A fast method for solving a class of tridiagonal linear systems

Abstract

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A Fresh Look at Generalized Sampling

“…We describe an algorithm adapted from Malcolm and Palmer [1974] for solving the common case where ϕ = p, q . , p is symmetric with support width of 3 elements.…”

Asymptotic properties of the QR factorization of banded Hessenberg–Toeplitz matrices

“…With these relations, we can obtain from (6) and (9) with some algebraic manipulations (13) and (14). Combining the limit of {|x after some ÿnite steps of the QR factorization, we can get a vector x (k) which satisÿes the above condition.…”

“…In fact, if one can show that the entries in the QR factorization converge rapidly to machine precision, one can avoid the computation of the entire factorization and use the limits directly instead in the factors, as soon as convergence is achieved, leading to signiÿcant computational savings in solving the linear system Ax = c, or the least squares problem min Ax − d 2 (where A = is still a Toeplitz matrix) by the QR factorization. If A is a symmetric, diagonally dominant, tridiagonal Toeplitz matrix, it is shown in Reference [14] that the diagonals of the LU factors of A converge and computational savings are possible. Similar properties also hold for cyclic reduction, see Reference [15]; but for the QR factorization we are not aware of any results in the literature.…”

Fast Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinates

“…In this paper, we develop a second-order finite difference approximation scheme and solve the resulting large algebraic system of linear equations systematically using block tridiagonal system [14] and extend the Hockney's method [15] to solve the three dimensional Poisson's equation on Cylindrical coordinates system.…”