Summary It is known that in many functions of banded and, more generally, sparse Hermitian positive definite matrices, the entries exhibit a rapid decay away from the sparsity pattern. This is particularly true for the inverse, and based on results for the inverse, bounds for Cauchy–Stieltjes functions of Hermitian positive definite matrices have recently been obtained. We add to the known results by considering certain types of normal matrices, for which fewer and typically less satisfactory results exist so far. Starting from a very general estimate based on approximation properties of Chebyshev polynomials on ellipses, we obtain as special cases insightful decay bounds for various classes of normal matrices, including (shifted) skew‐Hermitian and Hermitian indefinite matrices. In addition, some of our results improve over known bounds when applied to the Hermitian positive definite case.

It is well known that the entries of the inverse of a Hermitian positive definite, banded matrix exhibit a decay away from the main diagonal if the condition number of the matrix is not too large compared to the matrix size. There is a rich literature on bounds which predict and explain this decay behavior. However, all the widely known results on exponential decay lead to a Toeplitz matrix of bounds, i.e., they yield the same bound for all entries along a sub-or superdiagonal. In general, there is no reason to expect the inverse of A to have a Toeplitz structure so that this is an obvious shortcoming of these decay bounds. We construct an example of a tridiagonal matrix for which the difference between these decay bounds and the actual decay is especially pronounced and then show how these bounds can be adapted to better reflect the actual decay by investigating certain (modified) submatrices of A. As a by-product, we also investigate how the distribution of all eigenvalues of A rather than just the spectral interval influences the decay behavior. Here, our results hold for matrices with a general, not necessarily banded, sparsity structure.

The computation of matrix functions f (A), or related quantities like their trace, is an important but challenging task, in particular for large and sparse matrices A. In recent years, probing methods have become an often considered tool in this context, as they allow to replace the computation of f (A) or tr(f (A)) by the evaluation of (a small number of) quantities of the form f (A)v or v T f (A)v, respectively. These tasks can then efficiently be solved by standard techniques like, e.g., Krylov subspace methods. It is well-known that probing methods are particularly efficient when f (A) is approximately sparse, e.g., when the entries of f (A) show a strong off-diagonal decay, but a rigorous error analysis is lacking so far. In this paper we develop new theoretical results on the existence of sparse approximations for f (A) and error bounds for probing methods based on graph colorings. As a by-product, by carefully inspecting the proofs of these error bounds, we also gain new insights into when to stop the Krylov iteration used for approximating f (A)v or v T f (A)v, thus allowing for a practically efficient implementation of the probing methods.

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