Iterative Solutions of Systems of Linear Equations Whose Coefficient Matrix Is Positive Real
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.
Day, Jane M. and Melvin Henriksen. 1986. Iterative solutions of systems of linear equations whose coefficient matrix is positive real. Linear and Multilinear Algebra. 19(3):267-285. < http://www.informaworld.com/10.1080/03081088608817721 >.