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 * @since 1.2
047 * @see MullerSolver
048 */
049public class MullerSolver2 extends AbstractUnivariateSolver {
050
051    /** Default absolute accuracy. */
052    private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
053
054    /**
055     * Construct a solver with default accuracy (1e-6).
056     */
057    public MullerSolver2() {
058        this(DEFAULT_ABSOLUTE_ACCURACY);
059    }
060    /**
061     * Construct a solver.
062     *
063     * @param absoluteAccuracy Absolute accuracy.
064     */
065    public MullerSolver2(double absoluteAccuracy) {
066        super(absoluteAccuracy);
067    }
068    /**
069     * Construct a solver.
070     *
071     * @param relativeAccuracy Relative accuracy.
072     * @param absoluteAccuracy Absolute accuracy.
073     */
074    public MullerSolver2(double relativeAccuracy,
075                        double absoluteAccuracy) {
076        super(relativeAccuracy, absoluteAccuracy);
077    }
078
079    /**
080     * {@inheritDoc}
081     */
082    @Override
083    protected double doSolve()
084        throws TooManyEvaluationsException,
085               NumberIsTooLargeException,
086               NoBracketingException {
087        final double min = getMin();
088        final double max = getMax();
089
090        verifyInterval(min, max);
091
092        final double relativeAccuracy = getRelativeAccuracy();
093        final double absoluteAccuracy = getAbsoluteAccuracy();
094        final double functionValueAccuracy = getFunctionValueAccuracy();
095
096        // x2 is the last root approximation
097        // x is the new approximation and new x2 for next round
098        // x0 < x1 < x2 does not hold here
099
100        double x0 = min;
101        double y0 = computeObjectiveValue(x0);
102        if (FastMath.abs(y0) < functionValueAccuracy) {
103            return x0;
104        }
105        double x1 = max;
106        double y1 = computeObjectiveValue(x1);
107        if (FastMath.abs(y1) < functionValueAccuracy) {
108            return x1;
109        }
110
111        if(y0 * y1 > 0) {
112            throw new NoBracketingException(x0, x1, y0, y1);
113        }
114
115        double x2 = 0.5 * (x0 + x1);
116        double y2 = computeObjectiveValue(x2);
117
118        double oldx = Double.POSITIVE_INFINITY;
119        while (true) {
120            // quadratic interpolation through x0, x1, x2
121            final double q = (x2 - x1) / (x1 - x0);
122            final double a = q * (y2 - (1 + q) * y1 + q * y0);
123            final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0;
124            final double c = (1 + q) * y2;
125            final double delta = b * b - 4 * a * c;
126            double x;
127            final double denominator;
128            if (delta >= 0.0) {
129                // choose a denominator larger in magnitude
130                double dplus = b + FastMath.sqrt(delta);
131                double dminus = b - FastMath.sqrt(delta);
132                denominator = FastMath.abs(dplus) > FastMath.abs(dminus) ? dplus : dminus;
133            } else {
134                // take the modulus of (B +/- FastMath.sqrt(delta))
135                denominator = FastMath.sqrt(b * b - delta);
136            }
137            if (denominator != 0) {
138                x = x2 - 2.0 * c * (x2 - x1) / denominator;
139                // perturb x if it exactly coincides with x1 or x2
140                // the equality tests here are intentional
141                while (x == x1 || x == x2) {
142                    x += absoluteAccuracy;
143                }
144            } else {
145                // extremely rare case, get a random number to skip it
146                x = min + FastMath.random() * (max - min);
147                oldx = Double.POSITIVE_INFINITY;
148            }
149            final double y = computeObjectiveValue(x);
150
151            // check for convergence
152            final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);
153            if (FastMath.abs(x - oldx) <= tolerance ||
154                FastMath.abs(y) <= functionValueAccuracy) {
155                return x;
156            }
157
158            // prepare the next iteration
159            x0 = x1;
160            y0 = y1;
161            x1 = x2;
162            y1 = y2;
163            x2 = x;
164            y2 = y;
165            oldx = x;
166        }
167    }
168}