edu.ucla.stat.SOCR.util
Class BesselFunction

java.lang.Object
  edu.ucla.stat.SOCR.util.BesselConstants
      edu.ucla.stat.SOCR.util.BesselFunction

public class BesselFunction
extends BesselConstants

BesselFunction and Airy functions.

Field Summary

protected static double[]	A_i0 Chebyshev coefficients for exp(-x) I0(x) in the interval [0,8].
protected static double[]	A_i1 Chebyshev coefficients for exp(-x) I1(x) / x in the interval [0,8].
protected static double[]	A_k0 COEFFICIENTS FOR METHODS k0, k0e *
protected static double[]	A_k1 COEFFICIENTS FOR METHODS k1, k1e *
protected static double[]	B_i0 Chebyshev coefficients for exp(-x) sqrt(x) I0(x) in the inverted interval [8,infinity].
protected static double[]	B_i1
protected static double[]	B_k0
protected static double[]	B_k1

Fields inherited from class edu.ucla.stat.SOCR.util.BesselConstants

big, biginv, LOGPI, MACHEP, MAXGAM, MAXLOG, MINLOG, SQRTH, SQTPI

Constructor Summary

protected

BesselFunction() Makes this class non instantiable, but still let's others inherit from it.

Method Summary

static double	i0(double x) Returns the modified BesselFunction function of order 0 of the argument.
static double	i0e(double x) Returns the exponentially scaled modified BesselFunction function of order 0 of the argument.
static double	i1(double x) Returns the modified BesselFunction function of order 1 of the argument.
static double	i1e(double x) Returns the exponentially scaled modified BesselFunction function of order 1 of the argument.
static double	in(int alpha, double y) Returns the modified BesselFunction function of the first kind of order nn of the argument.
static double	j0(double x) Returns the BesselFunction function of the first kind of order 0 of the argument.
static double	j1(double x) Returns the BesselFunction function of the first kind of order 1 of the argument.
static double	jn(int n, double x) Returns the BesselFunction function of the first kind of order n of the argument.
static double	k0(double x) Returns the modified BesselFunction function of the third kind of order 0 of the argument.
static double	k0e(double x) Returns the exponentially scaled modified BesselFunction function of the third kind of order 0 of the argument.
static double	k1(double x) Returns the modified BesselFunction function of the third kind of order 1 of the argument.
static double	k1e(double x) Returns the exponentially scaled modified BesselFunction function of the third kind of order 1 of the argument.
static double	kn(int nn, double x) Returns the modified BesselFunction function of the third kind of order nn of the argument.
static double	y0(double x) Returns the BesselFunction function of the second kind of order 0 of the argument.
static double	y1(double x) Returns the BesselFunction function of the second kind of order 1 of the argument.
static double	yn(int n, double x) Returns the BesselFunction function of the second kind of order n of the argument.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

A_i0

protected static final double[] A_i0

Chebyshev coefficients for exp(-x) I0(x) in the interval [0,8]. lim(x->0){ exp(-x) I0(x) } = 1.

B_i0

protected static final double[] B_i0

Chebyshev coefficients for exp(-x) sqrt(x) I0(x) in the inverted interval [8,infinity]. lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).

A_i1

protected static final double[] A_i1

Chebyshev coefficients for exp(-x) I1(x) / x in the interval [0,8]. lim(x->0){ exp(-x) I1(x) / x } = 1/2.

B_i1

protected static final double[] B_i1

A_k0

protected static final double[] A_k0

COEFFICIENTS FOR METHODS k0, k0e *

B_k0

protected static final double[] B_k0

A_k1

protected static final double[] A_k1

COEFFICIENTS FOR METHODS k1, k1e *

B_k1

protected static final double[] B_k1

Constructor Detail

BesselFunction

protected BesselFunction()

Makes this class non instantiable, but still let's others inherit from it.

Method Detail

i0

public static double i0(double x)
                 throws java.lang.ArithmeticException

Returns the modified BesselFunction function of order 0 of the argument.

The function is defined as i0(x) = j0( ix ).

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

i0e

public static double i0e(double x)
                  throws java.lang.ArithmeticException

Returns the exponentially scaled modified BesselFunction function of order 0 of the argument.

The function is defined as i0e(x) = exp(-|x|) j0( ix ).

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

i1

public static double i1(double x)
                 throws java.lang.ArithmeticException

Returns the modified BesselFunction function of order 1 of the argument.

The function is defined as i1(x) = -i j1( ix ).

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

i1e

public static double i1e(double x)
                  throws java.lang.ArithmeticException

Returns the exponentially scaled modified BesselFunction function of order 1 of the argument.

The function is defined as i1(x) = -i exp(-|x|) j1( ix ).

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

j0

public static double j0(double x)
                 throws java.lang.ArithmeticException

Returns the BesselFunction function of the first kind of order 0 of the argument.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

j1

public static double j1(double x)
                 throws java.lang.ArithmeticException

Returns the BesselFunction function of the first kind of order 1 of the argument.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

jn

public static double jn(int n,
                        double x)
                 throws java.lang.ArithmeticException

Returns the BesselFunction function of the first kind of order n of the argument.

Parameters:

n - the order of the BesselFunction function.

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

k0

public static double k0(double x)
                 throws java.lang.ArithmeticException

Returns the modified BesselFunction function of the third kind of order 0 of the argument.

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

k0e

public static double k0e(double x)
                  throws java.lang.ArithmeticException

Returns the exponentially scaled modified BesselFunction function of the third kind of order 0 of the argument.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

k1

public static double k1(double x)
                 throws java.lang.ArithmeticException

Returns the modified BesselFunction function of the third kind of order 1 of the argument.

The range is partitioned into the two intervals [0,2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

k1e

public static double k1e(double x)
                  throws java.lang.ArithmeticException

Returns the exponentially scaled modified BesselFunction function of the third kind of order 1 of the argument.

k1e(x) = exp(x) * k1(x).

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

in

public static double in(int alpha,
                        double y)
                 throws java.lang.ArithmeticException

Returns the modified BesselFunction function of the first kind of order nn of the argument.

This is an approximate solution using the formula here: http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution $I\_\backslash nu(z)$ is a modified [[Bessel function]] of the first kind given by $I\_a(y)\; =\; (y/2)^a\; \backslash sum\_\{j=0\}^\backslash infty\; \backslash frac\{\; (y^2/4)^j\}\{j!\; \backslash Gamma(a+j+1)\}\; .$

Parameters:

nn - the order of the BesselFunction function.

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

kn

public static double kn(int nn,
                        double x)
                 throws java.lang.ArithmeticException

Returns the modified BesselFunction function of the third kind of order nn of the argument.

The range is partitioned into the two intervals [0,9.55] and (9.55, infinity). An ascending power series is used in the low range, and an asymptotic expansion in the high range.

Parameters:

nn - the order of the BesselFunction function.

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

y0

public static double y0(double x)
                 throws java.lang.ArithmeticException

Returns the BesselFunction function of the second kind of order 0 of the argument.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

y1

public static double y1(double x)
                 throws java.lang.ArithmeticException

Returns the BesselFunction function of the second kind of order 1 of the argument.

Parameters:

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException

yn

public static double yn(int n,
                        double x)
                 throws java.lang.ArithmeticException

Returns the BesselFunction function of the second kind of order n of the argument.

Parameters:

n - the order of the BesselFunction function.

x - the value to compute the bessel function of.

Throws:

java.lang.ArithmeticException