Developer Reference for Intel® oneAPI Math Kernel Library for C
mkl_?getrfnpi
Performs LU factorization (complete or incomplete) of a general matrix without pivoting.
Syntax
lapack_int LAPACKE_mkl_sgetrfnpi ( intmatrix_layout , lapack_intm , lapack_intn , lapack_intnfact , float*a , lapack_intlda );
lapack_int LAPACKE_mkl_dgetrfnpi ( intmatrix_layout , lapack_intm , lapack_intn , lapack_intnfact , double*a , lapack_intlda );
lapack_int LAPACKE_mkl_cgetrfnpi ( intmatrix_layout , lapack_intm , lapack_intn , lapack_intnfact , lapack_complex_float*a , lapack_intlda );
lapack_int LAPACKE_mkl_zgetrfnpi ( intmatrix_layout , lapack_intm , lapack_intn , lapack_intnfact , lapack_complex_double*a , lapack_intlda );
Include Files
mkl.h
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:
\(A = LU\) ,
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:
\(A = LU + \tilde{A}\) ,
where L is lower trapezoidal with unit diagonal elements, U is upper trapezoidal, and \(\tilde{A}\) is the unfactored part of matrix \(A\) . See the application notes section for further details.
Input Parameters
A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
m
The number of rows in matrix A ; m ≥ 0.
n
The number of columns in matrix A ; n ≥ 0.
nfact
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
Array of size (lda,*) at least lda * n for column major layout and at least lda * m for row major layout . Contains the matrix A .
lda
The leading dimension of array a . lda ≥ max(1, m ) for column major layout and lda ≥ max(1, n ) for row major layout .
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 \(\tilde{A}_{22}\). See the application notes section for further details.
Return Values
This function returns a value info .
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
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
The computed \(L\) and \(U\) are the exact factors of a perturbed matrix \(A\) + \(E\) , with
\[|E| \leq c(min(m, n))\varepsilon |L||U|\]
where c(n) is a modest linear function of n , and \(\varepsilon\) is the machine precision.
The approximate number of floating-point operations for real flavors is
\((2/3) n^3\), If m = n = nfact
\((1/3) n^2 (3m - n)\), if m > n = nfact
\((1/3) m^2 (3n - m)\), if m = nfact < n
\((2/3) n^3 - ( n - nfact)^3\), if m = n, nfact < min(m, n)
\((1/3)( n^2 (3m - n) - ( n - nfact )^{2}(3m - 2 nfact - n) )\), if m > n > nfact
\((1/3)( m^2 (3n - m) - ( m - nfact)^{2}(3n - 2 nfact - m ) )\), 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:
\[A = \left [ \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} \right ]\]
where
\(A_{11}\) is a square matrix of order nfact ,
\(A_{21}\) is an ( m - nfact )-by- nfact matrix,
\(A_{12}\) is an nfact -by-( n - nfact ) matrix, and
\(A_{22}\) is an ( m - nfact )-by-( n - nfact ) matrix.
The result is
\[A = \left [ \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{matrix} \right ] = \left [ \begin{matrix} L_1 \\ L_2 \\ \end{matrix} \right ] \left [ \begin{matrix} U_1 & U_2 \\ \end{matrix} \right ] + \left [ \begin{matrix} 0 & 0 \\ 0 & \tilde{A}_{22} \\ \end{matrix} \right ]\]
\(L_1\) is a lower triangular square matrix of order nfact with unit diagonal and \(U_1\) is an upper triangular square matrix of order nfact . \(L_1\) and \(U_1\) result from LU factorization of matrix \(A_{11}\) : \(A_{11}\) = \(L_1\)\(U_1\) .
\(L_2\) is an ( m - nfact )-by- nfact matrix and \(L_2 = A_{21} U_1^{-1}\) . \(U_2\) is an nfact -by-( n - nfact ) matrix and \(U_2 = L_1^{-1} A_{12}\) .
\(\tilde{A}_{22}\) is an ( m - nfact )-by-( n - nfact ) matrix and \(\tilde{A}_{22}\) = \(A_{22}\) - \(L_2\)\(U_2\) .
On exit, elements of the upper triangle \(U_1\) are stored in place of the upper triangle of block \(A_{11}\) in array A ; elements of the lower triangle \(L_1\) are stored in the lower triangle of block \(A_{11}\) in array A (unit diagonal elements are not stored). Elements of \(L_2\) replace elements of \(A_{21}\) ; \(U_2\) replaces elements of \(A_{12}\) and \(\tilde{A}_{22}\) replaces elements of \(A_{22}\) .
