| /* |
| * 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.math4.legacy.analysis.solvers; |
| |
| import org.apache.commons.math4.legacy.exception.NoBracketingException; |
| import org.apache.commons.math4.legacy.exception.NumberIsTooLargeException; |
| import org.apache.commons.math4.legacy.exception.TooManyEvaluationsException; |
| import org.apache.commons.math4.core.jdkmath.JdkMath; |
| |
| /** |
| * 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 (JdkMath.abs(fMin) < functionValueAccuracy) { |
| return min; |
| } |
| final double fMax = computeObjectiveValue(max); |
| if (JdkMath.abs(fMax) < functionValueAccuracy) { |
| return max; |
| } |
| final double fInitial = computeObjectiveValue(initial); |
| if (JdkMath.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 + JdkMath.sqrt(delta)); |
| final double xminus = x1 + (-2.0 * y1) / (c1 - JdkMath.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 = JdkMath.max(relativeAccuracy * JdkMath.abs(x), absoluteAccuracy); |
| if (JdkMath.abs(x - oldx) <= tolerance || |
| JdkMath.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 (JdkMath.signum(y0) + JdkMath.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; |
| } |
| } |
| } |
| } |