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The research conducted under this effort was centered about general matrix methods and applications of matrix theory in solving large systems of linear equations. In particular, the classification of certain factorizations of M matrices was undertaken. An extension of the Stein-Rosenberg theorem comparing the spectral radii of matrices useful in constructing iterative solution techniques was obtained. The use of summability methods and approximated conformal mapping techniques in the study of iterative methods was pursued. Finally, a study of SOR and SSOR iterative methods was accomplished using the theory of H matrices. Eleven papers appeared in the referred literature during this effort. Keywords: General matrix methods; Large systems of linear equations; Stein-Rosenberg theorem.

Research conducted by the principal investigator during the period June 1, 1979 to May 31, 1980, resulted in the following research articles which have either appeared in print, or have been accepted in refereed mathematical journals, in this period: Inequalities for polynomials with a prescribed zero; On the Enestrom-Kakeya Theorem and Its Sharpness; Bounds for incomplete polynomials vanishing at both endpoints of an interval; Remarks on some conjectures of G. G. Lorentz; On incomplete polynomials; An extension of the Enestrom-Kakeya Theorem and its sharpness; On zeros of generalized Bessel polynomials; Incomplete factorizations of matrices and connections with H- matrices; On two conjectures on the zeros of generalized Bessel polynomials; Incomplete polynomials: an electrostatic approach; An introduction to the Convergence Theory of Pade' Approximants.

The basic aim of the research proposed for this AFOSR Grant was to investigate, in depth, the optimization of iterative methods for solution of nonsymmetric nonpositive-definite matrix equations which arise in the numerical approximation (by finite difference or finite elements) of typical convection- diffusion problems. In the opinion of this Principal Investigator, the research results obtained, with AFOSR support, were of sufficient quality to be published in leading technical journals. We list below published research papers (each carrying an acknowledgment of AFOSR support), associated in this period with this current research effort.

The objective of this project was to study numerical methods for solving sparse linear systems of equations of the type that arise from discretized partial differential equations. Such systems arise in mathematical models of numerous physical processes including turbulent flow, chemical reactive flow, semiconductor device simulation, and structural mechanics. For many such problems, analytic solutions are not available, so the only way of obtaining insight into the model is through numerical approximation and solution. The solution of sparse linear systems is often the most costly task, in terms of both computer time and storage, required by this process, so that improving the efficiency of such computations is of critical importance for the construction of accurate numerical models. Emphasis has been on sparse iterative methods for solving these systems, and implementation of linear system solvers on parallel computers. Major results are summarized in the final report.

This research conducted by the principal investigator during the period October 1, 1979 to June 30, 1980, resulted in the following research articles which have either appeared in print, or have been accepted in refereed mathematical journals, in this period: Inequalities for polynomials with a prescribed zero; Theorems of Stein-Rosenberg Type; On the Enestrom-Kakeya Theorem and Its Sharpness; Bounds for incomplete polynomials vanishing at both endpoints of an interval; Hermite-Birkhoff interpolation in the n-th roots of unity; Remarks on some conjectures of G.G. Lorentz; On incomplete polynomials. II; Interpolation in the roots of unity: an extension of a Theorem of J. L. Walsh; On the sharpness of some upper bounds for spectral radii of S.O.R. iteration matrices; An extension of the Enestrom-Kakeya Theorem and its sharpness; Lacunary trigonometric interpolation on equidistant nodes.

This research conducted by the principal investigator during the period October 1, 1979 to June 30, 1980, resulted in the following research articles which have either appeared in print, or have been accepted in refereed mathematical journals, in this period: Inequalities for polynomials with a prescribed zero; Theorems of Stein-Rosenberg Type; On the Enestrom-Kakeya Theorem and Its Sharpness; Bounds for incomplete polynomials vanishing at both endpoints of an interval; Hermite-Birkhoff interpolation in the n-th roots of unity; Remarks on some conjectures of G.G. Lorentz; On incomplete polynomials. II; Interpolation in the roots of unity: an extension of a Theorem of J. L. Walsh; On the sharpness of some upper bounds for spectral radii of S.O.R. iteration matrices; An extension of the Enestrom-Kakeya Theorem and its sharpness; Lacunary trigonometric interpolation on equidistant nodes.

This effort involved research investigating improved iterative methods for solving sparse systems of linear equations. Research papers appearing in print during this period included such titles as On the minimum module of normalized polynomials, Extended numerical computations on the 1/9 conjecture in rational approximation Theory, and A study of semi-iterative methods for nonsymmetric systems of linear equations. A major accomplishment during this period of effort was the use of summability methods and conformal mapping techniques in the study of iterative methods, thus enhancing the theoretical foundations of such methods.