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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.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.
*
*/
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<UnivariateFunction> 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 simply calls {@link #bracket(UnivariateFunction, double, double, double,
* double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
* with {@code q} and {@code r} set to 1.0 and {@code maximumIterations} set to {@code Integer.MAX_VALUE}.
* <strong>Note: </strong> this method can take
* <code>Integer.MAX_VALUE</code> iterations to throw a
* <code>ConvergenceException.</code> Unless you are confident that there
* is a root between <code>lowerBound</code> and <code>upperBound</code>
* near <code>initial,</code> it is better to use
* {@link #bracket(UnivariateFunction, double, double, double, double,
* double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)},
* explicitly specifying the maximum number of iterations.</p>
*
* @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, 1.0, 1.0, Integer.MAX_VALUE);
}
/**
* This method simply calls {@link #bracket(UnivariateFunction, double, double, double,
* double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
* with {@code q} and {@code r} set to 1.0.
* @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 {
return bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, maximumIterations);
}
/**
* This method attempts to find two values a and b satisfying <ul>
* <li> {@code lowerBound <= a < initial < b <= upperBound} </li>
* <li> {@code f(a) * f(b) <= 0} </li>
* </ul>
* If {@code f} is continuous on {@code [a,b]}, this means that {@code a}
* and {@code b} bracket a root of {@code f}.
* <p>
* The algorithm checks the sign of \( f(l_k) \) and \( f(u_k) \) for increasing
* values of k, where \( l_k = max(lower, initial - \delta_k) \),
* \( u_k = min(upper, initial + \delta_k) \), using recurrence
* \( \delta_{k+1} = r \delta_k + q, \delta_0 = 0\) and starting search with \( k=1 \).
* The algorithm stops when one of the following happens: <ul>
* <li> at least one positive and one negative value have been found -- success!</li>
* <li> both endpoints have reached their respective limits -- NoBracketingException </li>
* <li> {@code maximumIterations} iterations elapse -- NoBracketingException </li></ul></p>
* <p>
* If different signs are found at first iteration ({@code k=1}), then the returned
* interval will be \( [a, b] = [l_1, u_1] \). If different signs are found at a later
* iteration ({code k>1}, then the returned interval will be either
* \( [a, b] = [l_{k+1}, l_{k}] \) or \( [a, b] = [u_{k}, u_{k+1}] \). A root solver called
* with these parameters will therefore start with the smallest bracketing interval known
* at this step.
* </p>
* <p>
* Interval expansion rate is tuned by changing the recurrence parameters {@code r} and
* {@code q}. When the multiplicative factor {@code r} is set to 1, the sequence is a
* simple arithmetic sequence with linear increase. When the multiplicative factor {@code r}
* is larger than 1, the sequence has an asymptotically exponential rate. Note than the
* additive parameter {@code q} should never be set to zero, otherwise the interval would
* degenerate to the single initial point for all values of {@code k}.
* </p>
* <p>
* As a rule of thumb, when the location of the root is expected to be approximately known
* within some error margin, {@code r} should be set to 1 and {@code q} should be set to the
* order of magnitude of the error margin. When the location of the root is really a wild guess,
* then {@code r} should be set to a value larger than 1 (typically 2 to double the interval
* length at each iteration) and {@code q} should be set according to half the initial
* search interval length.
* </p>
* <p>
* As an example, if we consider the trivial function {@code f(x) = 1 - x} and use
* {@code initial = 4}, {@code r = 1}, {@code q = 2}, the algorithm will compute
* {@code f(4-2) = f(2) = -1} and {@code f(4+2) = f(6) = -5} for {@code k = 1}, then
* {@code f(4-4) = f(0) = +1} and {@code f(4+4) = f(8) = -7} for {@code k = 2}. Then it will
* return the interval {@code [0, 2]} as the smallest one known to be bracketing the root.
* As shown by this example, the initial value (here {@code 4}) may lie outside of the returned
* bracketing interval.
* </p>
* @param function function to check
* @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 q additive offset used to compute bounds sequence (must be strictly positive)
* @param r multiplicative factor used to compute bounds sequence
* @param maximumIterations Maximum number of iterations to perform
* @return a two element array holding the bracketing values.
* @exception NoBracketingException if function cannot be bracketed in the search interval
*/
public static double[] bracket(final UnivariateFunction function, final double initial,
final double lowerBound, final double upperBound,
final double q, final double r, final int maximumIterations)
throws NoBracketingException {
if (function == null) {
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
if (q <= 0) {
throw new NotStrictlyPositiveException(q);
}
if (maximumIterations <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.INVALID_MAX_ITERATIONS, maximumIterations);
}
verifySequence(lowerBound, initial, upperBound);
// initialize the recurrence
double a = initial;
double b = initial;
double fa = Double.NaN;
double fb = Double.NaN;
double delta = 0;
for (int numIterations = 0;
(numIterations < maximumIterations) && (a > lowerBound || b < upperBound);
++numIterations) {
final double previousA = a;
final double previousFa = fa;
final double previousB = b;
final double previousFb = fb;
delta = r * delta + q;
a = FastMath.max(initial - delta, lowerBound);
b = FastMath.min(initial + delta, upperBound);
fa = function.value(a);
fb = function.value(b);
if (numIterations == 0) {
// at first iteration, we don't have a previous interval
// we simply compare both sides of the initial interval
if (fa * fb <= 0) {
// the first interval already brackets a root
return new double[] { a, b };
}
} else {
// we have a previous interval with constant sign and expand it,
// we expect sign changes to occur at boundaries
if (fa * previousFa <= 0) {
// sign change detected at near lower bound
return new double[] { a, previousA };
} else if (fb * previousFb <= 0) {
// sign change detected at near upper bound
return new double[] { previousB, b };
}
}
}
// no bracketing found
throw new NoBracketingException(a, b, fa, fb);
}
/**
* 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));
}
}
}