The Use of Parallel Polynomial Preconditioners in the Solution of Systems of Linear Equations

This thesis mainly explores the use of polynomial preconditioners in iterative solvers for large-scale sparse linear systems Ax = b. It is well known that preconditioners can significantly improve the convergence of solvers, particularly when the coefficient matrix is ill-conditioned. Further, polynomial preconditioners have several advantages over other popular preconditioners — they may be implemented easily, they are highly parallel, and they are extremely agile. Due to the intrinsic disadvantages of polynomial methods (e.g., spectrum information is needed, poor stability of the large-degree polynomial preconditioning) and the limitation of computing technologies, the polynomial preconditioning technique was somehow ignored in the past ten years. Fortunately, at present, polynomial preconditioners are attracting more and more attention with the development of computer science. The construction of appropriate polynomial preconditioners can be transformed into constrained optimization problem, namely that of finding am-degree polynomial in matrix A, Pm(A), such that Pm(A) ≈ A. Three typical polynomial preconditioners arise, Neumann-series, Least-squares and Minimax. In this thesis Symmetric linear equations will be mainly discussed, where the spectrum of the coefficient matrix is expressed by a real compact set. In the case of SID (Symmetric Indefinite) linear systems, only Least-squares and Chebyshev methods are applicable. The implementation and detailed performance analysis of the Generalized Least-squares and the Generalized Minimax preconditioners are the main effort of this work. The convergence property of polynomial preconditioned iterative solvers is closely dependent on an accurate estimate of the eigenspectrum of the coefficient matrix. Consequently, it is natural to combine polynomial preconditioners with GMRES, the well-known standard algorithm for the computation of the solution of systems of linear equations which incorporates the so-called Arnoldi process. Thus, the use