naginterfaces.library.lapackeig.dgeqp3

naginterfaces.library.lapackeig.dgeqp3(a, jpvt)[source]

dgeqp3 computes the factorization, with column pivoting, of a real matrix.

For full information please refer to the NAG Library document for f08bf

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f08/f08bff.html

Parameters
afloat, array-like, shape

The matrix .

jpvtint, array-like, shape

If , the th column of is moved to the beginning of before the decomposition is computed and is fixed in place during the computation. Otherwise, the th column of is a free column (i.e., one which may be interchanged during the computation with any other free column).

Returns
afloat, ndarray, shape

If , the elements below the diagonal are overwritten by details of the orthogonal matrix and the upper triangle is overwritten by the corresponding elements of the upper triangular matrix .

If , the strictly lower triangular part is overwritten by details of the orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the upper trapezoidal matrix .

jpvtint, ndarray, shape

Details of the permutation matrix . More precisely, if , the th column of is moved to become the th column of ; in other words, the columns of are the columns of in the order .

taufloat, ndarray, shape

The scalar factors of the elementary reflectors.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

dgeqp3 forms the factorization, with column pivoting, of an arbitrary rectangular real matrix.

If , the factorization is given by:

where is an upper triangular matrix, is an orthogonal matrix and is an permutation matrix. It is sometimes more convenient to write the factorization as

which reduces to

where consists of the first columns of , and the remaining columns.

If , is trapezoidal, and the factorization can be written

where is upper triangular and is rectangular.

The matrix is not formed explicitly but is represented as a product of elementary reflectors (see the F08 Introduction for details). Functions are provided to work with in this representation (see Further Comments).

Note also that for any , the information returned in the first columns of the array represents a factorization of the first columns of the permuted matrix .

The function allows specified columns of to be moved to the leading columns of at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the th stage the pivot column is chosen to be the column which maximizes the -norm of elements to over columns to .

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore