A modified lanczos method for the numerical solution of large sparse matrix pencil systems

In several physical applications of mathematics, it becomes necessary to solve matrix pencil systems of the form (A + λB)x = c for several values of the real parameter λ. To solve this problem, an algorithm is developed, which is a modification of the Lanczos algorithm. The algorithm is presented in such a way that if parallel processors axe available, the solution to the matrix pencil system can be obtained for several values of λ simultaneously. After this iterative method is derived, a comparison is made between the modified Lanczos method and the Preconditioned Conjugate Gradient Method.

Convergence of the iterative method is haunted by the loss of orthogonality of supposedly orthogonal matrices Q_j which axe used in the algorithm. To combat this problem, the topics of reorthogonalization and selective orthogonalization are discussed. Numerical results axe included, and listings of Fortran programs utilizing these algorithms appear in the appendixes.