Padé and Gregory error estimates for the logarithm of block triangular matrices

Authors:
João R. Cardoso;F. Silva Leite
Affiliations:
Instituto Superior de Engenharia de Coimbra, Rua Pedro Nunes, Coimbra, Portugal;Departamento de Matemática, Universidade de Coimbra, Coimbra, Portugal
Venue:
Applied Numerical Mathematics
Year:
2006

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Abstract

In this paper we give bounds for the error arising in the approximation of the logarithm of a block triangular matrix T by Padé approximants of the function f(x) = log[(1 + x)/(1 - x)] and partial sums of Gregory's series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform B = (T - I)(T + I)-1 is sufficiently close to zero, then both approximation methods are accurate. This will contribute for reducing the number of successive square roots of T needed in the inverse scaling and squaring procedure for the matrix logarithm.