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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.math3.analysis.solvers;
import org.apache.commons.math3.complex.Complex;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
/**
* Implements the
* Laguerre's Method for root finding of real coefficient polynomials.
* For reference, see
*
* A First Course in Numerical Analysis
* ISBN 048641454X, chapter 8.
*
* 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.
* The algorithm requires a bracketing condition.
*
* @version $Id: LaguerreSolver.java 1296557 2012-03-03 02:07:07Z erans $
* @since 1.2
*/
public class LaguerreSolver extends AbstractPolynomialSolver {
/** Default absolute accuracy. */
private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
/** Complex solver. */
private final ComplexSolver complexSolver = new ComplexSolver();
/**
* Construct a solver with default accuracy (1e-6).
*/
public LaguerreSolver() {
this(DEFAULT_ABSOLUTE_ACCURACY);
}
/**
* Construct a solver.
*
* @param absoluteAccuracy Absolute accuracy.
*/
public LaguerreSolver(double absoluteAccuracy) {
super(absoluteAccuracy);
}
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
*/
public LaguerreSolver(double relativeAccuracy,
double absoluteAccuracy) {
super(relativeAccuracy, absoluteAccuracy);
}
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
* @param functionValueAccuracy Function value accuracy.
*/
public LaguerreSolver(double relativeAccuracy,
double absoluteAccuracy,
double functionValueAccuracy) {
super(relativeAccuracy, absoluteAccuracy, functionValueAccuracy);
}
/**
* {@inheritDoc}
*/
@Override
public double doSolve() {
double min = getMin();
double max = getMax();
double initial = getStartValue();
final double functionValueAccuracy = getFunctionValueAccuracy();
verifySequence(min, initial, max);
// Return the initial guess if it is good enough.
double yInitial = computeObjectiveValue(initial);
if (FastMath.abs(yInitial) <= functionValueAccuracy) {
return initial;
}
// Return the first endpoint if it is good enough.
double yMin = computeObjectiveValue(min);
if (FastMath.abs(yMin) <= functionValueAccuracy) {
return min;
}
// Reduce interval if min and initial bracket the root.
if (yInitial * yMin < 0) {
return laguerre(min, initial, yMin, yInitial);
}
// Return the second endpoint if it is good enough.
double yMax = computeObjectiveValue(max);
if (FastMath.abs(yMax) <= functionValueAccuracy) {
return max;
}
// Reduce interval if initial and max bracket the root.
if (yInitial * yMax < 0) {
return laguerre(initial, max, yInitial, yMax);
}
throw new NoBracketingException(min, max, yMin, yMax);
}
/**
* Find a real root in the given interval.
*
* Despite the bracketing condition, the root returned by
* {@link LaguerreSolver.ComplexSolver#solve(Complex[],Complex)} may
* not be a real zero inside {@code [min, max]}.
* For example, p(x) = x3 + 1,
* with {@code min = -2}, {@code max = 2}, {@code initial = 0}.
* When it occurs, this code calls
* {@link LaguerreSolver.ComplexSolver#solveAll(Complex[],Complex)}
* in order to obtain all roots and picks up one real root.
*
* @param lo Lower bound of the search interval.
* @param hi Higher bound of the search interval.
* @param fLo Function value at the lower bound of the search interval.
* @param fHi Function value at the higher bound of the search interval.
* @return the point at which the function value is zero.
*/
public double laguerre(double lo, double hi,
double fLo, double fHi) {
double coefficients[] = getCoefficients();
Complex c[] = new Complex[coefficients.length];
for (int i = 0; i < coefficients.length; i++) {
c[i] = new Complex(coefficients[i], 0);
}
Complex initial = new Complex(0.5 * (lo + hi), 0);
Complex z = complexSolver.solve(c, initial);
if (complexSolver.isRoot(lo, hi, z)) {
return z.getReal();
} else {
double r = Double.NaN;
// Solve all roots and select the one we are seeking.
Complex[] root = complexSolver.solveAll(c, initial);
for (int i = 0; i < root.length; i++) {
if (complexSolver.isRoot(lo, hi, root[i])) {
r = root[i].getReal();
break;
}
}
return r;
}
}
/**
* Class for searching all (complex) roots.
*/
private class ComplexSolver {
/**
* Check whether the given complex root is actually a real zero
* in the given interval, within the solver tolerance level.
*
* @param min Lower bound for the interval.
* @param max Upper bound for the interval.
* @param z Complex root.
* @return {@code true} if z is a real zero.
*/
public boolean isRoot(double min, double max, Complex z) {
if (isSequence(min, z.getReal(), max)) {
double tolerance = FastMath.max(getRelativeAccuracy() * z.abs(), getAbsoluteAccuracy());
return (FastMath.abs(z.getImaginary()) <= tolerance) ||
(z.abs() <= getFunctionValueAccuracy());
}
return false;
}
/**
* Find all complex roots for the polynomial with the given
* coefficients, starting from the given initial value.
*
* @param coefficients Polynomial coefficients.
* @param initial Start value.
* @return the point at which the function value is zero.
* @throws org.apache.commons.math3.exception.TooManyEvaluationsException
* if the maximum number of evaluations is exceeded.
* @throws NullArgumentException if the {@code coefficients} is
* {@code null}.
* @throws NoDataException if the {@code coefficients} array is empty.
*/
public Complex[] solveAll(Complex coefficients[], Complex initial) {
if (coefficients == null) {
throw new NullArgumentException();
}
int n = coefficients.length - 1;
if (n == 0) {
throw new NoDataException(LocalizedFormats.POLYNOMIAL);
}
// Coefficients for deflated polynomial.
Complex c[] = new Complex[n + 1];
for (int i = 0; i <= n; i++) {
c[i] = coefficients[i];
}
// Solve individual roots 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]));
}
}
return root;
}
/**
* Find a complex root for the polynomial with the given coefficients,
* starting from the given initial value.
*
* @param coefficients Polynomial coefficients.
* @param initial Start value.
* @return the point at which the function value is zero.
* @throws org.apache.commons.math3.exception.TooManyEvaluationsException
* if the maximum number of evaluations is exceeded.
* @throws NullArgumentException if the {@code coefficients} is
* {@code null}.
* @throws NoDataException if the {@code coefficients} array is empty.
*/
public Complex solve(Complex coefficients[], Complex initial) {
if (coefficients == null) {
throw new NullArgumentException();
}
int n = coefficients.length - 1;
if (n == 0) {
throw new NoDataException(LocalizedFormats.POLYNOMIAL);
}
final double absoluteAccuracy = getAbsoluteAccuracy();
final double relativeAccuracy = getRelativeAccuracy();
final double functionValueAccuracy = getFunctionValueAccuracy();
Complex N = new Complex(n, 0.0);
Complex N1 = new Complex(n - 1, 0.0);
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 (true) {
// 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) {
return z;
}
if (pv.abs() <= functionValueAccuracy) {
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));
}
incrementEvaluationCount();
}
}
}
}