naginterfaces.library.matop.complex_​tri_​matrix_​sqrt

naginterfaces.library.matop.complex_tri_matrix_sqrt(a)[source]

complex_tri_matrix_sqrt computes the principal matrix square root, , of a complex upper triangular matrix .

For full information please refer to the NAG Library document for f01fp

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01fpf.html

Parameters
acomplex, array-like, shape

The upper triangular matrix .

Returns
acomplex, ndarray, shape

Contains, if no exception or warning is raised, the principal matrix square root, . Alternatively, if = 1, contains an non-principal square root of .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

has a defective vanishing eigenvalue. The square root cannot be found in this case.

(errno )

An internal error occurred. It is likely that the function was called incorrectly.

Warns
NagAlgorithmicWarning
(errno )

has negative or semisimple, vanishing eigenvalues. The principal square root is not defined in this case; a non-principal square root is returned.

Notes

A square root of a matrix is a solution to the equation . A nonsingular matrix has multiple square roots. For a matrix with no eigenvalues on the closed negative real line, the principal square root, denoted by , is the unique square root whose eigenvalues lie in the open right half-plane.

complex_tri_matrix_sqrt computes , where is an upper triangular matrix. is also upper triangular.

The algorithm used by complex_tri_matrix_sqrt is described in Björck and Hammarling (1983). In addition a blocking scheme described in Deadman et al. (2013) is used.

References

Björck, Å and Hammarling, S, 1983, A Schur method for the square root of a matrix, Linear Algebra Appl. (52/53), 127–140

Deadman, E, Higham, N J and Ralha, R, 2013, Blocked Schur Algorithms for Computing the Matrix Square Root, Applied Parallel and Scientific Computing: 11th International Conference, (PARA 2012, Helsinki, Finland), P. Manninen and P. Öster, Eds, Lecture Notes in Computer Science (7782), 171–181, Springer–Verlag

Higham, N J, 2008, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, USA