pracma (version 1.4.5)

lu: LU Matrix Factorization

Description

LU decomposition of a positive definite matrix as Gaussian factorization (without pivoting).

Usage

lu(A, scheme = c("kji", "jki", "ijk"))

Arguments

A

square positive definite numeric matrix (will not be checked).

scheme

order of row and column operations.

Value

• Returns a list with components L and U, the two lower and upper triangular matrices such that A=L%*%U.

Details

For a given matrix A, the LU decomposition exists and is unique iff its principal submatrices of order i=1,...,n-1 are nonsingular. The procedure here is a simple Gauss elimination without pivoting.

The scheme abbreviations refer to the order in which the cycles of row- and column-oriented operations are processed. The ``ijk'' scheme is one of the two compact forms, here the Doolite factorization (the Crout factorization would be similar).

References

Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second edition, Springer-Verlag, Berlin Heidelberg.

See Also

qr

Examples

A <- magic(5)
LU <- lu(A, scheme = "ijk")     # Doolittle scheme
LU$L %*% LU$U
##      [,1] [,2] [,3] [,4] [,5]
## [1,]   17   24    1    8   15
## [2,]   23    5    7   14   16
## [3,]    4    6   13   20   22
## [4,]   10   12   19   21    3
## [5,]   11   18   25    2    9