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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.analysis.solvers;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
/**
* Utility routines for {@link UnivariateSolver} objects.
*
* @version $Id: UnivariateSolverUtils.java 1400850 2012-10-22 11:57:17Z erans $
*/
public class UnivariateSolverUtils {
/**
* Class contains only static methods.
*/
private UnivariateSolverUtils() {}
/**
* Convenience method to find a zero of a univariate real function. A default
* solver is used.
*
* @param function Function.
* @param x0 Lower bound for the interval.
* @param x1 Upper bound for the interval.
* @return a value where the function is zero.
* @throws NoBracketingException if the function has the same sign at the
* endpoints.
* @throws NullArgumentException if {@code function} is {@code null}.
*/
public static double solve(UnivariateFunction function, double x0, double x1)
throws NullArgumentException,
NoBracketingException {
if (function == null) {
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
final UnivariateSolver solver = new BrentSolver();
return solver.solve(Integer.MAX_VALUE, function, x0, x1);
}
/**
* Convenience method to find a zero of a univariate real function. A default
* solver is used.
*
* @param function Function.
* @param x0 Lower bound for the interval.
* @param x1 Upper bound for the interval.
* @param absoluteAccuracy Accuracy to be used by the solver.
* @return a value where the function is zero.
* @throws NoBracketingException if the function has the same sign at the
* endpoints.
* @throws NullArgumentException if {@code function} is {@code null}.
*/
public static double solve(UnivariateFunction function,
double x0, double x1,
double absoluteAccuracy)
throws NullArgumentException,
NoBracketingException {
if (function == null) {
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
final UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
return solver.solve(Integer.MAX_VALUE, function, x0, x1);
}
/** Force a root found by a non-bracketing solver to lie on a specified side,
* as if the solver was a bracketing one.
* @param maxEval maximal number of new evaluations of the function
* (evaluations already done for finding the root should have already been subtracted
* from this number)
* @param f function to solve
* @param bracketing bracketing solver to use for shifting the root
* @param baseRoot original root found by a previous non-bracketing solver
* @param min minimal bound of the search interval
* @param max maximal bound of the search interval
* @param allowedSolution the kind of solutions that the root-finding algorithm may
* accept as solutions.
* @return a root approximation, on the specified side of the exact root
* @throws NoBracketingException if the function has the same sign at the
* endpoints.
*/
public static double forceSide(final int maxEval, final UnivariateFunction f,
final BracketedUnivariateSolver bracketing,
final double baseRoot, final double min, final double max,
final AllowedSolution allowedSolution)
throws NoBracketingException {
if (allowedSolution == AllowedSolution.ANY_SIDE) {
// no further bracketing required
return baseRoot;
}
// find a very small interval bracketing the root
final double step = FastMath.max(bracketing.getAbsoluteAccuracy(),
FastMath.abs(baseRoot * bracketing.getRelativeAccuracy()));
double xLo = FastMath.max(min, baseRoot - step);
double fLo = f.value(xLo);
double xHi = FastMath.min(max, baseRoot + step);
double fHi = f.value(xHi);
int remainingEval = maxEval - 2;
while (remainingEval > 0) {
if ((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0)) {
// compute the root on the selected side
return bracketing.solve(remainingEval, f, xLo, xHi, baseRoot, allowedSolution);
}
// try increasing the interval
boolean changeLo = false;
boolean changeHi = false;
if (fLo < fHi) {
// increasing function
if (fLo >= 0) {
changeLo = true;
} else {
changeHi = true;
}
} else if (fLo > fHi) {
// decreasing function
if (fLo <= 0) {
changeLo = true;
} else {
changeHi = true;
}
} else {
// unknown variation
changeLo = true;
changeHi = true;
}
// update the lower bound
if (changeLo) {
xLo = FastMath.max(min, xLo - step);
fLo = f.value(xLo);
remainingEval--;
}
// update the higher bound
if (changeHi) {
xHi = FastMath.min(max, xHi + step);
fHi = f.value(xHi);
remainingEval--;
}
}
throw new NoBracketingException(LocalizedFormats.FAILED_BRACKETING,
xLo, xHi, fLo, fHi,
maxEval - remainingEval, maxEval, baseRoot,
min, max);
}
/**
* This method attempts to find two values a and b satisfying
* -
lowerBound <= a < initial < b <= upperBound
* -
f(a) * f(b) < 0
*
* If f is continuous on [a,b], this means that a
* and b bracket a root of f.
*
* The algorithm starts by setting
* a := initial -1; b := initial +1, examines the value of the
* function at a and b and keeps moving
* the endpoints out by one unit each time through a loop that terminates
* when one of the following happens:
* -
f(a) * f(b) < 0 -- success!
* -
a = lower and b = upper
* -- NoBracketingException
* -
Integer.MAX_VALUE iterations elapse
* -- NoBracketingException
*
*
* Note: this method can take
* Integer.MAX_VALUE iterations to throw a
* ConvergenceException. Unless you are confident that there
* is a root between lowerBound and upperBound
* near initial, it is better to use
* {@link #bracket(UnivariateFunction, double, double, double, int)},
* explicitly specifying the maximum number of iterations.
*
* @param function Function.
* @param initial Initial midpoint of interval being expanded to
* bracket a root.
* @param lowerBound Lower bound (a is never lower than this value)
* @param upperBound Upper bound (b never is greater than this
* value).
* @return a two-element array holding a and b.
