# sparseLU-class: Sparse LU decomposition of a square sparse matrix In Matrix: Sparse and Dense Matrix Classes and Methods

 sparseLU-class R Documentation

## Sparse LU decomposition of a square sparse matrix

### Description

Objects of this class contain the components of the LU decomposition of a sparse square matrix.

### Objects from the Class

Objects can be created by calls of the form ```new("sparseLU", ...)``` but are more commonly created by function `lu()` applied to a sparse matrix, such as a matrix of class `dgCMatrix`.

### Slots

`L`:

Object of class `"dtCMatrix"`, the lower triangular factor from the left.

`U`:

Object of class `"dtCMatrix"`, the upper triangular factor from the right.

`p`:

Object of class `"integer"`, permutation applied from the left.

`q`:

Object of class `"integer"`, permutation applied from the right.

`Dim`:

the dimension of the original matrix; inherited from class `MatrixFactorization`.

### Extends

Class `"LU"`, directly. Class `"MatrixFactorization"`, by class `"LU"`.

### Methods

expand

`signature(x = "sparseLU")` Returns a list with components `P`, `L`, `U`, and `Q`, where P and Q represent fill-reducing permutations, and L, and U the lower and upper triangular matrices of the decomposition. The original matrix corresponds to the product P'LUQ.

### Note

The decomposition is of the form

A = P'LUQ,

or equivalently PAQ' = LU, where all matrices are sparse and of size n by n. The matrices P and Q, and their transposes P' and Q' are permutation matrices, L is lower triangular and U is upper triangular.

`lu`, `solve`, `dgCMatrix`

### Examples

```## Extending the one in   examples(lu), calling the matrix  A,
## and confirming the factorization identities :
package = "Matrix")),
"CsparseMatrix")
## with dimnames(.) - to see that they propagate to L, U :
dimnames(A) <- list(paste0("r", seq_len(nrow(A))),
paste0("C", seq_len(ncol(A))))
str(luA <- lu(A)) # p is a 0-based permutation of the rows
# q is a 0-based permutation of the columns
xA <- expand(luA)
## which is simply doing
stopifnot(identical(xA\$ L, luA@L),
identical(xA\$ U, luA@U),
identical(xA\$ P, as(luA@p +1L, "pMatrix")),
identical(xA\$ Q, as(luA@q +1L, "pMatrix")))

P.LUQ <- with(xA, t(P) %*% L %*% U %*% Q)
stopifnot(all.equal(unname(A), unname(P.LUQ), tolerance = 1e-12))

## permute rows and columns of original matrix
pA <- A[luA@p + 1L, luA@q + 1L]
PAQ. <- with(xA, P %*% A %*% t(Q))
stopifnot(all.equal(unname(pA), unname(PAQ.), tolerance = 1e-12))

pLU <- drop0(luA@L %*% luA@U) # L %*% U -- dropping extra zeros
stopifnot(all.equal(pA, pLU, tolerance = 1e-12))
```

Matrix documentation built on Nov. 11, 2022, 9:06 a.m.