(Generalized) Triangular Decomposition of a Matrix
Description
Computes (generalized) triangular decompositions of square (sparse or dense) and nonsquare dense matrices.
Usage
1 2 3 4 5 6 7 8 
Arguments
x 
a dense or sparse matrix, in the latter case of square dimension. No missing values or IEEE special values are allowed. 
warnSing 
(when 
errSing 
(when 
order 
logical or integer, used to choose which fillreducing permutation technique will be used internally. Do not change unless you know what you are doing. 
tol 
positive number indicating the pivoting tolerance used in

keep.dimnames 
logical indicating that 
... 
further arguments passed to or from other methods. 
Details
lu()
is a generic function with special methods for different types
of matrices. Use showMethods("lu")
to list all the methods
for the lu
generic.
The method for class dgeMatrix
(and all dense
matrices) is based on LAPACK's "dgetrf"
subroutine. It returns
a decomposition also for singular and nonsquare matrices.
The method for class dgCMatrix
(and all sparse
matrices) is based on functions from the CSparse library. It signals
an error (or returns NA
, when errSing = FALSE
, see
above) when the decomposition algorithm fails, as when x
is
(too close to) singular.
Value
An object of class "LU"
, i.e., "denseLU"
(see its separate help page),
or "sparseLU"
, see sparseLU
; this is
a representation of a triangular decomposition of x
.
Note
Because the underlying algorithm differ entirely,
in the dense case (class denseLU
), the
decomposition is
A = P L U,
where as in the sparse case (class
sparseLU
), it is
A = P' L U Q.
References
Golub, G., and Van Loan, C. F. (1989). Matrix Computations, 2nd edition, Johns Hopkins, Baltimore.
Timothy A. Davis (2006) Direct Methods for Sparse Linear Systems, SIAM Series “Fundamentals of Algorithms”.
See Also
Class definitions denseLU
and sparseLU
and function expand
;
qr
, chol
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  ## Dense 
x < Matrix(rnorm(9), 3, 3)
lu(x)
dim(x2 < round(10 * x[,3]))# nonsquare
expand(lu2 < lu(x2))
## Sparse (see more in ?"sparseLUclass") % ./sparseLUclass.Rd
pm < as(readMM(system.file("external/pores_1.mtx",
package = "Matrix")),
"CsparseMatrix")
str(pmLU < lu(pm)) # p is a 0based permutation of the rows
# q is a 0based permutation of the columns
## permute rows and columns of original matrix
ppm < pm[pmLU@p + 1L, pmLU@q + 1L]
pLU < drop0(pmLU@L %*% pmLU@U) # L %*% U  dropping extra zeros
## equal up to "rounding"
ppm[1:14, 1:5]
pLU[1:14, 1:5]