* @throws NoBracketingException if a root cannot be bracketted.
* @throws NotStrictlyPositiveException if {@code maximumIterations <= 0}.
* @throws NullArgumentException if {@code function} is {@code null}.
*/
public static double[] bracket(UnivariateFunction function,
double initial,
double lowerBound, double upperBound)
throws NullArgumentException,
NotStrictlyPositiveException,
NoBracketingException {
return bracket(function, initial, lowerBound, upperBound, Integer.MAX_VALUE);
}
/**
* This method attempts to find two values a and b satisfying
* -
lowerBound <= a < initial < b <= upperBound
* -
f(a) * f(b) <= 0
*
* If f is continuous on [a,b], this means that a
* and b bracket a root of f.
*
* The algorithm starts by setting
* a := initial -1; b := initial +1, examines the value of the
* function at a and b and keeps moving
* the endpoints out by one unit each time through a loop that terminates
* when one of the following happens:
* -
f(a) * f(b) <= 0 -- success!
* -
a = lower and b = upper
* -- NoBracketingException
* -
maximumIterations iterations elapse
* -- NoBracketingException
*
* @param function Function.
* @param initial Initial midpoint of interval being expanded to
* bracket a root.
* @param lowerBound Lower bound (a is never lower than this value).
* @param upperBound Upper bound (b never is greater than this
* value).
* @param maximumIterations Maximum number of iterations to perform
* @return a two element array holding a and b.
* @throws NoBracketingException if the algorithm fails to find a and b
* satisfying the desired conditions.
* @throws NotStrictlyPositiveException if {@code maximumIterations <= 0}.
* @throws NullArgumentException if {@code function} is {@code null}.
*/
public static double[] bracket(UnivariateFunction function,
double initial,
double lowerBound, double upperBound,
int maximumIterations)
throws NullArgumentException,
NotStrictlyPositiveException,
NoBracketingException {
if (function == null) {
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
if (maximumIterations <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.INVALID_MAX_ITERATIONS, maximumIterations);
}
verifySequence(lowerBound, initial, upperBound);
double a = initial;
double b = initial;
double fa;
double fb;
int numIterations = 0;
do {
a = FastMath.max(a - 1.0, lowerBound);
b = FastMath.min(b + 1.0, upperBound);
fa = function.value(a);
fb = function.value(b);
++numIterations;
} while ((fa * fb > 0.0) && (numIterations < maximumIterations) &&
((a > lowerBound) || (b < upperBound)));
if (fa * fb > 0.0) {
throw new NoBracketingException(LocalizedFormats.FAILED_BRACKETING,
a, b, fa, fb,
numIterations, maximumIterations, initial,
lowerBound, upperBound);
}
return new double[] {a, b};
}
/**
* Compute the midpoint of two values.
*
* @param a first value.
* @param b second value.
* @return the midpoint.
*/
public static double midpoint(double a, double b) {
return (a + b) * 0.5;
}
/**
* Check whether the interval bounds bracket a root. That is, if the
* values at the endpoints are not equal to zero, then the function takes
* opposite signs at the endpoints.
*
* @param function Function.
* @param lower Lower endpoint.
* @param upper Upper endpoint.
* @return {@code true} if the function values have opposite signs at the
* given points.
* @throws NullArgumentException if {@code function} is {@code null}.
*/
public static boolean isBracketing(UnivariateFunction function,
final double lower,
final double upper)
throws NullArgumentException {
if (function == null) {
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
final double fLo = function.value(lower);
final double fHi = function.value(upper);
return (fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0);
}
/**
* Check whether the arguments form a (strictly) increasing sequence.
*
* @param start First number.
* @param mid Second number.
* @param end Third number.
* @return {@code true} if the arguments form an increasing sequence.
*/
public static boolean isSequence(final double start,
final double mid,
final double end) {
return (start < mid) && (mid < end);
}
/**
* Check that the endpoints specify an interval.
*
* @param lower Lower endpoint.
* @param upper Upper endpoint.
* @throws NumberIsTooLargeException if {@code lower >= upper}.
*/
public static void verifyInterval(final double lower,
final double upper)
throws NumberIsTooLargeException {
if (lower >= upper) {
throw new NumberIsTooLargeException(LocalizedFormats.ENDPOINTS_NOT_AN_INTERVAL,
lower, upper, false);
}
}
/**
* Check that {@code lower < initial < upper}.
*
* @param lower Lower endpoint.
* @param initial Initial value.
* @param upper Upper endpoint.
* @throws NumberIsTooLargeException if {@code lower >= initial} or
* {@code initial >= upper}.
*/
public static void verifySequence(final double lower,
final double initial,
final double upper)
throws NumberIsTooLargeException {
verifyInterval(lower, initial);
verifyInterval(initial, upper);
}
/**
* Check that the endpoints specify an interval and the end points
* bracket a root.
*
* @param function Function.
* @param lower Lower endpoint.
* @param upper Upper endpoint.
* @throws NoBracketingException if the function has the same sign at the
* endpoints.
* @throws NullArgumentException if {@code function} is {@code null}.
*/
public static void verifyBracketing(UnivariateFunction function,
final double lower,
final double upper)
throws NullArgumentException,
NoBracketingException {
if (function == null) {
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
verifyInterval(lower, upper);
if (!isBracketing(function, lower, upper)) {
throw new NoBracketingException(lower, upper,
function.value(lower),
function.value(upper));
}
}
}