Том 216, № 6 (2016)

Certain classes of normal Toeplitz-plus-Hankel matrices whose eigenvalues can be computed as efficiently as those of circulants are indicated.

The classical results by Kräuter and Seifter concerning the divisibility of permanents for (±1)-matrices by large powers of 2 are useful in testing whether the permanent function is nonvanishing. This paper suggests a new approach to this problem, allowing one to obtain a short combinatorial proof of the results by Kräuter and Seifter.

A new version of the parallel Alternating Direction Implicit (ADI) method by Peaceman and Rachford for solving systems of linear algebraic equations with positive-definite coefficient matrices represented as sums of two commuting terms is suggested. The algorithms considered are suited for solving two-dimensional grid boundary-value problems with separable variables, as well as the Sylvester and Lyapunov matrix equations. The approach to rising parallel efficiency proposed in the paper is based on representing rational functions as sums of partial fractions. An additive version of the factorized ADI method for solving Sylvester’s equation is described. Estimates of the speedup resulting from increasing the number of computer units are presented. These estimates demonstrate a potential advantage of using the additive algorithms when implemented on a supercomputer with large number of processors or cores.

Two-sided estimates for the continuously differentiable coordinate splines of the second order are established, and sufficient conditions of their nonnegativity are provided. The results obtained are applied to trigonometric splines.

Let A and B be square nonsingular n-by-n matrices with entries being rational or rational Gaussian numbers. The paper describes a method for verifying whether these matrices are congruent. The method uses a finite number of arithmetic (and, in the complex case, also conjugation) operations.

The paper considers some modern problems arising in developing parallel algorithms for solving large systems of linear algebraic equations with sparse matrices occurring in mathematical modeling of real-life processes and phenomena on a multiprocessor computer system (MCS). Two main requirements to methods and technologies under consideration are fast convergence of iterations and scalable parallelism, which are intrinsically contradictory and need a special investigation. The paper analyzes main trends is developing preconditioned iterative methods in Krylov’s subspaces based on algebraic domain decomposition and principles of their program implementation on a heterogeneous MCS with hierarchical memory structure.

The paper suggests upper bounds for the l_∞ norm of the inverses to block matrices belonging to certain subclasses of the class of block \( \mathrm{\mathscr{H}} \)-matrices, which improve and supplement known results.