mkl_?getrfnpi

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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mkl_?getrfnpi

Performs LU factorization (complete or incomplete) of a general matrix without pivoting.

Syntax

call mkl_sgetrfnpi (m, n, nfact, a, lda, info)

call mkl_dgetrfnpi (m, n, nfact, a, lda, info )

call mkl_cgetrfnpi (m, n, nfact, a, lda, info )

call mkl_zgetrfnpi (m, n, nfact, a, lda, info )

call mkl_getrfnpi ( a [, nfact] [, info] )

Include Files

mkl.fi, lapack.f90

Description

The routine computes the LU factorization of a general m-by-n matrix A without using pivoting. It supports incomplete factorization. The factorization has the form:

A = L*U,

where L is lower triangular with unit diagonal elements (lower trapezoidal if m > n) and U is upper triangular (upper trapezoidal if m < n).

Incomplete factorization has the form:

where L is lower trapezoidal with unit diagonal elements, U is upper trapezoidal, and is the unfactored part of matrix A. See the application notes section for further details.

NOTE:

Use ?getrf if it is possible that the matrix is not diagonal dominant.

Input Parameters

The data types are given for the Fortran interface.

m

INTEGER. The number of rows in matrix A; m≥ 0.

n

INTEGER. The number of columns in matrix A; n≥ 0.

nfact

INTEGER. The number of rows and columns to factor; 0 ≤nfact≤ min(m, n). Note that if nfact < min(m, n), incomplete factorization is performed.

a

REAL for mkl_sgetrfnpi

DOUBLE PRECISION for mkl_dgetrfnpi

COMPLEX for mkl_cgetrfnpi

DOUBLE COMPLEX for mkl_zgetrfnpi

Array of size (lda,*). Contains the matrix A. The second dimension of a must be at least max(1, n).

lda

INTEGER. The leading dimension of array a. lda≥ max(1, m).

Output Parameters

a

Overwritten by L and U. The unit diagonal elements of L are not stored.

When incomplete factorization is specified by setting nfact < min(m, n), a also contains the unfactored submatrix . See the application notes section for further details.

info

INTEGER. If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = i, u_ii is 0. The requested factorization has been completed, but U is exactly singular. Division by 0 will occur if factorization is completed and factor U is used for solving a system of linear equations.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine getrf interface are as follows:

a: Holds the matrix A of size (m,n).

Application Notes

The computed L and U are the exact factors of a perturbed matrix A + E, with

|E| ≤ c(min(m, n))ε |L||U|

where c(n) is a modest linear function of n, and ε is the machine precision.

The approximate number of floating-point operations for real flavors is

(2/3)`n`³	If `m = n` = nfact
(1/3)`n`²(3`m`-`n`)	If `m>n = nfact`
(1/3)`m`²(3`n`-`m`)	If `m = nfact<n`
(2/3)`n`³ - (`n`-`nfact`)³	If `m = n,nfact< min(m, n)`
(1/3)(`n`²(3`m`-`n`) - (`n`-nfact)²(3`m` - 2nfact - `n`))	If `m>n > nfact`
(1/3)(`m`²(3`n`-`m`) - (`m`-nfact)²(3`n` - 2nfact - `m`))	If `nfact < m < n`

The number of operations for complex flavors is four times greater.

When incomplete factorization is specified, the first nfact rows and columns are factored, with the update of the remaining rows and columns of A as follows:

If matrix A is represented as a block 2-by-2 matrix:

where

A₁₁ is a square matrix of order nfact,
A₂₁ is an (m - nfact)-by-nfact matrix,
A₁₂ is an nfact-by-(n - nfact) matrix, and
A₂₂ is an (m - nfact)-by-(n - nfact) matrix.

The result is

L₁ is a lower triangular square matrix of order nfact with unit diagonal and U₁ is an upper triangular square matrix of order nfact. L₁ and U₁ result from LU factorization of matrix A₁₁: A₁₁ = L₁U₁.

L₂ is an (m - nfact)-by-nfact matrix and L₂ = A₂₁U₁^-1. U₂ is an nfact-by-(n - nfact) matrix and U₂ = L₁^-1A₁₂.

is an (m - nfact)-by-(n - nfact) matrix and = A₂₂ - L₂U₂.

On exit, elements of the upper triangle U₁ are stored in place of the upper triangle of block A₁₁ in array a; elements of the lower triangle L₁ are stored in the lower triangle of block A₁₁ in array a (unit diagonal elements are not stored). Elements of L₂ replace elements of A₂₁; U₂ replaces elements of A₁₂ and replaces elements of A₂₂.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

mkl_?getrfnpi