# lu: (Generalized) Triangular Decomposition of a Matrix In Matrix: Sparse and Dense Matrix Classes and Methods

## Description

Computes (generalized) triangular decompositions of square (sparse or dense) and non-square dense matrices.

## Usage

 ```1 2 3 4 5 6 7 8``` ```lu(x, ...) ## S4 method for signature 'matrix' lu(x, warnSing = TRUE, ...) ## S4 method for signature 'dgeMatrix' lu(x, warnSing = TRUE, ...) ## S4 method for signature 'dgCMatrix' lu(x, errSing = TRUE, order = TRUE, tol = 1, keep.dimnames = TRUE, ...) ```

## 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 `x` is a `"denseMatrix"`) logical specifying if a `warning` should be signalled when `x` is singular. `errSing` (when `x` is a `"sparseMatrix"`) logical specifying if an error (see `stop`) should be signalled when `x` is singular. When `x` is singular, `lu(x, errSing=FALSE)` returns `NA` instead of an LU decomposition. No warning is signalled and the useR should be careful in that case. `order` logical or integer, used to choose which fill-reducing 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 `cs_lu`. Do only change with much care. `keep.dimnames` logical indicating that `dimnames` should be propagated to the result, i.e., “kept”. This was hardcoded to `FALSE` in upto Matrix version 1.2-0. Setting to `FALSE` may gain some performance. `...` 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 non-square 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”.

Class definitions `denseLU` and `sparseLU` and function `expand`; `qr`, `chol`.
 ``` 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]))# non-square expand(lu2 <- lu(x2)) ##--- Sparse (see more in ?"sparseLU-class")----- % ./sparseLU-class.Rd pm <- as(readMM(system.file("external/pores_1.mtx", package = "Matrix")), "CsparseMatrix") str(pmLU <- lu(pm)) # p is a 0-based permutation of the rows # q is a 0-based 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] ```