The relationship between the Jacobi and the successive overrelaxation (SOR) matrices of a k-cyclic matrix

AbstractLet A be a (k−l, l)-generalized consistently ordered matrix with T and Lω its associated Jacobi and SOR matrices whose eigenvalues μ and λ satisfy the well-known relationship (λ+ω−1)k=ωkμkλk−1. For a subclass of the above matrices A we prove that the matrix analogue of the previous relationship holds. Exploiting the matrix relationship we show that the SOR method is equivalent to a certain monoparametric k-step iterative one when used for the solution of the fixed-point problem x=Tx+c