mirror of https://github.com/jlizier/jidt
497 lines
21 KiB
Java
497 lines
21 KiB
Java
/*
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* Java Information Dynamics Toolkit (JIDT)
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* Copyright (C) 2017, Joseph T. Lizier
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*
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* This program is free software: you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation, either version 3 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program. If not, see <http://www.gnu.org/licenses/>.
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*/
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/*
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* This class was originally distributed as part of the Apache Commons
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* Math3 library (3.6.1), under the Apache License Version 2.0, which is
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* copied below. This Apache 2 software is now included as a derivative
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* work in the GPLv3 licensed JIDT project, as per:
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* http://www.apache.org/licenses/GPL-compatibility.html
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*
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* The original Apache source code has been modified as follows:
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* -- We have modified package names to sit inside the JIDT structure.
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*/
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package infodynamics.utils.commonsmath3.analysis.solvers;
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import infodynamics.utils.commonsmath3.analysis.UnivariateFunction;
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import infodynamics.utils.commonsmath3.exception.NoBracketingException;
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import infodynamics.utils.commonsmath3.exception.NotStrictlyPositiveException;
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import infodynamics.utils.commonsmath3.exception.NullArgumentException;
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import infodynamics.utils.commonsmath3.exception.NumberIsTooLargeException;
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import infodynamics.utils.commonsmath3.exception.util.LocalizedFormats;
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import infodynamics.utils.commonsmath3.util.FastMath;
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/**
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* Utility routines for {@link UnivariateSolver} objects.
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*
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*/
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public class UnivariateSolverUtils {
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/**
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* Class contains only static methods.
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*/
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private UnivariateSolverUtils() {}
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/**
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* Convenience method to find a zero of a univariate real function. A default
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* solver is used.
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*
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* @param function Function.
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* @param x0 Lower bound for the interval.
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* @param x1 Upper bound for the interval.
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* @return a value where the function is zero.
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* @throws NoBracketingException if the function has the same sign at the
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* endpoints.
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* @throws NullArgumentException if {@code function} is {@code null}.
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*/
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public static double solve(UnivariateFunction function, double x0, double x1)
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throws NullArgumentException,
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NoBracketingException {
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if (function == null) {
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throw new NullArgumentException(LocalizedFormats.FUNCTION);
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}
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final UnivariateSolver solver = new BrentSolver();
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return solver.solve(Integer.MAX_VALUE, function, x0, x1);
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}
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/**
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* Convenience method to find a zero of a univariate real function. A default
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* solver is used.
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*
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* @param function Function.
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* @param x0 Lower bound for the interval.
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* @param x1 Upper bound for the interval.
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* @param absoluteAccuracy Accuracy to be used by the solver.
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* @return a value where the function is zero.
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* @throws NoBracketingException if the function has the same sign at the
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* endpoints.
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* @throws NullArgumentException if {@code function} is {@code null}.
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*/
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public static double solve(UnivariateFunction function,
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double x0, double x1,
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double absoluteAccuracy)
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throws NullArgumentException,
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NoBracketingException {
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if (function == null) {
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throw new NullArgumentException(LocalizedFormats.FUNCTION);
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}
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final UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
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return solver.solve(Integer.MAX_VALUE, function, x0, x1);
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}
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/**
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* Force a root found by a non-bracketing solver to lie on a specified side,
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* as if the solver were a bracketing one.
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*
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* @param maxEval maximal number of new evaluations of the function
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* (evaluations already done for finding the root should have already been subtracted
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* from this number)
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* @param f function to solve
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* @param bracketing bracketing solver to use for shifting the root
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* @param baseRoot original root found by a previous non-bracketing solver
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* @param min minimal bound of the search interval
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* @param max maximal bound of the search interval
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* @param allowedSolution the kind of solutions that the root-finding algorithm may
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* accept as solutions.
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* @return a root approximation, on the specified side of the exact root
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* @throws NoBracketingException if the function has the same sign at the
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* endpoints.
