Relationship of eigenvalue between GMPSD iterative matrix and Jacobi iterative matrix (Q981054)

The paper focuses on the eigenvalue relationship between the generalized modified preconditioned simultaneous displacement (GMPSD) iterative matrix and the Jacobi iterative matrix for the block \(p\)-cyclic case. The first part is an introduction concerning \(p\)-cyclic matrices, the modified preconditioned simultaneous displacement (MPSD) and the GMSD. The main results are proved in the second section. Thus, in the first part, a theorem is presented which gives the relationship which holds between the eigenvalues of the block Jacobi iterative matrix and the eigenvalues of the block GMPSD iterative matrix, for a given \(p\)-cyclic matrix. Based on this theorem, nine corollaries concerning the relationship of the eigenvalues are derived. Similar to the first part, in the second one, the authors derive a relationship of the eigenvalues for a given generalized consistently ordered \((q,p-q)\) matrix, where \(q\) is relatively prime to \(p\); they prove one main theorem and nine corollaries.