Vector and matrix sequence transformations based on biorthogonality Original Research Article

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 xn = x + Znα, xn = x + Znαn and xn = x + Znαn + Ynβ where Zn and Yn are known matrices, α, αn and β 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.