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*/
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public static double forceSide(final int maxEval, final UnivariateFunction f,
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final BracketedUnivariateSolver<UnivariateFunction> bracketing,
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final double baseRoot, final double min, final double max,
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final AllowedSolution allowedSolution)
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throws NoBracketingException {
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if (allowedSolution == AllowedSolution.ANY_SIDE) {
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// no further bracketing required
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return baseRoot;
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}
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// find a very small interval bracketing the root
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final double step = FastMath.max(bracketing.getAbsoluteAccuracy(),
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FastMath.abs(baseRoot * bracketing.getRelativeAccuracy()));
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double xLo = FastMath.max(min, baseRoot - step);
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double fLo = f.value(xLo);
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double xHi = FastMath.min(max, baseRoot + step);
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double fHi = f.value(xHi);
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int remainingEval = maxEval - 2;
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while (remainingEval > 0) {
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if ((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0)) {
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// compute the root on the selected side
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return bracketing.solve(remainingEval, f, xLo, xHi, baseRoot, allowedSolution);
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}
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// try increasing the interval
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boolean changeLo = false;
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boolean changeHi = false;
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if (fLo < fHi) {
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// increasing function
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if (fLo >= 0) {
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changeLo = true;
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} else {
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changeHi = true;
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}
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} else if (fLo > fHi) {
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// decreasing function
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if (fLo <= 0) {
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changeLo = true;
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} else {
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changeHi = true;
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}
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} else {
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// unknown variation
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changeLo = true;
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changeHi = true;
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}
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// update the lower bound
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if (changeLo) {
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xLo = FastMath.max(min, xLo - step);
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fLo = f.value(xLo);
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remainingEval--;
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}
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// update the higher bound
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if (changeHi) {
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xHi = FastMath.min(max, xHi + step);
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fHi = f.value(xHi);
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remainingEval--;
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}
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}
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throw new NoBracketingException(LocalizedFormats.FAILED_BRACKETING,
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xLo, xHi, fLo, fHi,
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maxEval - remainingEval, maxEval, baseRoot,
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min, max);
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}
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/**
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* This method simply calls {@link #bracket(UnivariateFunction, double, double, double,
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* double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
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* with {@code q} and {@code r} set to 1.0 and {@code maximumIterations} set to {@code Integer.MAX_VALUE}.
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* <p>
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* <strong>Note: </strong> this method can take {@code Integer.MAX_VALUE}
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* iterations to throw a {@code ConvergenceException.} Unless you are
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* confident that there is a root between {@code lowerBound} and
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* {@code upperBound} near {@code initial}, it is better to use
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* {@link #bracket(UnivariateFunction, double, double, double, double,double, int)
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* bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)},
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* explicitly specifying the maximum number of iterations.</p>
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*
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* @param function Function.
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* @param initial Initial midpoint of interval being expanded to
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* bracket a root.
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* @param lowerBound Lower bound (a is never lower than this value)
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* @param upperBound Upper bound (b never is greater than this
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* value).
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* @return a two-element array holding a and b.
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* @throws NoBracketingException if a root cannot be bracketted.
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* @throws NotStrictlyPositiveException if {@code maximumIterations <= 0}.
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* @throws NullArgumentException if {@code function} is {@code null}.
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*/
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public static double[] bracket(UnivariateFunction function,
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double initial,
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double lowerBound, double upperBound)
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throws NullArgumentException,
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NotStrictlyPositiveException,
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NoBracketingException {
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return bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, Integer.MAX_VALUE);
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}
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/**
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* This method simply calls {@link #bracket(UnivariateFunction, double, double, double,
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* double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
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* with {@code q} and {@code r} set to 1.0.
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* @param function Function.
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* @param initial Initial midpoint of interval being expanded to
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* bracket a root.
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* @param lowerBound Lower bound (a is never lower than this value).
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* @param upperBound Upper bound (b never is greater than this
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* value).
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* @param maximumIterations Maximum number of iterations to perform
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* @return a two element array holding a and b.
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* @throws NoBracketingException if the algorithm fails to find a and b
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* satisfying the desired conditions.
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* @throws NotStrictlyPositiveException if {@code maximumIterations <= 0}.
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* @throws NullArgumentException if {@code function} is {@code null}.
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*/
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public static double[] bracket(UnivariateFunction function,
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double initial,
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double lowerBound, double upperBound,
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int maximumIterations)
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throws NullArgumentException,
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NotStrictlyPositiveException,
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NoBracketingException {
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return bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, maximumIterations);
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}
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/**
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* This method attempts to find two values a and b satisfying <ul>
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* <li> {@code lowerBound <= a < initial < b <= upperBound} </li>
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* <li> {@code f(a) * f(b) <= 0} </li>
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* </ul>
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* If {@code f} is continuous on {@code [a,b]}, this means that {@code a}
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* and {@code b} bracket a root of {@code f}.
