| /* |
| * 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.math.analysis.solvers; |
| |
| import org.apache.commons.math.ConvergenceException; |
| import org.apache.commons.math.FunctionEvaluationException; |
| import org.apache.commons.math.MaxIterationsExceededException; |
| import org.apache.commons.math.analysis.UnivariateRealFunction; |
| import org.apache.commons.math.util.FastMath; |
| import org.apache.commons.math.util.MathUtils; |
| |
| /** |
| * 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. Methods solve() and solve2() find |
| * real zeros, using different ways to bypass complex arithmetics.</p> |
| * |
| * @version $Revision: 1070725 $ $Date: 2011-02-15 02:31:12 +0100 (mar. 15 févr. 2011) $ |
| * @since 1.2 |
| */ |
| public class MullerSolver extends UnivariateRealSolverImpl { |
| |
| /** |
| * Construct a solver for the given function. |
| * |
| * @param f function to solve |
| * @deprecated as of 2.0 the function to solve is passed as an argument |
| * to the {@link #solve(UnivariateRealFunction, double, double)} or |
| * {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)} |
| * method. |
| */ |
| @Deprecated |
| public MullerSolver(UnivariateRealFunction f) { |
| super(f, 100, 1E-6); |
| } |
| |
| /** |
| * Construct a solver. |
| * @deprecated in 2.2 (to be removed in 3.0). |
| */ |
| @Deprecated |
| public MullerSolver() { |
| super(100, 1E-6); |
| } |
| |
| /** {@inheritDoc} */ |
| @Deprecated |
| public double solve(final double min, final double max) |
| throws ConvergenceException, FunctionEvaluationException { |
| return solve(f, min, max); |
| } |
| |
| /** {@inheritDoc} */ |
| @Deprecated |
| public double solve(final double min, final double max, final double initial) |
| throws ConvergenceException, FunctionEvaluationException { |
| return solve(f, min, max, initial); |
| } |
| |
| /** |
| * Find a real root in the given interval with initial value. |
| * <p> |
| * Requires bracketing condition.</p> |
| * |
| * @param f the function to solve |
| * @param min the lower bound for the interval |
| * @param max the upper bound for the interval |
| * @param initial the start value to use |
| * @param maxEval Maximum number of evaluations. |
| * @return the point at which the function value is zero |
| * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| * or the solver detects convergence problems otherwise |
| * @throws FunctionEvaluationException if an error occurs evaluating the function |
| * @throws IllegalArgumentException if any parameters are invalid |
| */ |
| @Override |
| public double solve(int maxEval, final UnivariateRealFunction f, |
| final double min, final double max, final double initial) |
| throws MaxIterationsExceededException, FunctionEvaluationException { |
| setMaximalIterationCount(maxEval); |
| return solve(f, min, max, initial); |
| } |
| |
| /** |
| * Find a real root in the given interval with initial value. |
| * <p> |
| * Requires bracketing condition.</p> |
| * |
| * @param f the function to solve |
| * @param min the lower bound for the interval |
| * @param max the upper bound for the interval |
| * @param initial the start value to use |
| * @return the point at which the function value is zero |
| * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| * or the solver detects convergence problems otherwise |
| * @throws FunctionEvaluationException if an error occurs evaluating the function |
| * @throws IllegalArgumentException if any parameters are invalid |
| * @deprecated in 2.2 (to be removed in 3.0). |
| */ |
| @Deprecated |
| public double solve(final UnivariateRealFunction f, |
| final double min, final double max, final double initial) |
| throws MaxIterationsExceededException, FunctionEvaluationException { |
| |
| // check for zeros before verifying bracketing |
| if (f.value(min) == 0.0) { return min; } |
| if (f.value(max) == 0.0) { return max; } |
| if (f.value(initial) == 0.0) { return initial; } |
| |
| verifyBracketing(min, max, f); |
| verifySequence(min, initial, max); |
| if (isBracketing(min, initial, f)) { |
| return solve(f, min, initial); |
