Computes the RQ factorization of a general m-by-n matrix.
Syntax
FORTRAN 77:
call sgerqf(m, n, a, lda, tau, work, lwork, info)
call dgerqf(m, n, a, lda, tau, work, lwork, info)
call cgerqf(m, n, a, lda, tau, work, lwork, info)
call zgerqf(m, n, a, lda, tau, work, lwork, info)
Fortran 95:
call gerqf(a [, tau] [,info])
C:
lapack_int LAPACKE_<?>gerqf( int matrix_order, lapack_int m, lapack_int n, <datatype>* a, lapack_int lda, <datatype>* tau );
Include Files
The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.
Description
The routine forms the RQ factorization of a general m-by-n matrix A. 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.
Note
This routine supports the Progress Routine feature. See Progress Function for details.
Input Parameters
The data types are given for the Fortran interface. 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.
- m
INTEGER. The number of rows in the matrix A (m ≥ 0).
- n
INTEGER. The number of columns in A (n ≥ 0).
- a, work
REAL for sgerqf
DOUBLE PRECISION for dgerqf
COMPLEX for cgerqf
DOUBLE COMPLEX for zgerqf.
Arrays:
a(lda,*) contains the m-by-n 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
INTEGER. The leading dimension of a; at least max(1, m).
- lwork
INTEGER. The size of the work array;
lwork ≥ 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 on exit by the factorization data as follows:
if m ≤ n, the upper triangle of the subarray
a(1:m, n-m+1:n ) contains the m-by-m upper triangular matrix R;
if m ≥ n, the elements on and above the (m-n)th subdiagonal contain the m-by-n upper trapezoidal matrix R;
in both cases, the remaining elements, with the array tau, represent the orthogonal/unitary matrix Q as a product of min(m,n) elementary reflectors.
- tau
REAL for sgerqf
DOUBLE PRECISION for dgerqf
COMPLEX for cgerqf
DOUBLE COMPLEX for zgerqf.
Array, DIMENSION at least max (1, min(m, n)).
Contains scalar factors of the elementary reflectors for the matrix Q.
- work(1)
If info = 0, on exit work(1) contains the minimum value of lwork required for optimum performance.
- info
INTEGER.
If info = 0, the execution is successful.
If info = -i, the i-th parameter had an illegal value.
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 restorable arguments, see Fortran 95 Interface Conventions.
Specific details for the routine gerqf interface are the following:
- a
Holds the matrix A of size (m,n).
- tau
Holds the vector of length min(m,n).
Application Notes
For better performance, try using lwork =m*blocksize, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.
If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.
If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.
If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.
Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.
Related routines include: