# LU Factorization (G Dataflow)

Performs the LU factorization of a matrix.

## matrix A

A real matrix.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.

Default: Empty array

## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

## matrix L

A lower triangular matrix with ones on the diagonal.

## matrix U

An upper triangular matrix.

## matrix P

A permutation matrix.

## error out

Error information. The node produces this output according to standard error behavior.

## Algorithm for Calculating the LU Factorization of a Matrix

This node factors an m × n matrix A into the following types of matrices so that PA = LU:

• L is an m × min(m,n) matrix. When mn, L is a lower triangular matrix with ones on the diagonal. When m > n, L is a lower trapezoidal matrix with ones on the diagonal.
• U is a min(m,n) × n matrix. When mn, U is an upper triangular matrix. When m < n, U is an upper trapezoidal matrix.
• P is an m × n permutation matrix, which serves as the identity matrix with some rows exchanged.

For a singular matrix, this node completes the factorization and returns a warning, and there is at least one zero at the diagonal of U.

The following equation illustrates one useful property of LU factorization when A is a square matrix:

$\mathrm{det}\left(A\right)=\underset{k=0}{\overset{n-1}{\prod }}{u}_{kk}$

where det(A) is the determinant of A.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported