This paper is concerned with the acceleration, by Chebyshev acceleration or conjugate gradient acceleration, of basic iterative methods for solving systems of linear algebraic equations. It is shown that under certain conditions these acceleration procedures are equivalent to similar procedures applied to the “double method” corresponding to two applications of the original basic iterative method. This result is applied to show the equivalence of certain acceleration procedures applied to the Jacobi methods for “red/black” systems, and similar procedures applied to the “reduced system,” which is obtained from the original system by eliminating some of the unknowns. The result is also used to study the behavior of the generalized conjugate gradient procedure of Concus and Golub and of Widlund, for solving linear systems where the matrices are positive real rather than symmetric and positive definite.
An overview is given of the simplifications which arise when p-cyclic systems are solved by iterative methods. Besides the classic iterative methods, we treat the Chebyshev semi-iterative method and the OR and MR variants of the class of Krylov subspace methods. Particular emphasis is given to equivalent iterations applied to the cyclically reduced system.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.