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

• Autor(a) principal: Cardoso, João R.
• Data de Publicação: 2006
• Outros Autores: Silva Leite, F.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/4619; https://doi.org/10.1016/j.apnum.2005.04.003
• topic: Matrix logarithm; Inverse scaling and squaring; Padé approximants and Gregory's series

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.