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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.math.analysis.solvers; import org.apache.commons.math.ConvergenceException; import org.apache.commons.math.FunctionEvaluationException; import org.apache.commons.math.MathRuntimeException; import org.apache.commons.math.MaxIterationsExceededException; import org.apache.commons.math.analysis.UnivariateRealFunction; import org.apache.commons.math.analysis.polynomials.PolynomialFunction; import org.apache.commons.math.complex.Complex; import org.apache.commons.math.exception.util.LocalizedFormats; import org.apache.commons.math.util.FastMath; /** * Implements the <a href="http://mathworld.wolfram.com/LaguerresMethod.html"> * Laguerre's Method</a> for root finding of real coefficient polynomials. * For reference, see <b>A First Course in Numerical Analysis</b>, * ISBN 048641454X, chapter 8. * <p> * Laguerre's method is global in the sense that it can start with any initial * approximation and be able to solve all roots from that point.</p> * * @version $Revision: 1070725 $ $Date: 2011-02-15 02:31:12 +0100 (mar. 15 fvr. 2011) $ * @since 1.2 */ public class LaguerreSolver extends UnivariateRealSolverImpl { /** polynomial function to solve. * @deprecated as of 2.0 the function is not stored anymore in the instance */ @Deprecated private final PolynomialFunction p; /** * Construct a solver for the given function. * * @param f function to solve * @throws IllegalArgumentException if function is not polynomial * @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 LaguerreSolver(UnivariateRealFunction f) throws IllegalArgumentException { super(f, 100, 1E-6); if (f instanceof PolynomialFunction) { p = (PolynomialFunction) f; } else { throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.FUNCTION_NOT_POLYNOMIAL); } } /** * Construct a solver. * @deprecated in 2.2 (to be removed in 3.0) */ @Deprecated public LaguerreSolver() { super(100, 1E-6); p = null; } /** * Returns a copy of the polynomial function. * * @return a fresh copy of the polynomial function * @deprecated as of 2.0 the function is not stored anymore within the instance. */ @Deprecated public PolynomialFunction getPolynomialFunction() { return new PolynomialFunction(p.getCoefficients()); } /** {@inheritDoc} */ @Deprecated public double solve(final double min, final double max) throws ConvergenceException, FunctionEvaluationException { return solve(p, min, max); } /** {@inheritDoc} */ @Deprecated public double solve(final double min, final double max, final double initial) throws ConvergenceException, FunctionEvaluationException { return solve(p, min, max, initial); } /** * Find a real root in the given interval with initial value. * <p> * Requires bracketing condition.</p> * * @param f function to solve (must be polynomial) * @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 ConvergenceException 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 ConvergenceException, 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 function to solve (must be polynomial) * @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 ConvergenceException 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 ConvergenceException, 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> * Despite the bracketing condition, the root returned by solve(Complex[], * Complex) may not be a real zero inside [min, max]. For example, * p(x) = x^3 + 1, min = -2, max = 2, initial = 0. We can either try * another initial value, or, as we did here, call solveAll() to obtain * all roots and pick up the one that we're looking for.</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 ConvergenceException 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 ConvergenceException, FunctionEvaluationException { setMaximalIterationCount(maxEval); return solve(f, min, max); } /** * Find a real root in the given interval. * <p> * Despite the bracketing condition, the root returned by solve(Complex[], * Complex) may not be a real zero inside [min, max]. For example, * p(x) = x^3 + 1, min = -2, max = 2, initial = 0. We can either try * another initial value, or, as we did here, call solveAll() to obtain * all roots and pick up the one that we're looking for.</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 ConvergenceException 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 ConvergenceException, FunctionEvaluationException { // check function type if (!