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.util.FastMath;
020 import org.apache.commons.math3.exception.NumberIsTooLargeException;
021 import org.apache.commons.math3.exception.NoBracketingException;
022 import org.apache.commons.math3.exception.TooManyEvaluationsException;
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 * Muller's original method would have function evaluation at complex point.
035 * Since our f(x) is real, we have to find ways to avoid that. Bracketing
036 * condition is one way to go: by requiring bracketing in every iteration,
037 * the newly computed approximation is guaranteed to be real.</p>
038 * <p>
039 * Normally Muller's method converges quadratically in the vicinity of a
040 * zero, however it may be very slow in regions far away from zeros. For
041 * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use
042 * bisection as a safety backup if it performs very poorly.</p>
043 * <p>
044 * The formulas here use divided differences directly.</p>
045 *
046 * @version $Id: MullerSolver.java 1391927 2012-09-30 00:03:30Z erans $
047 * @since 1.2
048 * @see MullerSolver2
049 */
050 public class MullerSolver 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 MullerSolver() {
059 this(DEFAULT_ABSOLUTE_ACCURACY);
060 }
061 /**
062 * Construct a solver.
063 *
064 * @param absoluteAccuracy Absolute accuracy.
065 */
066 public MullerSolver(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 MullerSolver(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 final double initial = getStartValue();
091
092 final double functionValueAccuracy = getFunctionValueAccuracy();
093
094 verifySequence(min, initial, max);
095
096 // check for zeros before verifying bracketing
097 final double fMin = computeObjectiveValue(min);
098 if (FastMath.abs(fMin) < functionValueAccuracy) {
099 return min;
100 }
101 final double fMax = computeObjectiveValue(max);
102 if (FastMath.abs(fMax) < functionValueAccuracy) {
103 return max;
104 }
105 final double fInitial = computeObjectiveValue(initial);
106 if (FastMath.abs(fInitial) < functionValueAccuracy) {
107 return initial;
108 }
109
110 verifyBracketing(min, max);
111
112 if (isBracketing(min, initial)) {
113 return solve(min, initial, fMin, fInitial);
114 } else {
115 return solve(initial, max, fInitial, fMax);
116 }
117 }
118
119 /**
120 * Find a real root in the given interval.
121 *
122 * @param min Lower bound for the interval.
123 * @param max Upper bound for the interval.
124 * @param fMin function value at the lower bound.
125 * @param fMax function value at the upper bound.
126 * @return the point at which the function value is zero.
127 * @throws TooManyEvaluationsException if the allowed number of calls to
128 * the function to be solved has been exhausted.
129 */
130 private double solve(double min, double max,
131 double fMin, double fMax)
132 throws TooManyEvaluationsException {
133 final double relativeAccuracy = getRelativeAccuracy();
134 final double absoluteAccuracy = getAbsoluteAccuracy();
135 final double functionValueAccuracy = getFunctionValueAccuracy();
136
137 // [x0, x2] is the bracketing interval in each iteration
138 // x1 is the last approximation and an interpolation point in (x0, x2)
139 // x is the new root approximation and new x1 for next round
140 // d01, d12, d012 are divided differences
141
142 double x0 = min;
143 double y0 = fMin;
144 double x2 = max;
145 double y2 = fMax;
146 double x1 = 0.5 * (x0 + x2);
147 double y1 = computeObjectiveValue(x1);
148
149 double oldx = Double.POSITIVE_INFINITY;
150 while (true) {
151 // Muller's method employs quadratic interpolation through
152 // x0, x1, x2 and x is the zero of the interpolating parabola.
153 // Due to bracketing condition, this parabola must have two
154 // real roots and we choose one in [x0, x2] to be x.
155 final double d01 = (y1 - y0) / (x1 - x0);
156 final double d12 = (y2 - y1) / (x2 - x1);
157 final double d012 = (d12 - d01) / (x2 - x0);
158 final double c1 = d01 + (x1 - x0) * d012;
159 final double delta = c1 * c1 - 4 * y1 * d012;
160 final double xplus = x1 + (-2.0 * y1) / (c1 + FastMath.sqrt(delta));
161 final double xminus = x1 + (-2.0 * y1) / (c1 - FastMath.sqrt(delta));
162 // xplus and xminus are two roots of parabola and at least
163 // one of them should lie in (x0, x2)
164 final double x = isSequence(x0, xplus, x2) ? xplus : xminus;
165 final double y = computeObjectiveValue(x);
166
167 // check for convergence
168 final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);
169 if (FastMath.abs(x - oldx) <= tolerance ||
170 FastMath.abs(y) <= functionValueAccuracy) {
171 return x;
172 }
173
174 // Bisect if convergence is too slow. Bisection would waste
175 // our calculation of x, hopefully it won't happen often.
176 // the real number equality test x == x1 is intentional and
177 // completes the proximity tests above it
178 boolean bisect = (x < x1 && (x1 - x0) > 0.95 * (x2 - x0)) ||
179 (x > x1 && (x2 - x1) > 0.95 * (x2 - x0)) ||
180 (x == x1);
181 // prepare the new bracketing interval for next iteration
182 if (!bisect) {
183 x0 = x < x1 ? x0 : x1;
184 y0 = x < x1 ? y0 : y1;
185 x2 = x > x1 ? x2 : x1;
186 y2 = x > x1 ? y2 : y1;
187 x1 = x; y1 = y;
188 oldx = x;
189 } else {
190 double xm = 0.5 * (x0 + x2);
191 double ym = computeObjectiveValue(xm);
192 if (FastMath.signum(y0) + FastMath.signum(ym) == 0.0) {
193 x2 = xm; y2 = ym;
194 } else {
195 x0 = xm; y0 = ym;
196 }
197 x1 = 0.5 * (x0 + x2);
198 y1 = computeObjectiveValue(x1);
199 oldx = Double.POSITIVE_INFINITY;
200 }
201 }
202 }
203 }