Carmen Perea

SOME COMPARISON THEOREMS FOR WEAK NONNEGATIVE SPLITTINGS OF BOUNDED OPERATORS

Abstract The comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors. In this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general Banach space and rather general cones, and in a Hilbert space, which extend some of the results obtained by Woźnicki (Japan J. Indust. Appl. Math. 11(1994) 289–342) and Marek and Szyld (Numer. Math. 44(1984) 23–35). Furthermore, we present new theorems also for bounded operator which extend some results by Csordas and Varga (Numer. Math. 44. (1984) 23–35) for weak nonnegative splittings of matrices.

Convergence and comparison theorems for a generalized alternating iterative method

Using as a principal tool the convergence results of standard iterative process for the solution of linear systems, alternating iterative methods are studied. We extend the convergence theorem for the stationary alternating iterative method of Benzi and Szyld [Numererische Mathematik 76 (1997) 309], for weak nonnegative splittings of the first type of a monotone matrix to weak nonnegative splitting of the second type. On the other hand, we introduce a more general method, the nonstationary alternating iterative method, establishing convergence results for weak nonnegative splittings of a monotone matrix, and for P-regular splittings of a symmetric positive definite matrix.

Input-state-output representations and constructions of finite support 2D convolutional codes

Two-dimensional convolutional codes are considered, with codewords having compact support indexed in

Maximum Distance Separable 2D Convolutional Codes

\mathbb N

Comparison theorems for weak nonnegative splittings of K-monotone matrices

2 and taking values in

Convergence theorems for parallel alternating iterative methods

\mathbb F

A BSP recursive divide and conquer algorithm to solve a tridiagonal linear system

n, where

A Systems Theory Approach to Periodically Time-Varying Convolutional Codes by Means of Their Invariant Equivalent

\mathbb F

An overlapped two-way method for solving tridiagonal linear systems in a BSP computer

is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented.

Sequential and parallel synchronous alternating iterative methods

Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate k/n and degree δ, which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate k/n and degree δ that reach such bound when n k(((I(δ/k)J + 2)(I(δ/k)J + 3))/2) if k f δ, or n k((((δ/k) + 1)((δ/k) + 2))/2) if k | δ, by presenting a concrete constructive procedure.