Tridiagonal matrix algorithm

Abstract: A circulant tridiagonal system is a special type of Toeplitz system that appears in a variety of problems in scientific computation. In this paper we give a formula for the inverse of a symmetric circulant tridiagonal matrix as a product of a circulant matrix and its transpose, and discuss the utility of this approach for solving the associated system.

Abstract: The development of the discrete fractional Fourier transform (DFRFT) necessitates having orthonormal eigenvectors for the DFT matrix, F. The objective of having the DFRFT approximate its continuous counterpart can be met if the eigenvectors of F approximate samples of the Hermite-Gaussian functions. Orthonormal Hermite-Gaussian-like eigenvectors for F are rigorously derived by a detailed analysis of an almost tridiagonal matrix, S, which commutes with F. By an appropriate similarity transformation, S is reduced to a 2/spl times/2 block diagonal form and the elements of the two exactly tridiagonal matrices forming the two diagonal blocks are explicitly derived in terms of the elements of matrix S.

Abstract: We present a parallel algorithm for the solution of tridiagonal linear systems based on the method of partitioning and backward elimination. A given N × N system is partitioned into r blocks each of size n × n (N = rn n even). Within each block the system is rewritten by collecting pairs of equations from the top and bottom (written backwards); this makes it suitable to solve the subsystem by a UL-factorization of its coefficient matrix. Once a core system of size 2r × 2r has been solved, solutions of the r subsystems can be computed in parallel. Interestingly, the serial arithmetical operations count for the present algorithm is 0(1612;N); this makes it faster than the quadrant interlocking factorization method of Chawla and Passi [1], the partitioning method of Wang [7] and cyclic reduction (see Hockney and Jesshope [6, p. 479]), each of which has a serial count of 0(17N).

Abstract: A finite volume, time-marching for solving time-dependent viscoelastic flow in two space dimensions for Oldroyd-B and Phan Thien–Tanner fluids, is presented. A non-uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi-implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under-relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four-to-one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four-to-one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd-B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.