/*
* 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.integration.gauss;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.Pair;
import org.apache.commons.math3.util.FastMath;
Factory that creates a
Gauss-type quadrature rule using Hermite polynomials
of the first kind.
Such a quadrature rule allows the calculation of improper integrals
of a function
\(f(x) e^{-x^2}\)
Recurrence relation and weights computation follow
Abramowitz and Stegun, 1964.
The coefficients of the standard Hermite polynomials grow very rapidly.
In order to avoid overflows, each Hermite polynomial is normalized with
respect to the underlying scalar product.
The initial interval for the application of the bisection method is
based on the roots of the previous Hermite polynomial (interlacing).
Upper and lower bounds of these roots are provided by
I. Krasikov,
Nonnegative quadratic forms and bounds on orthogonal polynomials,
Journal of Approximation theory 111, 31-49
Since: 3.3
/**
* Factory that creates a
* <a href="http://en.wikipedia.org/wiki/Gauss-Hermite_quadrature">
* Gauss-type quadrature rule using Hermite polynomials</a>
* of the first kind.
* Such a quadrature rule allows the calculation of improper integrals
* of a function
* <p>
* \(f(x) e^{-x^2}\)
* </p><p>
* Recurrence relation and weights computation follow
* <a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun">
* Abramowitz and Stegun, 1964</a>.
* </p><p>
* The coefficients of the standard Hermite polynomials grow very rapidly.
* In order to avoid overflows, each Hermite polynomial is normalized with
* respect to the underlying scalar product.
* The initial interval for the application of the bisection method is
* based on the roots of the previous Hermite polynomial (interlacing).
* Upper and lower bounds of these roots are provided by </p>
* <blockquote>
* I. Krasikov,
* <em>Nonnegative quadratic forms and bounds on orthogonal polynomials</em>,
* Journal of Approximation theory <b>111</b>, 31-49
* </blockquote>
*
* @since 3.3
*/
public class HermiteRuleFactory extends BaseRuleFactory<Double> {
π1/2 /** π<sup>1/2</sup> */
private static final double SQRT_PI = 1.77245385090551602729;
π-1/4 /** π<sup>-1/4</sup> */
private static final double H0 = 7.5112554446494248286e-1;
π-1/4 √2 /** π<sup>-1/4</sup> √2 */
private static final double H1 = 1.0622519320271969145;
{@inheritDoc} /** {@inheritDoc} */
@Override
protected Pair<Double[], Double[]> computeRule(int numberOfPoints)
throws DimensionMismatchException {
if (numberOfPoints == 1) {
// Break recursion.
return new Pair<Double[], Double[]>(new Double[] { 0d },
new Double[] { SQRT_PI });
}
// Get previous rule.
// If it has not been computed yet it will trigger a recursive call
// to this method.
final int lastNumPoints = numberOfPoints - 1;
final Double[] previousPoints = getRuleInternal(lastNumPoints).getFirst();
// Compute next rule.
final Double[] points = new Double[numberOfPoints];
final Double[] weights = new Double[numberOfPoints];
final double sqrtTwoTimesLastNumPoints = FastMath.sqrt(2 * lastNumPoints);
final double sqrtTwoTimesNumPoints = FastMath.sqrt(2 * numberOfPoints);
// Find i-th root of H[n+1] by bracketing.
final int iMax = numberOfPoints / 2;
for (int i = 0; i < iMax; i++) {
// Lower-bound of the interval.
double a = (i == 0) ? -sqrtTwoTimesLastNumPoints : previousPoints[i - 1].doubleValue();
// Upper-bound of the interval.
double b = (iMax == 1) ? -0.5 : previousPoints[i].doubleValue();
// H[j-1](a)
double hma = H0;
// H[j](a)
double ha = H1 * a;
// H[j-1](b)
double hmb = H0;
// H[j](b)
double hb = H1 * b;
for (int j = 1; j < numberOfPoints; j++) {
// Compute H[j+1](a) and H[j+1](b)
final double jp1 = j + 1;
final double s = FastMath.sqrt(2 / jp1);
final double sm = FastMath.sqrt(j / jp1);
final double hpa = s * a * ha - sm * hma;
final double hpb = s * b * hb - sm * hmb;
hma = ha;
ha = hpa;
hmb = hb;
hb = hpb;
}
// Now ha = H[n+1](a), and hma = H[n](a) (same holds for b).
// Middle of the interval.
double c = 0.5 * (a + b);
// P[j-1](c)
double hmc = H0;
// P[j](c)
double hc = H1 * c;
boolean done = false;
while (!done) {
done = b - a <= Math.ulp(c);
hmc = H0;
hc = H1 * c;
for (int j = 1; j < numberOfPoints; j++) {
// Compute H[j+1](c)
final double jp1 = j + 1;
final double s = FastMath.sqrt(2 / jp1);
final double sm = FastMath.sqrt(j / jp1);
final double hpc = s * c * hc - sm * hmc;
hmc = hc;
hc = hpc;
}
// Now h = H[n+1](c) and hm = H[n](c).
if (!done) {
if (ha * hc < 0) {
b = c;
hmb = hmc;
hb = hc;
} else {
a = c;
hma = hmc;
ha = hc;
}
c = 0.5 * (a + b);
}
}
final double d = sqrtTwoTimesNumPoints * hmc;
final double w = 2 / (d * d);
points[i] = c;
weights[i] = w;
final int idx = lastNumPoints - i;
points[idx] = -c;
weights[idx] = w;
}
// If "numberOfPoints" is odd, 0 is a root.
// Note: as written, the test for oddness will work for negative
// integers too (although it is not necessary here), preventing
// a FindBugs warning.
if (numberOfPoints % 2 != 0) {
double hm = H0;
for (int j = 1; j < numberOfPoints; j += 2) {
final double jp1 = j + 1;
hm = -FastMath.sqrt(j / jp1) * hm;
}
final double d = sqrtTwoTimesNumPoints * hm;
final double w = 2 / (d * d);
points[iMax] = 0d;
weights[iMax] = w;
}
return new Pair<Double[], Double[]>(points, weights);
}
}