Vector and matrix sequence transformations based on biorthogonality

Sequence transformations are used for the purpose of convergence acceleration. An important algebraic property connected with a sequence transformation is its kernel, that is the set of sequences transformed into a constant sequence (usually the limit of the sequence). In this paper, we show how to construct transformations whose kernels are the sets of vector or matrix sequences of the forms $x_n=x+Z_n \alpha, x_n=x+Z_n \alpha_n$ and $x_n=x+Z_n \alpha_n+Y_n \beta$ where $Z_n$ and $Y_n$ are known matrices, $\alpha$, $\alpha_n$ and $\beta$ unknown vectors or matrices. Recursive algorithms for their implementation are given. Applications to the solution of systems of linear and nonlinear equations are also discussed.