| } else { |
| return solve(f, initial, max); |
| } |
| } |
| |
| /** |
| * Find a real root in the given interval. |
| * <p> |
| * Original Muller's 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> |
| * |
| * @param f the function to solve |
| * @param min the lower bound for the interval |
| * @param max the upper bound for the interval |
| * @param maxEval Maximum number of evaluations. |
| * @return the point at which the function value is zero |
| * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| * or the solver detects convergence problems otherwise |
| * @throws FunctionEvaluationException if an error occurs evaluating the function |
| * @throws IllegalArgumentException if any parameters are invalid |
| */ |
| @Override |
| public double solve(int maxEval, final UnivariateRealFunction f, |
| final double min, final double max) |
| throws MaxIterationsExceededException, FunctionEvaluationException { |
| setMaximalIterationCount(maxEval); |
| return solve(f, min, max); |
| } |
| |
| /** |
| * Find a real root in the given interval. |
| * <p> |
| * Original Muller's 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> |
| * |
| * @param f the function to solve |
| * @param min the lower bound for the interval |
| * @param max the upper bound for the interval |
| * @return the point at which the function value is zero |
| * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| * or the solver detects convergence problems otherwise |
| * @throws FunctionEvaluationException if an error occurs evaluating the function |
| * @throws IllegalArgumentException if any parameters are invalid |
| * @deprecated in 2.2 (to be removed in 3.0). |
| */ |
| @Deprecated |
| public double solve(final UnivariateRealFunction f, |
| final double min, final double max) |
| throws MaxIterationsExceededException, FunctionEvaluationException { |
| |
| // [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 = f.value(x0); |
| double x2 = max; |
| double y2 = f.value(x2); |
| double x1 = 0.5 * (x0 + x2); |
| double y1 = f.value(x1); |
| |
| // check for zeros before verifying bracketing |
| if (y0 == 0.0) { |
| return min; |
| } |
| if (y2 == 0.0) { |
| return max; |
| } |
| verifyBracketing(min, max, f); |
| |
| double oldx = Double.POSITIVE_INFINITY; |
| for (int i = 1; i <= maximalIterationCount; ++i) { |
| // 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 = f.value(x); |
| |
| // check for convergence |
| final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy); |
| if (FastMath.abs(x - oldx) <= tolerance) { |
| setResult(x, i); |
| return result; |
| } |
| if (FastMath.abs(y) <= functionValueAccuracy) { |
| setResult(x, i); |
| return result; |
| } |
| |
| // 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 = f.value(xm); |
| if (MathUtils.sign(y0) + MathUtils.sign(ym) == 0.0) { |
| x2 = xm; y2 = ym; |
| } else { |
| x0 = xm; y0 = ym; |
| } |
| x1 = 0.5 * (x0 + x2); |
| y1 = f.value(x1); |
| oldx = Double.POSITIVE_INFINITY; |
| } |
| } |
| throw new MaxIterationsExceededException(maximalIterationCount); |
| } |
| |
| /** |
| * Find a real root in the given interval. |
| * <p> |
| * solve2() differs from solve() in the way it avoids complex operations. |
| * Except for the initial [min, max], solve2() does not require bracketing |
| * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex |
| * number arises in the computation, we simply use its modulus as real |
| * approximation.</p> |
| * <p> |
| * Because the interval may not be bracketing, bisection alternative is |
| * not applicable here. However in practice our treatment usually works |
| * well, especially near real zeros where the imaginary part of complex |
| * approximation is often negligible.</p> |
| * <p> |
| * The formulas here do not use divided differences directly.</p> |
| * |
| * @param min the lower bound for the interval |
| * @param max the upper bound for the interval |
| * @return the point at which the function value is zero |
| * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| * or the solver detects convergence problems otherwise |
