Node:Special Functions, Next:Coordinate Transformations, Previous:Sums and Products, Up:Arithmetic
[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) | Loadable Function |
Compute Bessel or Hankel functions of various kinds:
If the argument opt is supplied, the result is scaled by the
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 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.
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[a, ierr] = airy (k, z, opt) | Loadable Function |
Compute Airy functions of the first and second kind, and their
derivatives.
K Function Scale factor (if a third argument 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 The result is the same size as z. If requested, ierr contains the following status information and is the same size as the result.
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beta (a, b) | Mapping Function |
Return the Beta function,
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b). |
betainc (x, a, b) | Mapping Function |
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. |
bincoeff (n, k) | Mapping Function |
Return the binomial coefficient of n and k, defined as
/ \ | n | n (n-1) (n-2) ... (n-k+1) | | = ------------------------- | k | k! \ / For example,
bincoeff (5, 2) => 10 |
erf (z) | Mapping Function |
Computes the error function,
z / erf (z) = (2/sqrt (pi)) | e^(-t^2) dt / t=0 |
erfc (z) | Mapping Function |
Computes the complementary error function,
1 - erf (z) .
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erfinv (z) | Mapping Function |
Computes the inverse of the error function. |
gamma (z) | Mapping Function |
Computes the Gamma function,
infinity / gamma (z) = | t^(z-1) exp (-t) dt. / t=0 |
gammainc (x, a) | Mapping Function |
Computes the incomplete gamma function,
x 1 / gammainc (x, a) = --------- | exp (-t) t^(a-1) dt gamma (a) / t=0 If a is scalar, then If neither x nor a is scalar, the sizes of x and a must agree, and gammainc is applied element-by-element. |
lgamma (a, x) | Mapping Function |
gammaln (a, x) | Mapping Function |
Return the natural logarithm of the gamma function. |
cross (x, y, dim) | Function File |
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 defiend by dim. |
commutation_matrix (m, n) | Function File |
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. |
duplication_matrix (n) | Function File |
Return the duplication matrix
Dn
which is the unique
n^2 by n*(n+1)/2
matrix such that
Dn vech (A) = vec (A)
for all symmetric
n by n
matrices
A.
See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics. |