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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.analysis.solvers;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.util.FastMath;
/**
* This class implements the <a href="http://mathworld.wolfram.com/MullersMethod.html">
* Muller's Method</a> for root finding of real univariate functions. For
* reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477,
* chapter 3.
* <p>
* Muller's method applies to both real and complex functions, but here we
* restrict ourselves to real functions.
* This class differs from {@link MullerSolver} in the way it avoids complex
* operations.</p><p>
* Muller's original method would have function evaluation at complex point.
* Since our f(x) is real, we have to find ways to avoid that. Bracketing
* condition is one way to go: by requiring bracketing in every iteration,
* the newly computed approximation is guaranteed to be real.</p>
* <p>
* Normally Muller's method converges quadratically in the vicinity of a
* zero, however it may be very slow in regions far away from zeros. For
* example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use
* bisection as a safety backup if it performs very poorly.</p>
* <p>
* The formulas here use divided differences directly.</p>
*
* @since 1.2
* @see MullerSolver2
*/
public class MullerSolver extends AbstractUnivariateSolver {
/** Default absolute accuracy. */
private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
/**
* Construct a solver with default accuracy (1e-6).
*/
public MullerSolver() {
this(DEFAULT_ABSOLUTE_ACCURACY);
}
/**
* Construct a solver.
*
* @param absoluteAccuracy Absolute accuracy.
*/
public MullerSolver(double absoluteAccuracy) {
super(absoluteAccuracy);
}
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
*/
public MullerSolver(double relativeAccuracy,
double absoluteAccuracy) {
super(relativeAccuracy, absoluteAccuracy);
}
/**
* {@inheritDoc}
*/
@Override
protected double doSolve()
throws TooManyEvaluationsException,
NumberIsTooLargeException,
NoBracketingException {
final double min = getMin();
final double max = getMax();
final double initial = getStartValue();
final double functionValueAccuracy = getFunctionValueAccuracy();
verifySequence(min, initial, max);
// check for zeros before verifying bracketing
final double fMin = computeObjectiveValue(min);
if (FastMath.abs(fMin) < functionValueAccuracy) {
return min;
}
final double fMax = computeObjectiveValue(max);
if (FastMath.abs(fMax) < functionValueAccuracy) {
return max;
}
final double fInitial = computeObjectiveValue(initial);
if (FastMath.abs(fInitial) < functionValueAccuracy) {
return initial;
}
verifyBracketing(min, max);
if (isBracketing(min, initial)) {
return solve(min, initial, fMin, fInitial);
} else {
return solve(initial, max, fInitial, fMax);
}
}
/**
* Find a real root in the given interval.
*
* @param min Lower bound for the interval.
* @param max Upper bound for the interval.
* @param fMin function value at the lower bound.
* @param fMax function value at the upper bound.
* @return the point at which the function value is zero.
* @throws TooManyEvaluationsException if the allowed number of calls to
* the function to be solved has been exhausted.
*/
private double solve(double min, double max,
double fMin, double fMax)
throws TooManyEvaluationsException {
final double relativeAccuracy = getRelativeAccuracy();
final double absoluteAccuracy = getAbsoluteAccuracy();
final double functionValueAccuracy = getFunctionValueAccuracy();
// [x0, x2] is the bracketing interval in each iteration
// x1 is the last approximation and an interpolation point in (x0, x2)
// x is the new root approximation and new x1 for next round
// d01, d12, d012 are divided differences
double x0 = min;
double y0 = fMin;
double x2 = max;
double y2 = fMax;
double x1 = 0.5 * (x0 + x2);
double y1 = computeObjectiveValue(x1);
double oldx = Double.POSITIVE_INFINITY;
while (true) {
// Muller's method employs quadratic interpolation through
// x0, x1, x2 and x is the zero of the interpolating parabola.
// Due to bracketing condition, this parabola must have two
// real roots and we choose one in [x0, x2] to be x.
final double d01 = (y1 - y0) / (x1 - x0);
final double d12 = (y2 - y1) / (x2 - x1);
final double d012 = (d12 - d01) / (x2 - x0);
final double c1 = d01 + (x1 - x0) * d012;
final double delta = c1 * c1 - 4 * y1 * d012;
final double xplus = x1 + (-2.0 * y1) / (c1 + FastMath.sqrt(delta));
final double xminus = x1 + (-2.0 * y1) / (c1 - FastMath.sqrt(delta));
// xplus and xminus are two roots of parabola and at least
// one of them should lie in (x0, x2)
final double x = isSequence(x0, xplus, x2) ? xplus : xminus;
final double y = computeObjectiveValue(x);
// check for convergence
final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);
if (FastMath.abs(x - oldx) <= tolerance ||
FastMath.abs(y) <= functionValueAccuracy) {
return x;
}
// Bisect if convergence is too slow. Bisection would waste
// our calculation of x, hopefully it won't happen often.
// the real number equality test x == x1 is intentional and
// completes the proximity tests above it
boolean bisect = (x < x1 && (x1 - x0) > 0.95 * (x2 - x0)) ||
(x > x1 && (x2 - x1) > 0.95 * (x2 - x0)) ||
(x == x1);
// prepare the new bracketing interval for next iteration
if (!bisect) {
x0 = x < x1 ? x0 : x1;
y0 = x < x1 ? y0 : y1;
x2 = x > x1 ? x2 : x1;
y2 = x > x1 ? y2 : y1;
x1 = x; y1 = y;
oldx = x;
} else {
double xm = 0.5 * (x0 + x2);
double ym = computeObjectiveValue(xm);
if (FastMath.signum(y0) + FastMath.signum(ym) == 0.0) {
x2 = xm; y2 = ym;
} else {
x0 = xm; y0 = ym;
}
x1 = 0.5 * (x0 + x2);
y1 = computeObjectiveValue(x1);
oldx = Double.POSITIVE_INFINITY;
}
}
}
}
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