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

naginterfaces.library.matop.real_tri_matrix_sqrt(a)

real_tri_matrix_sqrt computes the principal matrix square root, $A^{1/2}$, of a real upper quasi-triangular $n\times n$ matrix $A$.

For full information please refer to the NAG Library document for f01ep

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f01/f01epf.html

Parameters

afloat, array-like, shape $(n,n)$

The $n\times n$ upper quasi-triangular matrix $A$.

Returns

afloat, ndarray, shape $(n,n)$

The $n\times n$ principal matrix square root $A^{1/2}$.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

$A$ has negative or vanishing eigenvalues. The principal square root is not defined in this case. real_gen_matrix_sqrt() or complex_gen_matrix_sqrt() may be able to provide further information.

(errno $2$)

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

Notes

A square root of a matrix $A$ is a solution $X$ to the equation $X^2=A$. 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 $A^{1/2}$, is the unique square root whose eigenvalues lie in the open right half-plane.

real_tri_matrix_sqrt computes $A^{1/2}$, where $A$ is an upper quasi-triangular matrix, with $1\times 1$ and $2\times 2$ blocks on the diagonal. Such matrices arise from the Schur factorization of a real general matrix, as computed by lapackeig.dhseqr, for example. real_tri_matrix_sqrt does not require $A$ to be in the canonical Schur form described in lapackeig.dhseqr, it merely requires $A$ to be upper quasi-triangular. $A^{1/2}$ then has the same block triangular structure as $A$.

The algorithm used by real_tri_matrix_sqrt is described in Higham (1987). 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, 1987, Computing real square roots of a real matrix, Linear Algebra Appl. (88/89), 405–430

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