The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond

Abstract

Cyclic reduction is an algorithm invented by G. H. Golub and R. W. Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle. Among the algorithms of Gene Golub, it is one of the most versatile and powerful ever created. Recently, it has been applied to solve different problems from different applicative areas. In this paper we survey the main features of cyclic reduction, relate it to properties of analytic functions, recall its extension to solving more general finite and infinite linear systems, and different kinds of nonlinear matrix equations, including algebraic Riccati equations, with applications to Markov chains, queueing models and transport theory. Some new results concerning the convergence properties of cyclic reduction and its applicability are proved under very weak assumptions. New formulae for overcoming breakdown are provided.

Matrix structures and applications

“…This infinite system can be solved by means of the Cyclic Reduction (CR) method introduced by Gene Golub (see [18] for bibliographic references and for general properties of CR) for the numerical solution of the discrete Poisson equation over a rectangle and here adjusted to the infinite block Toeplitz case. The CR technique works this way:…”

“…A detailed treatment of this topic can be found in [18]. The same technique can be extended to matrix equations of the kind …”

A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices

“…This choice is dictated by the growing importance of parallel and vector computers for scientific and technical computing. Fine-grained parallelism is inherent in CR, which has prompted many authors to implement this method on a number of parallel and vector computers, or computer architectures (reviews of the method are available in [16][17][18][19]; a few example implementations of CR for scalar tridiagonal and block-tridiagonal systems are described in Refs. [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]).…”

Fast Poisson Solvers for Graphics Processing Units

“…The radix-q PSCR (Partial Solution variant of the Cyclic Reduction) method [14][15][16][17] represents a different kind of approach based on the partial solution technique [18,19]. Excellent surveys on these kind of methods can be found in [20] and [21].…”