Developer Reference for Intel® oneAPI Math Kernel Library for C
?gelqf
Computes the LQ factorization of a general m-by-n matrix.
Syntax
lapack_intLAPACKE_sgelqf ( intmatrix_layout , lapack_intm , lapack_intn , float*a , lapack_intlda , float*tau );
lapack_intLAPACKE_dgelqf ( intmatrix_layout , lapack_intm , lapack_intn , double*a , lapack_intlda , double*tau );
lapack_intLAPACKE_cgelqf ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_complex_float*tau );
lapack_intLAPACKE_zgelqf ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_complex_double*tau );
Include Files
mkl.h
Description
sgelqf dgelqf cgelqf zgelqf gelqf
The routine forms the LQ factorization of a general m -by- n matrix A (see Orthogonal Factorizations ). No pivoting is performed.
The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min( m , n ) elementary reflectors . Routines are provided to work with Q in this representation.
This routine supports the Progress Routine feature. See Progress Function for details.
Input Parameters
- 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 the matrix A ( m≥ 0 ).
n
The number of columns in A ( n≥ 0 ).
- a , work
-
REAL for sgelqf DOUBLE PRECISION for dgelqf COMPLEX for cgelqf DOUBLE COMPLEX for zgelqf . Arrays:
Array a of size max(1, lda * n ) for column major layout and max(1, lda * m ) for row major layout contains the matrix A .
The second dimension of a must be at least max(1, n ). work is a workspace array, its dimension max(1, lwork) .
lda
The leading dimension of a ; at least max(1, m ) for column major layout and max(1, n ) for row major layout .
lwork
The size of the work array; at least max(1, m ).
If lwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla . See Application Notes for the suggested value of lwork .
Output Parameters
- a
-
Overwritten by the factorization data as follows: The elements on and below the diagonal of the array contain the m -by-min( m , n ) lower trapezoidal matrix L ( L is lower triangular if m ≤ n ); the elements above the diagonal, with the array tau , represent the orthogonal matrix Q as a product of elementary reflectors.
- tau
-
REAL for sgelqf DOUBLE PRECISION for dgelqf COMPLEX for cgelqf DOUBLE COMPLEX for zgelqf . Array, size at least max(1, min(m, n)) . Contains scalars that define elementary reflectors for the matrix Q (see Orthogonal Factorizations ).
- work(1)
-
If info = 0 , on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.
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.
LAPACK 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 factorization is the exact factorization of a matrix \(A\) + \(E\) , where
\(||E||_{2} = O(\varepsilon) ||A||_{2}\) .
The approximate number of floating-point operations for real flavors is
\((4/3)n^{3}\) if m = n,
\((2/3)n^{2}(3m - n)\) if m > n, or
\((2/3)m^{2}(3n - m)\) if m < n.
The number of operations for complex flavors is 4 times greater.
To find the minimum-norm solution of an underdetermined least squares problem minimizing \(||A x - b||_{2}\) for all columns b of a given matrix \(B\) , you can call the following:
?gelqf (this routine)
to factorize \(A = L Q\) ;
trsm (a BLAS routine)
to solve \(L Y = B\) for \(Y\) ;
to compute \(X = (Q_{1})^{T} Y\) (for real matrices);
to compute \(X = (Q_{1})^{H} Y\) (for complex matrices).
(The columns of the computed \(X\) are the minimum-norm solution vectors x . Here \(A\) is an m -by- n matrix with m < n ; \(Q\)1 denotes the first m columns of \(Q\) ).
To compute the elements of \(Q\) explicitly, call
(for real matrices)
(for complex matrices).