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

naginterfaces.library.matop.complex_tri_matrix_sqrt(a)

complex_tri_matrix_sqrt computes the principal matrix square root, $A^{1/2}$, of a complex upper triangular $n\times n$ matrix $A$.

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

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01fpf.html

Parameters

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

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

Returns

acomplex, ndarray, shape $(n,n)$

Contains, if no exception or warning is raised, the $n\times n$ principal matrix square root, $A^{1/2}$. Alternatively, if $errno$ = 1, contains an $n\times n$ non-principal square root of $A$.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

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

(errno $3$)

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

Warns

NagAlgorithmicWarning

(errno $1$)

$A$ 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 $A$ 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.

complex_tri_matrix_sqrt computes $A^{1/2}$, where $A$ is an upper triangular matrix. $A^{1/2}$ 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