/*
* 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.analysis.solvers;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Precision;
This class implements the
Brent algorithm for finding zeros of real univariate functions. The function should be continuous but not necessarily smooth. The solve method returns a zero x of the function f in the given interval [a, b] to within a tolerance 2 eps abs(x) + t where eps is the relative accuracy and t is the absolute accuracy. The given interval must bracket the root.
The reference implementation is given in chapter 4 of
Algorithms for Minimization Without Derivatives,
Richard P. Brent,
Dover, 2002
See Also:
/**
* This class implements the <a href="http://mathworld.wolfram.com/BrentsMethod.html">
* Brent algorithm</a> for finding zeros of real univariate functions.
* The function should be continuous but not necessarily smooth.
* The {@code solve} method returns a zero {@code x} of the function {@code f}
* in the given interval {@code [a, b]} to within a tolerance
* {@code 2 eps abs(x) + t} where {@code eps} is the relative accuracy and
* {@code t} is the absolute accuracy.
* <p>The given interval must bracket the root.</p>
* <p>
* The reference implementation is given in chapter 4 of
* <blockquote>
* <b>Algorithms for Minimization Without Derivatives</b>,
* <em>Richard P. Brent</em>,
* Dover, 2002
* </blockquote>
*
* @see BaseAbstractUnivariateSolver
*/
public class BrentSolver extends AbstractUnivariateSolver {
Default absolute accuracy. /** Default absolute accuracy. */
private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
Construct a solver with default absolute accuracy (1e-6).
/**
* Construct a solver with default absolute accuracy (1e-6).
*/
public BrentSolver() {
this(DEFAULT_ABSOLUTE_ACCURACY);
}
Construct a solver.
Params: - absoluteAccuracy – Absolute accuracy.
/**
* Construct a solver.
*
* @param absoluteAccuracy Absolute accuracy.
*/
public BrentSolver(double absoluteAccuracy) {
super(absoluteAccuracy);
}
Construct a solver.
Params: - relativeAccuracy – Relative accuracy.
- absoluteAccuracy – Absolute accuracy.
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
*/
public BrentSolver(double relativeAccuracy,
double absoluteAccuracy) {
super(relativeAccuracy, absoluteAccuracy);
}
Construct a solver.
Params: - relativeAccuracy – Relative accuracy.
- absoluteAccuracy – Absolute accuracy.
- functionValueAccuracy – Function value accuracy.
See Also:
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
* @param functionValueAccuracy Function value accuracy.
*
* @see BaseAbstractUnivariateSolver#BaseAbstractUnivariateSolver(double,double,double)
*/
public BrentSolver(double relativeAccuracy,
double absoluteAccuracy,
double functionValueAccuracy) {
super(relativeAccuracy, absoluteAccuracy, functionValueAccuracy);
}
{@inheritDoc}
/**
* {@inheritDoc}
*/
@Override
protected double doSolve()
throws NoBracketingException,
TooManyEvaluationsException,
NumberIsTooLargeException {
double min = getMin();
double max = getMax();
final double initial = getStartValue();
final double functionValueAccuracy = getFunctionValueAccuracy();
verifySequence(min, initial, max);
// Return the initial guess if it is good enough.
double yInitial = computeObjectiveValue(initial);
if (FastMath.abs(yInitial) <= functionValueAccuracy) {
return initial;
}
// Return the first endpoint if it is good enough.
double yMin = computeObjectiveValue(min);
if (FastMath.abs(yMin) <= functionValueAccuracy) {
return min;
}
// Reduce interval if min and initial bracket the root.
if (yInitial * yMin < 0) {
return brent(min, initial, yMin, yInitial);
}
// Return the second endpoint if it is good enough.
double yMax = computeObjectiveValue(max);
if (FastMath.abs(yMax) <= functionValueAccuracy) {
return max;
}
// Reduce interval if initial and max bracket the root.
if (yInitial * yMax < 0) {
return brent(initial, max, yInitial, yMax);
}
throw new NoBracketingException(min, max, yMin, yMax);
}
Search for a zero inside the provided interval.
This implementation is based on the algorithm described at page 58 of
the book
Algorithms for Minimization Without Derivatives,
Richard P. Brent,
Dover 0-486-41998-3
Params: - lo – Lower bound of the search interval.
- hi – Higher bound of the search interval.
- fLo – Function value at the lower bound of the search interval.
- fHi – Function value at the higher bound of the search interval.
Returns: the value where the function is zero.
/**
* Search for a zero inside the provided interval.
* This implementation is based on the algorithm described at page 58 of
* the book
* <blockquote>
* <b>Algorithms for Minimization Without Derivatives</b>,
* <it>Richard P. Brent</it>,
* Dover 0-486-41998-3
* </blockquote>
*
* @param lo Lower bound of the search interval.
* @param hi Higher bound of the search interval.
* @param fLo Function value at the lower bound of the search interval.
* @param fHi Function value at the higher bound of the search interval.
* @return the value where the function is zero.
*/
private double brent(double lo, double hi,
double fLo, double fHi) {
double a = lo;
double fa = fLo;
double b = hi;
double fb = fHi;
double c = a;
double fc = fa;
double d = b - a;
double e = d;
final double t = getAbsoluteAccuracy();
final double eps = getRelativeAccuracy();
while (true) {
if (FastMath.abs(fc) < FastMath.abs(fb)) {
a = b;
b = c;
c = a;
fa = fb;
fb = fc;
fc = fa;
}
final double tol = 2 * eps * FastMath.abs(b) + t;
final double m = 0.5 * (c - b);
if (FastMath.abs(m) <= tol ||
Precision.equals(fb, 0)) {
return b;
}
if (FastMath.abs(e) < tol ||
FastMath.abs(fa) <= FastMath.abs(fb)) {
// Force bisection.
d = m;
e = d;
} else {
double s = fb / fa;
double p;
double q;
// The equality test (a == c) is intentional,
// it is part of the original Brent's method and
// it should NOT be replaced by proximity test.
if (a == c) {
// Linear interpolation.
p = 2 * m * s;
q = 1 - s;
} else {
// Inverse quadratic interpolation.
q = fa / fc;
final double r = fb / fc;
p = s * (2 * m * q * (q - r) - (b - a) * (r - 1));
q = (q - 1) * (r - 1) * (s - 1);
}
if (p > 0) {
q = -q;
} else {
p = -p;
}
s = e;
e = d;
if (p >= 1.5 * m * q - FastMath.abs(tol * q) ||
p >= FastMath.abs(0.5 * s * q)) {
// Inverse quadratic interpolation gives a value
// in the wrong direction, or progress is slow.
// Fall back to bisection.
d = m;
e = d;
} else {
d = p / q;
}
}
a = b;
fa = fb;
if (FastMath.abs(d) > tol) {
b += d;
} else if (m > 0) {
b += tol;
} else {
b -= tol;
}
fb = computeObjectiveValue(b);
if ((fb > 0 && fc > 0) ||
(fb <= 0 && fc <= 0)) {
c = a;
fc = fa;
d = b - a;
e = d;
}
}
}
}