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* 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.numbers.rootfinder;
import java.util.function.DoubleUnaryOperator;
/**
* This class implements the
* Brent algorithm 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.
* 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
*
*/
public class BrentSolver {
/** Relative accuracy. */
private final double relativeAccuracy;
/** Absolute accuracy. */
private final double absoluteAccuracy;
/** Function accuracy. */
private final double functionValueAccuracy;
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
* @param functionValueAccuracy Function value accuracy.
*/
public BrentSolver(double relativeAccuracy,
double absoluteAccuracy,
double functionValueAccuracy) {
this.relativeAccuracy = relativeAccuracy;
this.absoluteAccuracy = absoluteAccuracy;
this.functionValueAccuracy = functionValueAccuracy;
}
/**
* Search the function's zero within the given interval.
*
* @param func Function to solve.
* @param min Lower bound.
* @param max Upper bound.
* @return the root.
* @throws IllegalArgumentException if {@code min > max}.
* @throws IllegalArgumentException if the given interval does
* not bracket the root.
*/
public double findRoot(DoubleUnaryOperator func,
double min,
double max) {
// Avoid overflow computing the initial value: 0.5 * (min + max)
// Note: This sum is invalid if min == max == Double.MIN_VALUE
// so detect this edge case. It will raise a bracketing exception
// if min is not the root within the configured function accuracy;
// otherwise min is returned.
final double initial = min == max ? min : 0.5 * min + 0.5 * max;
return findRoot(func, min, initial, max);
}
/**
* Search the function's zero within the given interval,
* starting from the given estimate.
*
* @param func Function to solve.
* @param min Lower bound.
* @param initial Initial guess.
* @param max Upper bound.
* @return the root.
* @throws IllegalArgumentException if {@code min > max} or
* {@code initial} is not in the {@code [min, max]} interval.
* @throws IllegalArgumentException if the given interval does
* not bracket the root.
*/
public double findRoot(DoubleUnaryOperator func,
double min,
double initial,
double max) {
if (min > max) {
throw new SolverException(SolverException.TOO_LARGE, min, max);
}
if (initial < min ||
initial > max) {
throw new SolverException(SolverException.OUT_OF_RANGE, initial, min, max);
}
// Return the initial guess if it is good enough.
final double yInitial = func.applyAsDouble(initial);
if (Math.abs(yInitial) <= functionValueAccuracy) {
return initial;
}
// Return the first endpoint if it is good enough.
final double yMin = func.applyAsDouble(min);
if (Math.abs(yMin) <= functionValueAccuracy) {
return min;
}
// Reduce interval if min and initial bracket the root.
if (Double.compare(yInitial * yMin, 0.0) < 0) {
return brent(func, min, initial, yMin, yInitial);
}
// Return the second endpoint if it is good enough.
final double yMax = func.applyAsDouble(max);
if (Math.abs(yMax) <= functionValueAccuracy) {
return max;
}
// Reduce interval if initial and max bracket the root.
if (Double.compare(yInitial * yMax, 0.0) < 0) {
return brent(func, initial, max, yInitial, yMax);
}
throw new SolverException(SolverException.BRACKETING, min, yMin, max, 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
*
*
* @param func Function to solve.
* @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(DoubleUnaryOperator func,
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 = absoluteAccuracy;
final double eps = relativeAccuracy;
while (true) {
if (Math.abs(fc) < Math.abs(fb)) {
a = b;
b = c;
c = a;
fa = fb;
fb = fc;
fc = fa;
}
final double tol = 2 * eps * Math.abs(b) + t;
final double m = 0.5 * (c - b);
if (Math.abs(m) <= tol ||
equalsZero(fb)) {
return b;
}
if (Math.abs(e) < tol ||
Math.abs(fa) <= Math.abs(fb)) {
// Force bisection.
d = m;
e = d;
} else {
final 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;
}
if (p >= 1.5 * m * q - Math.abs(tol * q) ||
p >= Math.abs(0.5 * e * q)) {
// Inverse quadratic interpolation gives a value
// in the wrong direction, or progress is slow.
// Fall back to bisection.
d = m;
e = d;
} else {
e = d;
d = p / q;
}
}
a = b;
fa = fb;
if (Math.abs(d) > tol) {
b += d;
} else if (m > 0) {
b += tol;
} else {
b -= tol;
}
fb = func.applyAsDouble(b);
if ((fb > 0 && fc > 0) ||
(fb <= 0 && fc <= 0)) {
c = a;
fc = fa;
d = b - a;
e = d;
}
}
}
/**
* Return true if the value is within 1 ULP of zero.
*
* @param value Value
* @return true if zero within a 1 ULP tolerance
*/
private static boolean equalsZero(double value) {
return Math.abs(value) <= Double.MIN_VALUE;
}
}
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