# lu

LU matrix factorization

## Syntax

## Description

`[___] = lu(`

specifies thresholds for the pivoting strategy employed by
`S`

,`thresh`

)`lu`

using any of the previous output argument
combinations. Depending on the number of output arguments specified, the default
value and requirements for the `thresh`

input are different.
See the `thresh`

argument description for details.

`[___] = lu(___,`

returns `outputForm`

)`P`

and `Q`

in the form specified by
`outputForm`

. Specify `outputForm`

as
`'vector'`

to return `P`

and
`Q`

as permutation vectors. You can use any of the input
argument combinations in previous syntaxes.

## Examples

## Input Arguments

## Output Arguments

## Algorithms

The LU factorization is computed using a variant of Gaussian elimination. Computing an
accurate solution is dependent upon the value of the condition number of the original
matrix `cond(A)`

. If the matrix has a large condition number (it is
nearly singular), then the computed factorization might not be accurate.

The LU factorization is a key step in obtaining the inverse with
`inv`

and the determinant with `det`

. It is also
the basis for the linear equation solution or matrix division obtained with the
operators `\`

and `/`

. This necessarily means that the
numerical limitations of `lu`

are also present in these dependent
functions.

## References

[1] Gilbert, John R., and Tim
Peierls. “Sparse Partial Pivoting in Time Proportional to Arithmetic Operations.”
*SIAM Journal on Scientific and Statistical Computing* 9, no. 5
(September 1988): 862–874. https://doi.org/10.1137/0909058.

[2] Anderson, E., ed. *LAPACK Users’ Guide*. 3rd ed. Software, Environments,
Tools. Philadelphia: Society for Industrial and Applied Mathematics, 1999. https://doi.org/10.1137/1.9780898719604.

[3] Davis, Timothy A. "Algorithm
832: UMFPACK V4.3 – an unsymmetric-pattern multifrontal method." *ACM Transactions on Mathematical Software* 30, no. 2 (June
2004): 196–199. https://doi.org/10.1145/992200.992206.

## Extended Capabilities

## Version History

**Introduced before R2006a**