id: 06099298
dt: j
an: 06099298
au: Al-Mohy, Awad H.; Higham, Nicholas J.
ti: Improved inverse scaling and squaring algorithms for the matrix logarithm.
so: SIAM J. Sci. Comput. 34, No. 4, C153-C169 (2012).
py: 2012
pu: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
la: EN
cc: 
ut: 
ci: 
li: doi:10.1137/110852553
ab: Summary: A popular method for computing the matrix logarithm is the inverse
    scaling and squaring method, which essentially carries out the steps of
    the scaling and squaring method for the matrix exponential in reverse
    order. Here we make several improvements to the method, putting its
    development on a par with our recent version [the authors, SIAM J.
    Matrix Anal. Appl. 31(2009), No. 3, 970‒989 (2010; Zbl 1194.15021)]
    of the scaling and squaring method for the exponential. In particular,
    we introduce backward error analysis to replace the previous forward
    error analysis; obtain backward error bounds in terms of the quantities
    $\|A^p\|^{1/p}$, for several small integer $p$, instead of $\|A\|$; and
    use special techniques to compute the argument of the Padé approximant
    more accurately. We derive one algorithm that employs a Schur
    decomposition, and thereby works with triangular matrices, and another
    that requires only matrix multiplications and the solution of multiple
    right-hand side linear systems. Numerical experiments show the new
    algorithms to be generally faster and more accurate than their existing
    counterparts and suggest that the Schur-based method is the method of
    choice for computing the matrix logarithm.
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