Computation of a function of a matrix with close eigenvalues by means of the Newton interpolating polynomial

Abstract

An algorithm for computing an analytic function of a matrix A is described. The algorithm is intended for the case where A has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations.

Computation of Green’s Function of the Bounded Solutions Problem

“…If there are close eigenvalues, large rounding errors can occur, see the discussion of this phenomenon in [25]. In such a case, the method of calculating divided differences proposed in [20] can be applied.…”

Calculating a function of a matrix with a real spectrum

“…Finally, the function f of diagonal blocks (having small spectrum) can be calculated using the Taylor (or interpolating) polynomial; then the remaining blocks of F can be calculated using the block version of (1); note that the absence of t jj − t ii with close t jj and t ii in the neighboring blocks requires that the clusters be separated from each other. A modification of this algorithm that does not use a reordering was discussed in [36].…”

Green's function of the problem of bounded solutions in the case of a block triangular coefficient

“…This matrix representation is diagonal, but it can be 'bad' in the sense that the corresponding projectors have large norms; in such a case it may be convenient to replace one of the subspaces by the orthogonal (or close to orthogonal) complement of the other; as a result one will arrive at a triangular matrix representation of A. Similarly, the spectrum of A may be divided into clusters; so, it is again natural to use a diagonal or triangular matrix representation; the phenomenon of clusterization is discussed, e.g., in [21, lecture 12], [11,37]. Representation by triangular operator matrices is also natural for causal operators; in their turn, causal operators are widely used in control theory [13,16,53] and functional differential equations [34,35,36].…”