imsl.linalg.lu_factor

lu_factor(a)

Compute the pivoted LU factorization of a matrix.

Parameters:a ((N,N) array_like) – Square matrix to be factorized. Returns:

• pvt ((N,) ndarray) – The pivot sequence determined during the factorization.
• fac ((N,N) ndarray) – The LU factorization of the matrix.

Notes

The computed LU factorization of matrix A satisfies $L^{-1}A = U$. Let $F$ denote the matrix stored in fac. The triangular matrix $U$ is then stored in the upper triangle of $F$. The strict lower triangle of $F$ contains the information needed to reconstruct $L^{-1}$ using

$L^{-1} = L_{n-1}P_{n-1} \ldots L_1P_1.$

The factors $P_i$ and $L_i$ are defined by partial pivoting. $P_i$ is the identity matrix with rows i and pvt[i-1] interchanged. $L_i$ is the identity matrix with $F_{ji}$, for $j = i+1,\ldots n$, inserted below the diagonal in column i.

The factorization efficiency is based on a technique of “loop unrolling and jamming” by Dr. Leonard J. Harding of the University of Michigan, Ann Arbor, Michigan.

This function creates a temporary LU() instance and calls method LU.factor() on that instance.

An exception is raised if $U$, the upper triangular part of the factorization, has a zero diagonal element.