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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.optimization.univariate;
import org.apache.commons.math3.util.Precision;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.optimization.ConvergenceChecker;
import org.apache.commons.math3.optimization.GoalType;
/**
* Implements Richard Brent's algorithm (from his book "Algorithms for
* Minimization without Derivatives", p. 79) for finding minima of real
* univariate functions. This implementation is an adaptation partly
* based on the Python code from SciPy (module "optimize.py" v0.5).
* If the function is defined on some interval {@code (lo, hi)}, then
* this method finds an approximation {@code x} to the point at which
* the function attains its minimum.
*
* @version $Id: BrentOptimizer.java 1244107 2012-02-14 16:17:55Z erans $
* @since 2.0
*/
public class BrentOptimizer extends BaseAbstractUnivariateOptimizer {
/**
* Golden section.
*/
private static final double GOLDEN_SECTION = 0.5 * (3 - FastMath.sqrt(5));
/**
* Minimum relative tolerance.
*/
private static final double MIN_RELATIVE_TOLERANCE = 2 * FastMath.ulp(1d);
/**
* Relative threshold.
*/
private final double relativeThreshold;
/**
* Absolute threshold.
*/
private final double absoluteThreshold;
/**
* The arguments are used implement the original stopping criterion
* of Brent's algorithm.
* {@code abs} and {@code rel} define a tolerance
* {@code tol = rel |x| + abs}. {@code rel} should be no smaller than
* 2 macheps and preferably not much less than sqrt(macheps),
* where macheps is the relative machine precision. {@code abs} must
* be positive.
*
* @param rel Relative threshold.
* @param abs Absolute threshold.
* @param checker Additional, user-defined, convergence checking
* procedure.
* @throws NotStrictlyPositiveException if {@code abs <= 0}.
* @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
*/
public BrentOptimizer(double rel,
double abs,
ConvergenceChecker checker) {
super(checker);
if (rel < MIN_RELATIVE_TOLERANCE) {
throw new NumberIsTooSmallException(rel, MIN_RELATIVE_TOLERANCE, true);
}
if (abs <= 0) {
throw new NotStrictlyPositiveException(abs);
}
relativeThreshold = rel;
absoluteThreshold = abs;
}
/**
* The arguments are used implement the original stopping criterion
* of Brent's algorithm.
* {@code abs} and {@code rel} define a tolerance
* {@code tol = rel |x| + abs}. {@code rel} should be no smaller than
* 2 macheps and preferably not much less than sqrt(macheps),
* where macheps is the relative machine precision. {@code abs} must
* be positive.
*
* @param rel Relative threshold.
* @param abs Absolute threshold.
* @throws NotStrictlyPositiveException if {@code abs <= 0}.
* @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
*/
public BrentOptimizer(double rel,
double abs) {
this(rel, abs, null);
}
/** {@inheritDoc} */
@Override
protected UnivariatePointValuePair doOptimize() {
final boolean isMinim = getGoalType() == GoalType.MINIMIZE;
final double lo = getMin();
final double mid = getStartValue();
final double hi = getMax();
// Optional additional convergence criteria.
final ConvergenceChecker checker
= getConvergenceChecker();
double a;
double b;
if (lo < hi) {
a = lo;
b = hi;
} else {
a = hi;
b = lo;
}
double x = mid;
double v = x;
double w = x;
double d = 0;
double e = 0;
double fx = computeObjectiveValue(x);
if (!isMinim) {
fx = -fx;
}
double fv = fx;
double fw = fx;
UnivariatePointValuePair previous = null;
UnivariatePointValuePair current
= new UnivariatePointValuePair(x, isMinim ? fx : -fx);
int iter = 0;
while (true) {
final double m = 0.5 * (a + b);
final double tol1 = relativeThreshold * FastMath.abs(x) + absoluteThreshold;
final double tol2 = 2 * tol1;
// Default stopping criterion.
final boolean stop = FastMath.abs(x - m) <= tol2 - 0.5 * (b - a);
if (!stop) {
double p = 0;
double q = 0;
double r = 0;
double u = 0;
if (FastMath.abs(e) > tol1) { // Fit parabola.
r = (x - w) * (fx - fv);
q = (x - v) * (fx - fw);
p = (x - v) * q - (x - w) * r;
q = 2 * (q - r);
if (q > 0) {
p = -p;
} else {
q = -q;
}
r = e;
e = d;
if (p > q * (a - x) &&
p < q * (b - x) &&
FastMath.abs(p) < FastMath.abs(0.5 * q * r)) {
// Parabolic interpolation step.
d = p / q;
u = x + d;
// f must not be evaluated too close to a or b.
if (u - a < tol2 || b - u < tol2) {
if (x <= m) {
d = tol1;
} else {
d = -tol1;
}
}
} else {
// Golden section step.
if (x < m) {
e = b - x;
} else {
e = a - x;
}
d = GOLDEN_SECTION * e;
}
} else {
// Golden section step.
if (x < m) {
e = b - x;
} else {
e = a - x;
}
d = GOLDEN_SECTION * e;
}
// Update by at least "tol1".
if (FastMath.abs(d) < tol1) {
if (d >= 0) {
u = x + tol1;
} else {
u = x - tol1;
}
} else {
u = x + d;
}
double fu = computeObjectiveValue(u);
if (!isMinim) {
fu = -fu;
}
// Update a, b, v, w and x.
if (fu <= fx) {
if (u < x) {
b = x;
} else {
a = x;
}
v = w;
fv = fw;
w = x;
fw = fx;
x = u;
fx = fu;
} else {
if (u < x) {
a = u;
} else {
b = u;
}
if (fu <= fw ||
Precision.equals(w, x)) {
v = w;
fv = fw;
w = u;
fw = fu;
} else if (fu <= fv ||
Precision.equals(v, x) ||
Precision.equals(v, w)) {
v = u;
fv = fu;
}
}
previous = current;
current = new UnivariatePointValuePair(x, isMinim ? fx : -fx);
// User-defined convergence checker.
if (checker != null) {
if (checker.converged(iter, previous, current)) {
return current;
}
}
} else { // Default termination (Brent's criterion).
return current;
}
++iter;
}
}
}
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