Iterative Solutions of Systems of Linear Equations Whose Coefficient Matrix Is Positive Real

Document Type



Mathematics (HMC)

Publication Date



During the academic year 1981-82, we worked, together with a team of students and faculty under the auspices of the Claremont Mathematics Clinic, on a problem in computational aerodynamics that resulted in the report [L]. The problem came from an engineering group at Lockheed-California, who were modeling the flow of air over the surface of an aircraft. They obtained large full systems of linear equations of the form Ax = b, where A is a matrix with real entries such that xtAx>0 for all nonzero real vectors x, and were applying a version of successive over-relaxation (SOR). Such a matrix is called positive real in [Y] and can have complex eigenvalues, hence need not be similar to a (symmetric) positive definite matrix. Below we give some intervals of ∞-values for which the SOR iteration matrix Lw(A) has spectral radius less than 1, when A is positive real and satisfies other conditions. We supply a few pertinent examples and discuss the necessity of some of these conditions.

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© 1986 Gordon and Breach Science Publishers