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Diffstat (limited to 'src/main/java/org/apache/commons/math/analysis/solvers/MullerSolver.java')
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diff --git a/src/main/java/org/apache/commons/math/analysis/solvers/MullerSolver.java b/src/main/java/org/apache/commons/math/analysis/solvers/MullerSolver.java new file mode 100644 index 0000000..a6d03bc --- /dev/null +++ b/src/main/java/org/apache/commons/math/analysis/solvers/MullerSolver.java @@ -0,0 +1,415 @@ +/* + * 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); + } +} |