Geodesic
Monotone convergence of the Lanczos approximations to matrix functions of Hermitian matrices
Electronic transactions on numerical analysis, Tome 35 (2009), pp. 118-128
When A is a Hermitian matrix, the action f (A)b of a matrix function f (A) on a vector b can efficiently be approximated via the Lanczos method. In this note we use M -matrix theory to establish that the 2- norm of the error of the sequence of approximations is monotonically decreasing if f is a Stieltjes transform and A is positive definite. We discuss the relation of our approach to a recent, more general monotonicity result of Druskin for Laplace transforms. We also extend the class of functions to certain product type functions. This yields, for example, monotonicity when approximating $sign(A)$b with A indefinite if the Lanczos method is performed for A2 rather than A.
Classification : 6530, 65F10, 65F50
Keywords: matrix functions, Lanczos method, Galerkin approximation, monotone convergence, error estimates 
@article{ETNA_2009__35__a8,
     author = {Frommer,  Andreas},
     title = {Monotone convergence of the {Lanczos} approximations to matrix functions of {Hermitian} matrices},
     journal = {Electronic transactions on numerical analysis},
     pages = {118--128},
     year = {2009},
     volume = {35},
     zbl = {1190.65063},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2009__35__a8/}
}
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%0 Journal Article
%A Frommer,  Andreas
%T Monotone convergence of the Lanczos approximations to matrix functions of Hermitian matrices
%J Electronic transactions on numerical analysis
%D 2009
%P 118-128
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%U http://geodesic.mathdoc.fr/item/ETNA_2009__35__a8/
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Frommer,  Andreas. Monotone convergence of the Lanczos approximations to matrix functions of Hermitian matrices. Electronic transactions on numerical analysis, Tome 35 (2009), pp. 118-128. http://geodesic.mathdoc.fr/item/ETNA_2009__35__a8/