The algorithm structure and V-matrices

Abstract

This is a brief survey devoted to K-matrices, which can be used not only for describing the structure of an algorithm but can also provide some additional information about it. We show the way K-matrices appear in such applications as the roundoff error analysis, the fast computation of gradients, and the complexity evaluation of linear algorithms. A perfectly simple procedure for deriving a fast algorithm for computing gradients is proposed. We also discuss some lower bounds for linear algorithms based on a determinant inequality obtained in the paper.Structural properties of algorithms form the basis for research in several directions that appear at first glance to be distantly related. We can name here at least three such directions: the roundoff error analysis, the fast computation of gradients, and the complexity evaluation of linear algorithms. So far, these areas have not been considered interrelated because there were no cross-references from one area to another. The goal of this paper is to show that there is a natural unifying basis for these areas, namely F-matrices. It should be emphasized that K-matrices fully reflect the informational structure of an algorithm.Our intention was to write a survey of K-matrices and their application to the analysis of algorithms. In Section 1 we give the definition of F-matrices and a sketch of their use in the roundoff error analysis. Sections 2 and 3 are devoted to some applications of F-matrices that make use of different augmentations of F-matrices.In Section 2 we consider a very simple procedure for deriving a fast algorithm for computing gradients. In Section 3 we primarily discuss lower bounds for linear algorithms that are based on a new determinant inequality, Appendix contains the proof of the inequality.

An Efficient Method for Solving Problems of Acoustic Scattering on Three-Dimensional Transparent Structures

“…However, in many problems, in particular those discussed below, one needs to compute sums of the form Equation (10), in which the function A(l), −(N − 1) ≤ l ≤ (N − 1) is arbitrary in the specified range. Such sums arise when multiplying Toeplitz matrices by the vector [31,32]. This function A(l) is defined at the (2N − 1) integer point.…”