/*
* Open Source Physics software is free software as described near the bottom of this code file.
*
* For additional information and documentation on Open Source Physics please see:
* <http://www.opensourcephysics.org/>
*/
package org.opensourcephysics.numerics.specialfunctions;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.Map;
import org.opensourcephysics.numerics.Function;
import org.opensourcephysics.numerics.Polynomial;
/**
* Computes Laguerre polynomials and associated Laguerre polynomials.
*
* @author W. Christian
* @version 1.0
*/
public class Legendre {
static final ArrayList<Polynomial> legendreList;
static final Map<QNKey, AssociatedLegendreFunction> associatedMap = new HashMap<QNKey, AssociatedLegendreFunction>();
private Legendre() {} // all methods are static so prohibit instantiation
/**
* Gets the el-th Legendre polynomial. If the polynomial has already been calculated
* it is returned from the list. Uses the recurrence relationship to construct new polynomials
* based on lower order polynomials.
*/
public static synchronized Polynomial getPolynomial(int el) {
if(el<0) {
throw new IllegalArgumentException(Messages.getString("Legendre.neg_degree")); //$NON-NLS-1$
}
if(el<legendreList.size()) {
return legendreList.get(el);
}
Polynomial pk = new Polynomial(new double[] {0, (2.0*el-1.0)/el});
Polynomial p1 = getPolynomial(el-1).multiply(pk);
Polynomial p2 = getPolynomial(el-2).multiply((1.0-el)/el);
Polynomial p = p1.add(p2);
legendreList.add(p); // polynomial was not in the list so add it.
return p;
}
/**
* Gets the associated Legendre function. If the function has already been calculated
* it is returned from the map. Uses the Legendre polynomial recurrence relationship to construct new functions
* based on derivatives of lower order polynomials.
*/
public static synchronized Function getAssociatedFunction(int el, int m) {
if(m*m>el*el) {
throw new IllegalArgumentException(Messages.getString("Legendre.out_of_range_m")); //$NON-NLS-1$
}
QNKey key = new QNKey(el, m);
AssociatedLegendreFunction f = associatedMap.get(key);
if(f!=null) {
return f;
}
f = new AssociatedLegendreFunction(el, m);
associatedMap.put(key, f); // polynomial was not in the list so add it.
return f;
}
/**
* Evaluates the el-th Legendre polynomial at x.
*
* @return the value of the function
*/
public static double evaluate(int el, double x) {
return getPolynomial(el).evaluate(x);
}
/**
* Computes the AssoicatedLegendre function.
* @author Wolfgang Christian
*/
static class AssociatedLegendreFunction implements Function {
Polynomial p;
int n, m;
boolean oddPower, positiveM;
AssociatedLegendreFunction(int el, int m) {
this.n = el;
positiveM = (m>0);
m = Math.abs(m); // following definition valid for positive m
this.m = m;
oddPower = (m%2==1);
p = getPolynomial(n);
for(int i = 0; i<m; i++) {
p = p.derivative();
}
if(positiveM&&oddPower) { // odd positive m terms have negative sign
p = p.multiply(-1);
}
if(!positiveM) { // apply definition for negative m
p = p.multiply(Factorials.factorial(n-m)/Factorials.factorial(n+m));
}
}
/**
* Evaluates the associated Legendre function.
*/
public double evaluate(final double x) {
double val = Math.pow((1-x*x), m/2)*p.evaluate(x); // power function is more efficient with integer values
return oddPower ? val*Math.sqrt(1-x*x) : val;
}
}
static {
// seed the first two Legendre polynomials
legendreList = new ArrayList<Polynomial>();
Polynomial p0 = new Polynomial(new double[] {1.0});
legendreList.add(p0);
Polynomial p1 = new Polynomial(new double[] {0, 1.0});
legendreList.add(p1);
}
}
/*
* Open Source Physics software is free software; you can redistribute
* it and/or modify it under the terms of the GNU General Public License (GPL) as
* published by the Free Software Foundation; either version 2 of the License,
* or(at your option) any later version.
* Code that uses any portion of the code in the org.opensourcephysics package
* or any subpackage (subdirectory) of this package must must also be be released
* under the GNU GPL license.
*
* This software is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston MA 02111-1307 USA
* or view the license online at http://www.gnu.org/copyleft/gpl.html
*
* Copyright (c) 2007 The Open Source Physics project
* http://www.opensourcephysics.org
*/