## Computing a logarithm of a unitary matrix with general spectrum [dataset]

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/23450

Title

Computing a logarithm of a unitary matrix with general spectrum [dataset]

Author(s)

Loring, Terry A.

Abstract

\begin{abstract}
We analyze an algorithm for computing a skew-Hermitian logarithm
of a unitary matrix, and also skew-Hermitian approximate logarithms
for nearly unitary matrices. This algorithm is very easy
to implement using standard software and it works well even for unitary
matrices with no spectral conditions assumed. Certain examples, with
many eigenvalues near $-1$, lead to very non-Hermitian
output for other basic methods of calculating matrix logarithms.
Altering the output of these algorithms to force an Hermitian
output creates accuracy issues which are avoided by the
considered algorithm.
A modification is introduced to deal properly with the
$J$-skew symmetric unitary
matrices. Applications to numerical studies of
topological insulators in two symmetry classes are discussed.
\end{abstract}

Description

The Matlab files stored here were used to create the test data in
"Computing a logarithm of a unitary matrix with general spectrum"
which is expected to be published, pending minor revision, by
Numerical Linear Algebra with Application. These were run under
Matlab R2103b. The files here are:
testLogs.m
testLogsDual.m
PVL.m
dual.m
logTable.m
logTableDual.m
logTableEasy.m
testEasyLogs.m
Tables I and II were produced using logTable(1,'table1-2').
Table III was produced using logTable(2,'table3').
Table IV was produced using logTable(3,'table4').
Table V was produced using logTableEasy('table5').
Tables VI and VII were produced using logTableDual(1,'table6-7').
Table VIII was produced using logTableDual(2,'table8').
Table IX was produced using logTableDual(3,'table9').

Subject(s)

matrix logarithm

Schur decomposition

nearly unitary matrix

functional calculus

normal matrices

unitary matrices

self-dual matrices

Schur decomposition

nearly unitary matrix

functional calculus

normal matrices

unitary matrices

self-dual matrices