GNU Octave - Special Functions

GNU Octave Manual Version 3
by John W. Eaton, David Bateman, Søren Hauberg
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GNU Octave Manual Version 3

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17.5 Special Functions

Loadable Function: [j, ierr] = besselj (alpha, x, opt)

Loadable Function: [y, ierr] = bessely (alpha, x, opt)

Loadable Function: [i, ierr] = besseli (alpha, x, opt)

Loadable Function: [k, ierr] = besselk (alpha, x, opt)

Loadable Function: [h, ierr] = besselh (alpha, k, x, opt)

Compute Bessel or Hankel functions of various kinds:

besselj: Bessel functions of the first kind. If the argument opt is supplied, the result is multiplied by exp(-abs(imag(x))).
bessely: Bessel functions of the second kind. If the argument opt is supplied, the result is multiplied by exp(-abs(imag(x))).
besseli: Modified Bessel functions of the first kind. If the argument opt is supplied, the result is multiplied by exp(-abs(real(x))).
besselk: Modified Bessel functions of the second kind. If the argument opt is supplied, the result is multiplied by exp(x).
besselh: Compute Hankel functions of the first (k = 1) or second (k = 2) kind. If the argument opt is supplied, the result is multiplied by exp (-I*x) for k = 1 or exp (I*x) for k = 2.

If alpha is a scalar, the result is the same size as x. If x is a scalar, the result is the same size as alpha. If alpha is a row vector and x is a column vector, the result is a matrix with length (x) rows and length (alpha) columns. Otherwise, alpha and x must conform and the result will be the same size.

The value of alpha must be real. The value of x may be complex.

If requested, ierr contains the following status information and is the same size as the result.

Normal return.
Input error, return NaN.
Overflow, return Inf.
Loss of significance by argument reduction results in less than half of machine accuracy.
Complete loss of significance by argument reduction, return NaN.
Error--no computation, algorithm termination condition not met, return NaN.

Loadable Function: [a, ierr] = airy (k, z, opt)

Compute Airy functions of the first and second kind, and their derivatives. If the optional argument opt is non-zero the value is multiplied by the scale factor given in the table below.

 K   Function   Scale factor (if 'opt' is supplied)
---  --------   ---------------------------------------
 0   Ai (Z)     exp ((2/3) * Z * sqrt (Z))
 1   dAi(Z)/dZ  exp ((2/3) * Z * sqrt (Z))
 2   Bi (Z)     exp (-abs (real ((2/3) * Z *sqrt (Z))))
 3   dBi(Z)/dZ  exp (-abs (real ((2/3) * Z *sqrt (Z))))

The function call airy (z) is equivalent to airy (0, z).

The result is the same size as z.

If requested, ierr contains the following status information and is the same size as the result.

Normal return.
Input error, return NaN.
Overflow, return Inf.
Loss of significance by argument reduction results in less than half of machine accuracy.
Complete loss of significance by argument reduction, return NaN.
Error--no computation, algorithm termination condition not met, return NaN.

Mapping Function: beta (a, b)

Return the Beta function,

beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).

Mapping Function: betainc (x, a, b)

Return the incomplete Beta function,

                                      x
                                     /
betainc (x, a, b) = beta (a, b)^(-1) | t^(a-1) (1-t)^(b-1) dt.
                                     /
                                  t=0

If x has more than one component, both a and b must be scalars. If x is a scalar, a and b must be of compatible dimensions.

Mapping Function: betaln (a, b)

Return the log of the Beta function,

betaln (a, b) = gammaln (a) + gammaln (b) - gammaln (a + b)

See also gamma, lgamma

Function File: L = legendre (n, X)

Computes the Legendre Function of degree n and order m where all values for m = 0 ... n are returned. n must be a scalar in the range [0 ... 255]. The return value has one dimension more than x.

The Legendre Function of degree n and order m is defined as

 m        m       2  m/2   d^m
P(x) = (-1) * (1-x  )    * ----  P (x)
 n                         dx^m   n

where P_n(x) is the Legendre polynomial of degree n

          1     d^n   2    n
P (x) = ------ [----(x - 1)  ] 
 n      2^n n!  dx^n

For example, legendre(3,[-1.0 -0.9 -0.8]) returns a matrix of results corresponding to the following values,

 x  |   -1.0   |   -0.9   |  -0.8
------------------------------------
m=0 | -1.00000 | -0.47250 | -0.08000
m=1 |  0.00000 | -1.99420 | -1.98000
m=2 |  0.00000 | -2.56500 | -4.32000
m=3 |  0.00000 | -1.24229 | -3.24000

Mapping Function: lgamma (x)
Mapping Function: gammaln (x): Return the natural logarithm of the absolute value of the gamma function of x.
See also gamma, gammai

Function File: cross (x, y, dim)

Computes the vector cross product of the two 3-dimensional vectors x and y.

cross ([1,1,0], [0,1,1])
=> [ 1; -1; 1 ]

If x and y are matrices, the cross product is applied along the first dimension with 3 elements. The optional argument dim is used to force the cross product to be calculated along the dimension defined by dim.

Function File: commutation_matrix (m, n)

Return the commutation matrix K(m,n)

which is the unique m*n by m*n

matrix such that K(m,n) * vec(A) = vec(A')

for all m by n

matrices A.

If only one argument m is given, K(m,m)

is returned.

See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

Function File: duplication_matrix (n)

Return the duplication matrix Dn

which is the unique n^2 by n*(n+1)/2