001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.math3.analysis.solvers;
018
019import org.apache.commons.math3.exception.NoBracketingException;
020import org.apache.commons.math3.exception.NumberIsTooLargeException;
021import org.apache.commons.math3.exception.TooManyEvaluationsException;
022import org.apache.commons.math3.util.FastMath;
023
024/**
025 * This class implements the <a href="http://mathworld.wolfram.com/MullersMethod.html">
026 * Muller's Method</a> for root finding of real univariate functions. For
027 * reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477,
028 * chapter 3.
029 * <p>
030 * Muller's method applies to both real and complex functions, but here we
031 * restrict ourselves to real functions.
032 * This class differs from {@link MullerSolver} in the way it avoids complex
033 * operations.</p>
034 * Except for the initial [min, max], it does not require bracketing
035 * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex
036 * number arises in the computation, we simply use its modulus as real
037 * approximation.</p>
038 * <p>
039 * Because the interval may not be bracketing, bisection alternative is
040 * not applicable here. However in practice our treatment usually works
041 * well, especially near real zeroes where the imaginary part of complex
042 * approximation is often negligible.</p>
043 * <p>
044 * The formulas here do not use divided differences directly.</p>
045 *
046 * @version $Id: MullerSolver2.java 1379560 2012-08-31 19:40:30Z erans $
047 * @since 1.2
048 * @see MullerSolver
049 */
050public class MullerSolver2 extends AbstractUnivariateSolver {
051
052    /** Default absolute accuracy. */
053    private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
054
055    /**
056     * Construct a solver with default accuracy (1e-6).
057     */
058    public MullerSolver2() {
059        this(DEFAULT_ABSOLUTE_ACCURACY);
060    }
061    /**
062     * Construct a solver.
063     *
064     * @param absoluteAccuracy Absolute accuracy.
065     */
066    public MullerSolver2(double absoluteAccuracy) {
067        super(absoluteAccuracy);
068    }
069    /**
070     * Construct a solver.
071     *
072     * @param relativeAccuracy Relative accuracy.
073     * @param absoluteAccuracy Absolute accuracy.
074     */
075    public MullerSolver2(double relativeAccuracy,
076                        double absoluteAccuracy) {
077        super(relativeAccuracy, absoluteAccuracy);
078    }
079
080    /**
081     * {@inheritDoc}
082     */
083    @Override
084    protected double doSolve()
085        throws TooManyEvaluationsException,
086               NumberIsTooLargeException,
087               NoBracketingException {
088        final double min = getMin();
089        final double max = getMax();
090
091        verifyInterval(min, max);
092
093        final double relativeAccuracy = getRelativeAccuracy();
094        final double absoluteAccuracy = getAbsoluteAccuracy();
095        final double functionValueAccuracy = getFunctionValueAccuracy();
096
097        // x2 is the last root approximation
098        // x is the new approximation and new x2 for next round
099        // x0 < x1 < x2 does not hold here
100
101        double x0 = min;
102        double y0 = computeObjectiveValue(x0);
103        if (FastMath.abs(y0) < functionValueAccuracy) {
104            return x0;
105        }
106        double x1 = max;
107        double y1 = computeObjectiveValue(x1);
108        if (FastMath.abs(y1) < functionValueAccuracy) {
109            return x1;
110        }
111
112        if(y0 * y1 > 0) {
113            throw new NoBracketingException(x0, x1, y0, y1);
114        }
115
116        double x2 = 0.5 * (x0 + x1);
117        double y2 = computeObjectiveValue(x2);
118
119        double oldx = Double.POSITIVE_INFINITY;
120        while (true) {
121            // quadratic interpolation through x0, x1, x2
122            final double q = (x2 - x1) / (x1 - x0);
123            final double a = q * (y2 - (1 + q) * y1 + q * y0);
124            final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0;
125            final double c = (1 + q) * y2;
126            final double delta = b * b - 4 * a * c;
127            double x;
128            final double denominator;
129            if (delta >= 0.0) {
130                // choose a denominator larger in magnitude
131                double dplus = b + FastMath.sqrt(delta);
132                double dminus = b - FastMath.sqrt(delta);
133                denominator = FastMath.abs(dplus) > FastMath.abs(dminus) ? dplus : dminus;
134            } else {
135                // take the modulus of (B +/- FastMath.sqrt(delta))
136                denominator = FastMath.sqrt(b * b - delta);
137            }
138            if (denominator != 0) {
139                x = x2 - 2.0 * c * (x2 - x1) / denominator;
140                // perturb x if it exactly coincides with x1 or x2
141                // the equality tests here are intentional
142                while (x == x1 || x == x2) {
143                    x += absoluteAccuracy;
144                }
145            } else {
146                // extremely rare case, get a random number to skip it
147                x = min + FastMath.random() * (max - min);
148                oldx = Double.POSITIVE_INFINITY;
149            }
150            final double y = computeObjectiveValue(x);
151
152            // check for convergence
153            final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);
154            if (FastMath.abs(x - oldx) <= tolerance ||
155                FastMath.abs(y) <= functionValueAccuracy) {
156                return x;
157            }
158
159            // prepare the next iteration
160            x0 = x1;
161            y0 = y1;
162            x1 = x2;
163            y1 = y2;
164            x2 = x;
165            y2 = y;
166            oldx = x;
167        }
168    }
169}