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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.math.analysis.solvers;
import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.MathRuntimeException;
import org.apache.commons.math.MaxIterationsExceededException;
import org.apache.commons.math.analysis.UnivariateRealFunction;
/**
* Implements the
* Brent algorithm for finding zeros of real univariate functions.
*
* The function should be continuous but not necessarily smooth.
*
* @version $Revision:670469 $ $Date:2008-06-23 10:01:38 +0200 (lun., 23 juin 2008) $
*/
public class BrentSolver extends UnivariateRealSolverImpl {
/** Error message for non-bracketing interval. */
private static final String NON_BRACKETING_MESSAGE =
"function values at endpoints do not have different signs. " +
"Endpoints: [{0}, {1}], Values: [{2}, {3}]";
/** Serializable version identifier */
private static final long serialVersionUID = 7694577816772532779L;
/**
* Construct a solver for the given function.
*
* @param f function to solve.
* @deprecated as of 2.0 the function to solve is passed as an argument
* to the {@link #solve(UnivariateRealFunction, double, double)} or
* {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)}
* method.
*/
@Deprecated
public BrentSolver(UnivariateRealFunction f) {
super(f, 100, 1E-6);
}
/**
* Construct a solver.
*/
public BrentSolver() {
super(100, 1E-6);
}
/** {@inheritDoc} */
@Deprecated
public double solve(double min, double max)
throws MaxIterationsExceededException, FunctionEvaluationException {
return solve(f, min, max);
}
/** {@inheritDoc} */
@Deprecated
public double solve(double min, double max, double initial)
throws MaxIterationsExceededException, FunctionEvaluationException {
return solve(f, min, max, initial);
}
/**
* Find a zero in the given interval with an initial guess.
* Throws IllegalArgumentException if the values of the
* function at the three points have the same sign (note that it is
* allowed to have endpoints with the same sign if the initial point has
* opposite sign function-wise).
*
* @param f function to solve.
* @param min the lower bound for the interval.
* @param max the upper bound for the interval.
* @param initial the start value to use (must be set to min if no
* initial point is known).
* @return the value where the function is zero
* @throws MaxIterationsExceededException the maximum iteration count
* is exceeded
* @throws FunctionEvaluationException if an error occurs evaluating
* the function
* @throws IllegalArgumentException if initial is not between min and max
* (even if it is a root)
*/
public double solve(final UnivariateRealFunction f,
final double min, final double max, final double initial)
throws MaxIterationsExceededException, FunctionEvaluationException {
clearResult();
verifySequence(min, initial, max);
// return the initial guess if it is good enough
double yInitial = f.value(initial);
if (Math.abs(yInitial) <= functionValueAccuracy) {
setResult(initial, 0);
return result;
}
// return the first endpoint if it is good enough
double yMin = f.value(min);
if (Math.abs(yMin) <= functionValueAccuracy) {
setResult(yMin, 0);
return result;
}
// reduce interval if min and initial bracket the root
if (yInitial * yMin < 0) {
return solve(f, min, yMin, initial, yInitial, min, yMin);
}
// return the second endpoint if it is good enough
double yMax = f.value(max);
if (Math.abs(yMax) <= functionValueAccuracy) {
setResult(yMax, 0);
return result;
}
// reduce interval if initial and max bracket the root
if (yInitial * yMax < 0) {
return solve(f, initial, yInitial, max, yMax, initial, yInitial);
}
// full Brent algorithm starting with provided initial guess
return solve(f, min, yMin, max, yMax, initial, yInitial);
}
/**
* Find a zero in the given interval.
*
* Requires that the values of the function at the endpoints have opposite
* signs. An IllegalArgumentException is thrown if this is not
* the case.
*
* @param f the function to solve
* @param min the lower bound for the interval.
* @param max the upper bound for the interval.
