In this work we review the present status of numerical methods for partial differential equations on vector and parallel computers. A discussion of the relevant aspects of these computers and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial-boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. A brief discussion of application areas utilizing these computers is included.

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Solution of Partial Differential Equations on Vector and Parallel Computers

Spectral and computational properties of band symmetric toeplitz matrices

A new matrix algebra $\mathcal{H}$, including the set of real symmetric circulant matrices, is introduced. It is proved that all the matrices of $\mathcal{H}$ can be simultaneously diagonalized by the similarity transformation associated to the discrete Hartley transform. An application of this result to the solution of Toeplitz systems by means of the preconditioned conjugate gradient method is presented.

On a matrix algebra related to the discrete Hartley transform

Using the concept of approximate algorithm it is shown that $6\log n + 6$ parallel steps and $2n$ processors suffice to approximate, with any precision, the solution of a linear system with an $n \times n$ triangular Toeplitz matrix A. Moreover, $7\log n + 7$ steps are sufficient for an exact computation, whereas the number of processors is increased to $( \frac{5}{2} )n^2 $. If A is also banded and k is its bandwidth, the number of processors is reduced to $( \frac{5}{2} )n(k + 1)$. Two applications are shown. It is proved that if B is any matrix belonging to the algebra generated over the complex field by a given $n \times n$ matrix, then the system $Bx = b$ can be solved with no more than $9\log n + 4$ steps with $O(n^2 )$ processors. It is proved that, given a Toeplitz matrix $A = (a_{i,j} )$ such that $a_{i,j} = 0$ if $i - j > k$ or $j - i > h$, $a_{k,1} \ne 0$, then $13\log n + O(\log ^2 k)$ steps and $\max ( (\frac{5}{2} )n(k + h), n(n + 1)/ 2 )$ processors are sufficient to solve the system $Ax = ...

Parallel solution of certain toeplitz linear systems

The paper describes a recursive algorithm for solving Hermitian Toeplitz systems of linear equations, without any restriction on the ranks of their nested Toeplitz subsystems. Such a general algorithm is needed, e.g., to obtain the eigenfilters for signal processing applications, or to compute the inverse of a nondefinite Toeplitz matrix. The regular portion of the algorithm is made of the classical Levinson recursion. The singular portion requires solving some well-defined systems of linear equations with gradient structure. The dimension of each of these sytems equals the amplitude of the corresponding singularity.

A generalization of the Levinson algorithm for Hermitian Toeplitz matrices with any rank profile

In this paper we describe three algorithms for solving positive-definite banded Toeplitz systems of linear equations on parallel computers. Assuming we have $4mn$ processors, where n is the order of the system and m is the number of super- or subdiagonals, each system may be solved in $O(m\log n)$ time steps. Numerical experiments that compare the behavior of these algorithms in solving pentadiagonal Toeplitz systems are presented.

On Certain Parallel Toeplitz Linear System Solvers

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On Certain Parallel Toeplitz Linear System Solvers