Computing a logarithm of a unitary matrix with general spectrum

Details Computing a logarithm of a unitary matrix with general spectrum Computing a logarithm of a unitary matrix with general spectrum

2012-03-28

arxiv.org/abs/1203.6151v2

Mathematics

Numerical Analysis

Added a discussion of applications to 1D lattice models of non-interacting fermions

Scientific paper

In theory, a unitary matrix U has a skew-hermitian logarithm H. In a computing environment one expects only to know U^*U \approx I and might wish to compute H with e^H \approx U and H^*= -H. This is relatively easy to accomplish using the Schur decomposition. Reasonable error bounds are derived. In cases where the norm of U^*U-I is somewhat large we discuss the utility of pre-processing with Newton's method of approximating the polar decomposition. In the case of U being J-skew-symmetric, one can insist that H be J-skew-symmetric and skew-Hermitian.

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