A note on the SSOR and USSOR iterative methods applied top-cyclic matrices

Abstract

The purpose of this note is threefold:i) to derive the new functional equation, $$\begin{gathered} \left[ {\lambda - \left( {1 - \omega } \right)\left( {1 - \hat \omega } \right)} \right]^p = \lambda ^k \left[ {\lambda \omega + \hat \omega - \omega \hat \omega } \right]^{\left| {\zeta _L } \right| - k} \left[ {\lambda \hat \omega + \omega - \omega \hat \omega } \right]^{\left| {\zeta _U } \right| - k} \hfill \\ \cdot \left( {\omega + \hat \omega - \omega \hat \omega } \right)^{2k} \mu ^p , \hfill \\ \end{gathered} $$ which couples the nonzero eigenvalues of the USSOR iteration matrix $$T_{\omega ,\hat \omega } $$ with the eigenvalues μ of the associated block Jacobi matrixB in thep-cyclic case,ii) to interpret the exponentk in this equation by means of graph theory, andiii) to connect the above equation with known results in the literature.