* @return the value where the function is zero
* @throws MaxIterationsExceededException if the maximum iteration count is exceeded
* @throws FunctionEvaluationException if an error occurs evaluating the
* function
* @throws IllegalArgumentException if min is not less than max or the
* signs of the values of the function at the endpoints are not opposites
*/
public double solve(final UnivariateRealFunction f,
final double min, final double max)
throws MaxIterationsExceededException,
FunctionEvaluationException {
clearResult();
verifyInterval(min, max);
double ret = Double.NaN;
double yMin = f.value(min);
double yMax = f.value(max);
// Verify bracketing
double sign = yMin * yMax;
if (sign > 0) {
// check if either value is close to a zero
if (Math.abs(yMin) <= functionValueAccuracy) {
setResult(min, 0);
ret = min;
} else if (Math.abs(yMax) <= functionValueAccuracy) {
setResult(max, 0);
ret = max;
} else {
// neither value is close to zero and min and max do not bracket root.
throw MathRuntimeException.createIllegalArgumentException(
NON_BRACKETING_MESSAGE, min, max, yMin, yMax);
}
} else if (sign < 0){
// solve using only the first endpoint as initial guess
ret = solve(f, min, yMin, max, yMax, min, yMin);
} else {
// either min or max is a root
if (yMin == 0.0) {
ret = min;
} else {
ret = max;
}
}
return ret;
}
/**
* Find a zero starting search according to the three provided points.
* @param f the function to solve
* @param x0 old approximation for the root
* @param y0 function value at the approximation for the root
* @param x1 last calculated approximation for the root
* @param y1 function value at the last calculated approximation
* for the root
* @param x2 bracket point (must be set to x0 if no bracket point is
* known, this will force starting with linear interpolation)
* @param y2 function value at the bracket point.
* @return the value where the function is zero
* @throws MaxIterationsExceededException if the maximum iteration count
* is exceeded
* @throws FunctionEvaluationException if an error occurs evaluating
* the function
*/
private double solve(final UnivariateRealFunction f,
double x0, double y0,
double x1, double y1,
double x2, double y2)
throws MaxIterationsExceededException, FunctionEvaluationException {
double delta = x1 - x0;
double oldDelta = delta;
int i = 0;
while (i < maximalIterationCount) {
if (Math.abs(y2) < Math.abs(y1)) {
// use the bracket point if is better than last approximation
x0 = x1;
x1 = x2;
x2 = x0;
y0 = y1;
y1 = y2;
y2 = y0;
}
if (Math.abs(y1) <= functionValueAccuracy) {
// Avoid division by very small values. Assume
// the iteration has converged (the problem may
// still be ill conditioned)
setResult(x1, i);
return result;
}
double dx = x2 - x1;
double tolerance =
Math.max(relativeAccuracy * Math.abs(x1), absoluteAccuracy);
if (Math.abs(dx) <= tolerance) {
setResult(x1, i);
return result;
}
if ((Math.abs(oldDelta) < tolerance) ||
(Math.abs(y0) <= Math.abs(y1))) {
// Force bisection.
delta = 0.5 * dx;
oldDelta = delta;
} else {
double r3 = y1 / y0;
double p;
double p1;
// the equality test (x0 == x2) is intentional,
// it is part of the original Brent's method,
// it should NOT be replaced by proximity test
if (x0 == x2) {
// Linear interpolation.
p = dx * r3;
p1 = 1.0 - r3;
} else {
// Inverse quadratic interpolation.
double r1 = y0 / y2;
double r2 = y1 / y2;
p = r3 * (dx * r1 * (r1 - r2) - (x1 - x0) * (r2 - 1.0));
p1 = (r1 - 1.0) * (r2 - 1.0) * (r3 - 1.0);
}
if (p > 0.0) {
p1 = -p1;
} else {
p = -p;
}
if (2.0 * p >= 1.5 * dx * p1 - Math.abs(tolerance * p1) ||
p >= Math.abs(0.5 * oldDelta * p1)) {
// Inverse quadratic interpolation gives a value
// in the wrong direction, or progress is slow.
// Fall back to bisection.
delta = 0.5 * dx;
oldDelta = delta;
} else {
oldDelta = delta;
delta = p / p1;
}
}
// Save old X1, Y1
x0 = x1;
y0 = y1;
// Compute new X1, Y1
if (Math.abs(delta) > tolerance) {
x1 = x1 + delta;
} else if (dx > 0.0) {
x1 = x1 + 0.5 * tolerance;
} else if (dx <= 0.0) {
x1 = x1 - 0.5 * tolerance;
}
y1 = f.value(x1);
if ((y1 > 0) == (y2 > 0)) {
x2 = x0;
y2 = y0;
delta = x1 - x0;
oldDelta = delta;
}
i++;
}
throw new MaxIterationsExceededException(maximalIterationCount);
}
}