A matric approach to the stability of solutions of algebraic systems by arithmetic and analytic machines

The stability of solutions of systems of algebraic equations on arithmetic machines by iterative or indirect methods and directly on analytic machines, is considered in this paper. The concept of an Explicit Operation Matrix is introduced which simplifies the analysis of stability of solutions. A thorough study of the various existing procedures for solutions of systems of linear equations is made on this basis, and useful criteria have been obtained for the stability. The analytic machine is analysed as a particular case of the iterative arithmetic machine wherein the iterative sequences become continuous rather than being discrete integral, which as a consequence results in the transformation of the Explicit Operation Matrix into an Implicit Operation Matrix. These analyses show that arithmetic machines using iterative methods, and analytic machines giving directs solutions, have similar constraints as to the positive definiteness of the matrix of the algebraic system. However, it has been found that for stability, an analytic machine imposes less stringent conditions on the nature of Implicit Operation Matrix, unlike arithmetic machines, which impose more stringent conditions on the nature of Explicit Operation Matrix. Some experimental results obtained along these lines using the Matrix Computor are also included.