subtract(finalTime).getReal() <= 0) {
setStepSize(finalTime.subtract(getStepStart().getTime()));
} else {
setStepSize(step.negate());
}
}
// main integration loop
setIsLastStep(false);
do {
// first stage
y = equations.getMapper().mapState(getStepStart());
yDotK[0] = equations.getMapper().mapDerivative(getStepStart());
// next stages
for (int k = 1; k < stages; ++k) {
for (int j = 0; j < y0.length; ++j) {
T sum = yDotK[0][j].multiply(a[k-1][0]);
for (int l = 1; l < k; ++l) {
sum = sum.add(yDotK[l][j].multiply(a[k-1][l]));
}
yTmp[j] = y[j].add(getStepSize().multiply(sum));
}
yDotK[k] = computeDerivatives(getStepStart().getTime().add(getStepSize().multiply(c[k-1])), yTmp);
}
// estimate the state at the end of the step
for (int j = 0; j < y0.length; ++j) {
T sum = yDotK[0][j].multiply(b[0]);
for (int l = 1; l < stages; ++l) {
sum = sum.add(yDotK[l][j].multiply(b[l]));
}
yTmp[j] = y[j].add(getStepSize().multiply(sum));
}
final T stepEnd = getStepStart().getTime().add(getStepSize());
final T[] yDotTmp = computeDerivatives(stepEnd, yTmp);
final FieldODEStateAndDerivative stateTmp = new FieldODEStateAndDerivative(stepEnd, yTmp, yDotTmp);
// discrete events handling
System.arraycopy(yTmp, 0, y, 0, y0.length);
setStepStart(acceptStep(createInterpolator(forward, yDotK, getStepStart(), stateTmp, equations.getMapper()),
finalTime));
if (!isLastStep()) {
// stepsize control for next step
final T nextT = getStepStart().getTime().add(getStepSize());
final boolean nextIsLast = forward ?
(nextT.subtract(finalTime).getReal() >= 0) :
(nextT.subtract(finalTime).getReal() <= 0);
if (nextIsLast) {
setStepSize(finalTime.subtract(getStepStart().getTime()));
}
}
} while (!isLastStep());
final FieldODEStateAndDerivative finalState = getStepStart();
setStepStart(null);
setStepSize(null);
return finalState;
}
/** Fast computation of a single step of ODE integration.
* This method is intended for the limited use case of
* very fast computation of only one step without using any of the
* rich features of general integrators that may take some time
* to set up (i.e. no step handlers, no events handlers, no additional
* states, no interpolators, no error control, no evaluations count,
* no sanity checks ...). It handles the strict minimum of computation,
* so it can be embedded in outer loops.
*
* This method is not used at all by the {@link #integrate(FieldExpandableODE,
* FieldODEState, RealFieldElement)} method. It also completely ignores the step set at
* construction time, and uses only a single step to go from {@code t0} to {@code t}.
*
*
* As this method does not use any of the state-dependent features of the integrator,
* it should be reasonably thread-safe if and only if the provided differential
* equations are themselves thread-safe.
*
* @param equations differential equations to integrate
* @param t0 initial time
* @param y0 initial value of the state vector at t0
* @param t target time for the integration
* (can be set to a value smaller than {@code t0} for backward integration)
* @return state vector at {@code t}
*/
public T[] singleStep(final FirstOrderFieldDifferentialEquations equations,
final T t0, final T[] y0, final T t) {
// create some internal working arrays
final T[] y = y0.clone();
final int stages = c.length + 1;
final T[][] yDotK = MathArrays.buildArray(getField(), stages, -1);
final T[] yTmp = y0.clone();
// first stage
final T h = t.subtract(t0);
yDotK[0] = equations.computeDerivatives(t0, y);
// next stages
for (int k = 1; k < stages; ++k) {
for (int j = 0; j < y0.length; ++j) {
T sum = yDotK[0][j].multiply(a[k-1][0]);
for (int l = 1; l < k; ++l) {
sum = sum.add(yDotK[l][j].multiply(a[k-1][l]));
}
yTmp[j] = y[j].add(h.multiply(sum));
}
yDotK[k] = equations.computeDerivatives(t0.add(h.multiply(c[k-1])), yTmp);
}
// estimate the state at the end of the step
for (int j = 0; j < y0.length; ++j) {
T sum = yDotK[0][j].multiply(b[0]);
for (int l = 1; l < stages; ++l) {
sum = sum.add(yDotK[l][j].multiply(b[l]));
}
y[j] = y[j].add(h.multiply(sum));
}
return y;
}
}