Extrapolation vs. projection methods for linear systems of equations

It is shown that the four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation, and topological epsilon algorithm, when applied to linearly generated vector sequences, are Krylov subspace methods, and are equivalent to some well known conjugate gradient type methods. A unified recursive method that includes the conjugate gradient, conjugate residual, and generalized conjugate gradient methods is developed. Finally, the error analyses for these methods are unified, and some known and some new error bounds for them are given. Keywork Iterative methods, vector sequences, extrapolation, projection, Krylov subspace methods, conjugate gradient type methods, Chebyshev acceleration.