| /* |
| * 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 févr. 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); |
| } |
| } |
