Computes the QR factorization of a general m-by-n matrix with column pivoting using level 3 BLAS.
FORTRAN 77:
call sgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info)
call dgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info)
call cgeqp3(m, n, a, lda, jpvt, tau, work, lwork, rwork, info)
call zgeqp3(m, n, a, lda, jpvt, tau, work, lwork, rwork, info)
FORTRAN 95:
call geqp3(a, jpvt [,tau] [,info])
C:
lapack_int LAPACKE_<?>geqp3( int matrix_order, lapack_int m, lapack_int n, <datatype>* a, lapack_int lda, lapack_int* jpvt, <datatype>* tau );
The routine forms the QR factorization of a general m-by-n matrix A with column pivoting: A*P = Q*R (see Orthogonal Factorizations) using Level 3 BLAS. Here P denotes an n-by-n permutation matrix. Use this routine instead of geqpf for better performance.
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.
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.
INTEGER. The number of rows in the matrix A (m ≥ 0).
INTEGER. The number of columns in A (n ≥ 0).
REAL for sgeqp3
DOUBLE PRECISION for dgeqp3
COMPLEX for cgeqp3
DOUBLE COMPLEX for zgeqp3.
Arrays:
a (lda,*) 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).INTEGER. The leading dimension of a; at least max(1, m).
INTEGER. The size of the work array; must be at least max(1, 3*n+1) for real flavors, and at least max(1, n+1) for complex flavors.
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 below for details.
INTEGER.
Array, DIMENSION at least max(1, n).
On entry, if jpvt(i) ≠ 0, the i-th column of A is moved to the beginning of AP before the computation, and fixed in place during the computation.
If jpvt(i) = 0, the i-th column of A is a free column (that is, it may be interchanged during the computation with any other free column).
REAL for cgeqp3
DOUBLE PRECISION for zgeqp3.
A workspace array, DIMENSION at least max(1, 2*n). Used in complex flavors only.
Overwritten by the factorization data as follows:
If m ≥ n, the elements below the diagonal are overwritten by the details of the unitary (orthogonal) matrix Q, and the upper triangle is overwritten by the corresponding elements of the upper triangular matrix R.
If m < n, the strictly lower triangular part is overwritten by the details of the matrix Q, and the remaining elements are overwritten by the corresponding elements of the m-by-n upper trapezoidal matrix R.
REAL for sgeqp3
DOUBLE PRECISION for dgeqp3
COMPLEX for cgeqp3
DOUBLE COMPLEX for zgeqp3.
Array, DIMENSION at least max (1, min(m, n)). Contains scalar factors of the elementary reflectors for the matrix Q.
Overwritten by details of the permutation matrix P in the factorization A*P = Q*R. More precisely, the columns of AP are the columns of A in the following order:
jpvt(1), jpvt(2), ..., jpvt(n).
INTEGER.
If info = 0, the execution is successful.
If info = -i, the i-th parameter had an illegal value.
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 geqp3 interface are the following:
Holds the matrix A of size (m,n).
Holds the vector of length n.
Holds the vector of length min(m,n)
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:
?geqp3 (this routine) |
to factorize A*P = Q*R; |
to compute C = QT*B (for real matrices); |
|
to compute C = QH*B (for complex matrices); |
|
trsm (a BLAS routine) |
to solve R*X = C. |
(The columns of the computed X are the permuted least squares solution vectors x; the output array jpvt specifies the permutation order.)
To compute the elements of Q explicitly, call
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.