Geodesic
Monotone convergence of the Lanczos approximations to matrix functions of Hermitian matrices
Electronic transactions on numerical analysis, Tome 35 (2009), pp. 118-128
Zbl   EuDML
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 
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/
@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/}
}
TY  - JOUR
AU  - Frommer,  Andreas
TI  - Monotone convergence of the Lanczos approximations to matrix functions of Hermitian matrices
JO  - Electronic transactions on numerical analysis
PY  - 2009
SP  - 118
EP  - 128
VL  - 35
UR  - http://geodesic.mathdoc.fr/item/ETNA_2009__35__a8/
LA  - en
ID  - ETNA_2009__35__a8
ER  - 
%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
%V 35
%U http://geodesic.mathdoc.fr/item/ETNA_2009__35__a8/
%G en
%F ETNA_2009__35__a8