Developer Reference for Intel® oneAPI Math Kernel Library for C
?geqrf
Computes the QR factorization of a general m-by-n matrix.
Syntax
lapack_intLAPACKE_sgeqrf ( intmatrix_layout , lapack_intm , lapack_intn , float*a , lapack_intlda , float*tau );
lapack_intLAPACKE_dgeqrf ( intmatrix_layout , lapack_intm , lapack_intn , double*a , lapack_intlda , double*tau );
lapack_intLAPACKE_cgeqrf ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_complex_float*tau );
lapack_intLAPACKE_zgeqrf ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_complex_double*tau );
Include Files
mkl.h
Description
sgeqrf dgeqrf cgeqrf zgeqrf geqrf
The routine forms the QR 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.
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 sgeqrf DOUBLE PRECISION for dgeqrf COMPLEX for cgeqrf DOUBLE COMPLEX for zgeqrf . Arrays: a ( lda ,*) contains the matrix A . The second dimension of a must be at least max(1, n ). 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 . 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 at least max(1, n ) for row major layout .
lwork
The size of the work array ( lwork≥n ).
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 above the diagonal of the array contain the min( m , n )-by- n upper trapezoidal matrix R ( R is upper triangular if m ≥ n ); the elements below the diagonal, with the array tau , present the orthogonal matrix Q as a product of min( m , n ) elementary reflectors (see Orthogonal Factorizations ) .
- tau
-
REAL for sgeqrf DOUBLE PRECISION for dgeqrf COMPLEX for cgeqrf DOUBLE COMPLEX for zgeqrf . Array, size at least max (1, min( m , n )). Contains scalars that define elementary reflectors for the matrix Q in its decomposition in a product of elementary reflectors (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 solve a set of least squares problems minimizing \(||A x - b||_{2}\) for all columns b of a given matrix \(B\) , you can call the following:
?geqrf (this routine)
to factorize \(A = QR\) ;
to compute \(C = Q^{T} B\) (for real matrices);
to compute \(C = Q^{H} B\) (for complex matrices);
trsm (a BLAS routine)
to solve \(R X = C\) .
(The columns of the computed \(X\) are the least squares solution vectors x .)
To compute the elements of \(Q\) explicitly, call
(for real matrices)
(for complex matrices).