# naginterfaces.library.lapackeig.zggrqf¶

naginterfaces.library.lapackeig.zggrqf(a, b)[source]

zggrqf computes a generalized factorization of a complex matrix pair , where is an matrix and is a matrix.

For full information please refer to the NAG Library document for f08zt

https://www.nag.com/numeric/nl/nagdoc_29.2/flhtml/f08/f08ztf.html

Parameters
acomplex, array-like, shape

The matrix .

bcomplex, array-like, shape

The matrix .

Returns
acomplex, ndarray, shape

If , the upper triangle of the subarray contains the upper triangular matrix .

If , the elements on and above the th subdiagonal contain the upper trapezoidal matrix ; the remaining elements, with the array , represent the unitary matrix as a product of elementary reflectors (see the F08 Introduction).

tauacomplex, ndarray, shape

The scalar factors of the elementary reflectors which represent the unitary matrix .

bcomplex, ndarray, shape

The elements on and above the diagonal of the array contain the upper trapezoidal matrix ( is upper triangular if ); the elements below the diagonal, with the array , represent the unitary matrix as a product of elementary reflectors (see the F08 Introduction).

taubcomplex, ndarray, shape

The scalar factors of the elementary reflectors which represent the unitary matrix .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

zggrqf forms the generalized factorization of an matrix and a matrix

where is an unitary matrix, is a unitary matrix and and are of the form

with or upper triangular,

with upper triangular.

In particular, if is square and nonsingular, the generalized factorization of and implicitly gives the factorization of as

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

Anderson, E, Bai, Z and Dongarra, J, 1992, Generalized factorization and its applications, Linear Algebra Appl. (Volume 162–164), 243–271

Hammarling, S, 1987, The numerical solution of the general Gauss-Markov linear model, Mathematics in Signal Processing, (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore), 441–456, Oxford University Press

Paige, C C, 1990, Some aspects of generalized factorizations, . In Reliable Numerical Computation, (eds M G Cox and S Hammarling), 73–91, Oxford University Press