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IMA Journal of Numerical Analysis Advance Access published online on February 16, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn068
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Rational approximation of the unitary exponential

Marko Huhtanen{dagger}

Institute of Mathematics, Helsinki University of Technology, Box 1100 FIN-02015, Finland

{dagger} Email: marko.huhtanen{at}ttk.fi

Received on 29 October 2007. Accepted for publication 19 September 2008.


  Abstract

The exponential function maps the Lie algebra of skew-Hermitian matrices to the respective Lie group of unitary matrices. Alternatively, unitary matrices are obtained by applying the generalized Cayley transform to polynomials in skew-Hermitian matrices. For a given large skew-Hermitian matrix, the related problem of choosing the polynomial to approximate its exponential applied to a vector is studied. Optimal polynomials are found with respect to various criteria.

Key Words: unitary matrix exponential; generalized Cayley transform; rational approximation; energy conservation


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