?geqrfp

Computes the QR factorization of a general m-by-n matrix with non-negative diagonal elements.

Syntax

call sgeqrfp(m, n, a, lda, tau, work, lwork, info)

call dgeqrfp(m, n, a, lda, tau, work, lwork, info)

call cgeqrfp(m, n, a, lda, tau, work, lwork, info)

call zgeqrfp(m, n, a, lda, tau, work, lwork, info)

• mkl.fi

Description

The routine forms the QR factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No pivoting is performed. The diagonal entries of R are real and nonnegative.

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

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 sgeqrfp

DOUBLE PRECISION for dgeqrfp

COMPLEX for cgeqrfp

DOUBLE COMPLEX for zgeqrfp.

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).

lda

INTEGER. The leading dimension of a; at least max(1, m).

lwork

INTEGER. The size of the work array (lworkn).

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 mn); 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).

The diagonal elements of the matrix R are real and non-negative.

tau

REAL for sgeqrfp

DOUBLE PRECISION for dgeqrfp

COMPLEX for cgeqrfp

DOUBLE COMPLEX for zgeqrfp.

Array, size at least max (1, min(m, n)). Contains scalars that define elementary reflectors for the matrix Qin 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.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine geqrfp 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 = n*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.

The computed factorization is the exact factorization of a matrix A + E, where

||E||2 = O(ε)||A||2.

The approximate number of floating-point operations for real flavors is

 (4/3)n3 if m = n, (2/3)n2(3m-n) if m > n, (2/3)m2(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:

 ?geqrfp (this routine) to factorize A = QR; ormqr to compute C = QT*B (for real matrices); unmqr 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 least squares solution vectors x.)

To compute the elements of Q explicitly, call

orgqr

(for real matrices)

ungqr

(for complex matrices).