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* <p>
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* The algorithm checks the sign of \( f(l_k) \) and \( f(u_k) \) for increasing
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* values of k, where \( l_k = max(lower, initial - \delta_k) \),
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* \( u_k = min(upper, initial + \delta_k) \), using recurrence
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* \( \delta_{k+1} = r \delta_k + q, \delta_0 = 0\) and starting search with \( k=1 \).
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* The algorithm stops when one of the following happens: <ul>
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* <li> at least one positive and one negative value have been found -- success!</li>
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* <li> both endpoints have reached their respective limits -- NoBracketingException </li>
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* <li> {@code maximumIterations} iterations elapse -- NoBracketingException </li></ul>
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* <p>
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* If different signs are found at first iteration ({@code k=1}), then the returned
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* interval will be \( [a, b] = [l_1, u_1] \). If different signs are found at a later
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* iteration {@code k>1}, then the returned interval will be either
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* \( [a, b] = [l_{k+1}, l_{k}] \) or \( [a, b] = [u_{k}, u_{k+1}] \). A root solver called
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* with these parameters will therefore start with the smallest bracketing interval known
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* at this step.
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* </p>
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* <p>
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* Interval expansion rate is tuned by changing the recurrence parameters {@code r} and
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* {@code q}. When the multiplicative factor {@code r} is set to 1, the sequence is a
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* simple arithmetic sequence with linear increase. When the multiplicative factor {@code r}
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* is larger than 1, the sequence has an asymptotically exponential rate. Note than the
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* additive parameter {@code q} should never be set to zero, otherwise the interval would
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* degenerate to the single initial point for all values of {@code k}.
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* </p>
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* <p>
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* As a rule of thumb, when the location of the root is expected to be approximately known
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* within some error margin, {@code r} should be set to 1 and {@code q} should be set to the
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* order of magnitude of the error margin. When the location of the root is really a wild guess,
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* then {@code r} should be set to a value larger than 1 (typically 2 to double the interval
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* length at each iteration) and {@code q} should be set according to half the initial
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* search interval length.
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* </p>
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* <p>
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* As an example, if we consider the trivial function {@code f(x) = 1 - x} and use
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* {@code initial = 4}, {@code r = 1}, {@code q = 2}, the algorithm will compute
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* {@code f(4-2) = f(2) = -1} and {@code f(4+2) = f(6) = -5} for {@code k = 1}, then
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* {@code f(4-4) = f(0) = +1} and {@code f(4+4) = f(8) = -7} for {@code k = 2}. Then it will
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* return the interval {@code [0, 2]} as the smallest one known to be bracketing the root.
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* As shown by this example, the initial value (here {@code 4}) may lie outside of the returned
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* bracketing interval.
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* </p>
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* @param function function to check
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* @param initial Initial midpoint of interval being expanded to
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* bracket a root.
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* @param lowerBound Lower bound (a is never lower than this value).
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* @param upperBound Upper bound (b never is greater than this
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* value).
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* @param q additive offset used to compute bounds sequence (must be strictly positive)
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* @param r multiplicative factor used to compute bounds sequence
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* @param maximumIterations Maximum number of iterations to perform
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* @return a two element array holding the bracketing values.