| * @throws FunctionEvaluationException if an error occurs evaluating the function |
| * @throws IllegalArgumentException if any parameters are invalid |
| * @deprecated replaced by {@link #solve2(UnivariateRealFunction, double, double)} |
| * since 2.0 |
| */ |
| @Deprecated |
| public double solve2(final double min, final double max) |
| throws MaxIterationsExceededException, FunctionEvaluationException { |
| return solve2(f, min, max); |
| } |
| |
| /** |
| * Find a real root in the given interval. |
| * <p> |
| * solve2() differs from solve() in the way it avoids complex operations. |
| * Except for the initial [min, max], solve2() does not require bracketing |
| * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex |
| * number arises in the computation, we simply use its modulus as real |
| * approximation.</p> |
| * <p> |
| * Because the interval may not be bracketing, bisection alternative is |
| * not applicable here. However in practice our treatment usually works |
| * well, especially near real zeros where the imaginary part of complex |
| * approximation is often negligible.</p> |
| * <p> |
| * The formulas here do not use divided differences directly.</p> |
| * |
| * @param f the function to solve |
| * @param min the lower bound for the interval |
| * @param max the upper bound for the interval |
| * @return the point at which the function value is zero |
| * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| * or the solver detects convergence problems otherwise |
| * @throws FunctionEvaluationException if an error occurs evaluating the function |
| * @throws IllegalArgumentException if any parameters are invalid |
| * @deprecated in 2.2 (to be removed in 3.0). |
| */ |
| @Deprecated |
| public double solve2(final UnivariateRealFunction f, |
| final double min, final double max) |
| throws MaxIterationsExceededException, FunctionEvaluationException { |
| |
| // x2 is the last root approximation |
| // x is the new approximation and new x2 for next round |
| // x0 < x1 < x2 does not hold here |
| |
| double x0 = min; |
| double y0 = f.value(x0); |
| double x1 = max; |
| double y1 = f.value(x1); |
| double x2 = 0.5 * (x0 + x1); |
| double y2 = f.value(x2); |
| |
| // check for zeros before verifying bracketing |
| if (y0 == 0.0) { return min; } |
| if (y1 == 0.0) { return max; } |
| verifyBracketing(min, max, f); |
| |
| double oldx = Double.POSITIVE_INFINITY; |
| for (int i = 1; i <= maximalIterationCount; ++i) { |
| // quadratic interpolation through x0, x1, x2 |
| final double q = (x2 - x1) / (x1 - x0); |
| final double a = q * (y2 - (1 + q) * y1 + q * y0); |
| final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0; |
| final double c = (1 + q) * y2; |
| final double delta = b * b - 4 * a * c; |
| double x; |
| final double denominator; |
| if (delta >= 0.0) { |
| // choose a denominator larger in magnitude |
| double dplus = b + FastMath.sqrt(delta); |
| double dminus = b - FastMath.sqrt(delta); |
| denominator = FastMath.abs(dplus) > FastMath.abs(dminus) ? dplus : dminus; |
| } else { |
| // take the modulus of (B +/- FastMath.sqrt(delta)) |
| denominator = FastMath.sqrt(b * b - delta); |
| } |
| if (denominator != 0) { |
| x = x2 - 2.0 * c * (x2 - x1) / denominator; |
| // perturb x if it exactly coincides with x1 or x2 |
| // the equality tests here are intentional |
| while (x == x1 || x == x2) { |
| x += absoluteAccuracy; |
| } |
| } else { |
| // extremely rare case, get a random number to skip it |
| x = min + FastMath.random() * (max - min); |
| oldx = Double.POSITIVE_INFINITY; |
| } |
| final double y = f.value(x); |
| |
| // check for convergence |
| final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy); |
| if (FastMath.abs(x - oldx) <= tolerance) { |
| setResult(x, i); |
| return result; |
| } |
| if (FastMath.abs(y) <= functionValueAccuracy) { |
| setResult(x, i); |
| return result; |
| } |
| |
| // prepare the next iteration |
| x0 = x1; |
| y0 = y1; |
| x1 = x2; |
| y1 = y2; |
| x2 = x; |
| y2 = y; |
| oldx = x; |
| } |
| throw new MaxIterationsExceededException(maximalIterationCount); |
| } |
| } |
