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 */
017 package org.apache.commons.math3.analysis.solvers;
018
019 import org.apache.commons.math3.exception.NoBracketingException;
020 import org.apache.commons.math3.exception.NumberIsTooLargeException;
021 import org.apache.commons.math3.exception.TooManyEvaluationsException;
022 import 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 */
050 public 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 }