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* @exception NoBracketingException if function cannot be bracketed in the search interval
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*/
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public static double[] bracket(final UnivariateFunction function, final double initial,
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final double lowerBound, final double upperBound,
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final double q, final double r, final int maximumIterations)
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throws NoBracketingException {
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if (function == null) {
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throw new NullArgumentException(LocalizedFormats.FUNCTION);
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}
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if (q <= 0) {
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throw new NotStrictlyPositiveException(q);
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}
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if (maximumIterations <= 0) {
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throw new NotStrictlyPositiveException(LocalizedFormats.INVALID_MAX_ITERATIONS, maximumIterations);
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}
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verifySequence(lowerBound, initial, upperBound);
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// initialize the recurrence
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double a = initial;
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double b = initial;
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double fa = Double.NaN;
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double fb = Double.NaN;
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double delta = 0;
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for (int numIterations = 0;
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(numIterations < maximumIterations) && (a > lowerBound || b < upperBound);
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++numIterations) {
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final double previousA = a;
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final double previousFa = fa;
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final double previousB = b;
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final double previousFb = fb;
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delta = r * delta + q;
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a = FastMath.max(initial - delta, lowerBound);
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b = FastMath.min(initial + delta, upperBound);
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fa = function.value(a);
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fb = function.value(b);
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if (numIterations == 0) {
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// at first iteration, we don't have a previous interval
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// we simply compare both sides of the initial interval
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if (fa * fb <= 0) {
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// the first interval already brackets a root
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return new double[] { a, b };
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}
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} else {
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// we have a previous interval with constant sign and expand it,
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// we expect sign changes to occur at boundaries
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if (fa * previousFa <= 0) {
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// sign change detected at near lower bound
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return new double[] { a, previousA };
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} else if (fb * previousFb <= 0) {
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// sign change detected at near upper bound
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return new double[] { previousB, b };
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}
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}
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}
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// no bracketing found
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throw new NoBracketingException(a, b, fa, fb);
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}
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/**
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* Compute the midpoint of two values.
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*
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* @param a first value.
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* @param b second value.
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* @return the midpoint.
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*/
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public static double midpoint(double a, double b) {
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return (a + b) * 0.5;
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}
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/**
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* Check whether the interval bounds bracket a root. That is, if the
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* values at the endpoints are not equal to zero, then the function takes
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* opposite signs at the endpoints.
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*
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* @param function Function.
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* @param lower Lower endpoint.
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* @param upper Upper endpoint.
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* @return {@code true} if the function values have opposite signs at the
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* given points.
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* @throws NullArgumentException if {@code function} is {@code null}.
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*/
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public static boolean isBracketing(UnivariateFunction function,
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final double lower,
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final double upper)
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throws NullArgumentException {
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if (function == null) {
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throw new NullArgumentException(LocalizedFormats.FUNCTION);
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}
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final double fLo = function.value(lower);
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final double fHi = function.value(upper);
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return (fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0);
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}
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/**
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* Check whether the arguments form a (strictly) increasing sequence.
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*
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* @param start First number.
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* @param mid Second number.
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* @param end Third number.
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* @return {@code true} if the arguments form an increasing sequence.
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*/
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public static boolean isSequence(final double start,
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final double mid,
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final double end) {
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return (start < mid) && (mid < end);
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}
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/**
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* Check that the endpoints specify an interval.
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*
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* @param lower Lower endpoint.
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* @param upper Upper endpoint.
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* @throws NumberIsTooLargeException if {@code lower >= upper}.
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*/
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public static void verifyInterval(final double lower,
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final double upper)
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throws NumberIsTooLargeException {
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if (lower >= upper) {
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throw new NumberIsTooLargeException(LocalizedFormats.ENDPOINTS_NOT_AN_INTERVAL,
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lower, upper, false);
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}
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}
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/**
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* Check that {@code lower < initial < upper}.
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*
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* @param lower Lower endpoint.
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* @param initial Initial value.
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* @param upper Upper endpoint.
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* @throws NumberIsTooLargeException if {@code lower >= initial} or
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* {@code initial >= upper}.
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*/
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public static void verifySequence(final double lower,
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final double initial,
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final double upper)
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throws NumberIsTooLargeException {
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verifyInterval(lower, initial);
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verifyInterval(initial, upper);
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}
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/**
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* Check that the endpoints specify an interval and the end points
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* bracket a root.
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*
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* @param function Function.
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* @param lower Lower endpoint.
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* @param upper Upper endpoint.
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* @throws NoBracketingException if the function has the same sign at the
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* endpoints.
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* @throws NullArgumentException if {@code function} is {@code null}.
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|
*/
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|
public static void verifyBracketing(UnivariateFunction function,
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|
final double lower,
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|
final double upper)
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|
throws NullArgumentException,
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|
NoBracketingException {
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|
if (function == null) {
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|
throw new NullArgumentException(LocalizedFormats.FUNCTION);
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|
}
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|
verifyInterval(lower, upper);
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|
if (!isBracketing(function, lower, upper)) {
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|
throw new NoBracketingException(lower, upper,
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|
function.value(lower),
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|
function.value(upper));
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|
}
|
|
}
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|
}
|