Abstract
Extensions have been developed, of several variants of the stride of two cyclic reduction method. The extensions refer to quasi-tridiagonal linear equation systems involving two additional nonzero elements in the first and last rows of the equation matrix, adjacent to the main three diagonals. Equations of this kind arise, for example, in the simulations of biosensors or other electrochemical systems by solving relevant ordinary or partial differential equations by finite difference methods, when boundary derivatives are approximated by one-sided, multipoint finite differences. The correctness of the algorithms developed has been verified using example matrices with pseudo-random coefficients, under conditions of both sequential and parallel execution.
Highlights
In this paper we describe a specialised cyclic reduction (CR) algorithm for numerically solving linear algebraic equation systems: Ax = r, (1)
(e) From the programming point of view, single instances of data structures containing matrix A and vectors r and x are sufficient in ordinary CR, since all reduction and back-substitution steps can be realised by gradually transforming the initial A and r
Errors er ∞ / r ∞ are expected to be greater than ν, due to machine errors generated in arithmetic operations involved in formula (7)
Summary
Coefficients in the first and last rows of A result, in turn, from one-sided three- or four-point, standard or compact approximations to the first spatial derivatives occurring in boundary conditions The latter discretisations are perhaps not very popular, since many authors use just two-point (one-sided or central) approximations to the boundary derivatives, which lead to purely tridiagonal matrices. Algorithms of solving equations similar to Eq (1), the Reader is referred to Bieniasz [12,13] In contrast to those serial algorithms, the numerical algorithm to be described here is an adaptation of the CR method for tridiagonal matrices, first described by Hockney [14] and Buzbee et al [15] for block-tridiagonal matrices, and attributed to Golub and Hockney.
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