(f instanceof PolynomialFunction)) { throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.FUNCTION_NOT_POLYNOMIAL); } // check for zeros before verifying bracketing if (f.value(min) == 0.0) { return min; } if (f.value(max) == 0.0) { return max; } verifyBracketing(min, max, f); double coefficients[] = ((PolynomialFunction) f).getCoefficients(); Complex c[] = new Complex[coefficients.length]; for (int i = 0; i < coefficients.length; i++) { c[i] = new Complex(coefficients[i], 0.0); } Complex initial = new Complex(0.5 * (min + max), 0.0); Complex z = solve(c, initial); if (isRootOK(min, max, z)) { setResult(z.getReal(), iterationCount); return result; } // solve all roots and select the one we're seeking Complex[] root = solveAll(c, initial); for (int i = 0; i < root.length; i++) { if (isRootOK(min, max, root[i])) { setResult(root[i].getReal(), iterationCount); return result; } } // should never happen throw new ConvergenceException(); } /** * Returns true iff the given complex root is actually a real zero * in the given interval, within the solver tolerance level. * * @param min the lower bound for the interval * @param max the upper bound for the interval * @param z the complex root * @return true iff z is the sought-after real zero */ protected boolean isRootOK(double min, double max, Complex z) { double tolerance = FastMath.max(relativeAccuracy * z.abs(), absoluteAccuracy); return (isSequence(min, z.getReal(), max)) && (FastMath.abs(z.getImaginary()) <= tolerance || z.abs() <= functionValueAccuracy); } /** * Find all complex roots for the polynomial with the given coefficients, * starting from the given initial value. * * @param coefficients the polynomial coefficients array * @param initial the start value to use * @return the point at which the function value is zero * @throws ConvergenceException 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. */ @Deprecated public Complex[] solveAll(double coefficients[], double initial) throws ConvergenceException, FunctionEvaluationException { Complex c[] = new Complex[coefficients.length]; Complex z = new Complex(initial, 0.0); for (int i = 0; i < c.length; i++) { c[i] = new Complex(coefficients[i], 0.0); } return solveAll(c, z); } /** * Find all complex roots for the polynomial with the given coefficients, * starting from the given initial value. * * @param coefficients the polynomial coefficients array * @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. */ @Deprecated public Complex[] solveAll(Complex coefficients[], Complex initial) throws MaxIterationsExceededException, FunctionEvaluationException { int n = coefficients.length - 1; int iterationCount = 0; if (n < 1) { throw MathRuntimeException .createIllegalArgumentException(LocalizedFormats.NON_POSITIVE_POLYNOMIAL_DEGREE, n); } Complex c[] = new Complex[n + 1]; // coefficients for deflated polynomial for (int i = 0; i <= n; i++) { c[i] = coefficients[i]; } // solve individual root successively Complex root[] = new Complex[n]; for (int i = 0; i < n; i++) { Complex subarray[] = new Complex[n - i + 1]; System.arraycopy(c, 0, subarray, 0, subarray.length); root[i] = solve(subarray, initial); // polynomial deflation using synthetic division Complex newc = c[n - i]; Complex oldc = null; for (int j = n - i - 1; j >= 0; j--) { oldc = c[j]; c[j] = newc; newc = oldc.add(newc.multiply(root[i])); } iterationCount += this.iterationCount; } resultComputed = true; this.iterationCount = iterationCount; return root; } /** * Find a complex root for the polynomial with the given coefficients, * starting from the given initial value. * * @param coefficients the polynomial coefficients array * @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. */ @Deprecated public Complex solve(Complex coefficients[], Complex initial) throws MaxIterationsExceededException, FunctionEvaluationException { int n = coefficients.length - 1; if (n < 1) { throw MathRuntimeException .createIllegalArgumentException(LocalizedFormats.NON_POSITIVE_POLYNOMIAL_DEGREE, n); } Complex N = new Complex(n, 0.0); Complex N1 = new Complex(n - 1, 0.0); int i = 1; Complex pv = null; Complex dv = null; Complex d2v = null; Complex G = null; Complex G2 = null; Complex H = null; Complex delta = null; Complex denominator = null; Complex z = initial; Complex oldz = new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY); while (i <= maximalIterationCount) { // Compute pv (polynomial value), dv (derivative value), and // d2v (second derivative value) simultaneously. pv = coefficients[n]; dv = Complex.ZERO; d2v = Complex.ZERO; for (int j = n - 1; j >= 0; j--) { d2v = dv.add(z.multiply(d2v)); dv = pv.add(z.multiply(dv)); pv = coefficients[j].add(z.multiply(pv)); } d2v = d2v.multiply(new Complex(2.0, 0.0)); // check for convergence double tolerance = FastMath.max(relativeAccuracy * z.abs(), absoluteAccuracy); if ((z.subtract(oldz)).abs() <= tolerance) { resultComputed = true; iterationCount = i; return z; } if (pv.abs() <= functionValueAccuracy) { resultComputed = true; iterationCount = i; return z; } // now pv != 0, calculate the new approximation G = dv.divide(pv); G2 = G.multiply(G); H = G2.subtract(d2v.divide(pv)); delta = N1.multiply((N.multiply(H)).subtract(G2)); // choose a denominator larger in magnitude Complex deltaSqrt = delta.sqrt(); Complex dplus = G.add(deltaSqrt); Complex dminus = G.subtract(deltaSqrt); denominator = dplus.abs() > dminus.abs() ? dplus : dminus; // Perturb z if denominator is zero, for instance, // p(x) = x^3 + 1, z = 0. if (denominator.equals(new Complex(0.0, 0.0))) { z = z.add(new Complex(absoluteAccuracy, absoluteAccuracy)); oldz = new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY); } else { oldz = z; z = z.subtract(N.divide(denominator)); } i++; } throw new MaxIterationsExceededException(maximalIterationCount); } }