ABSTRACT
This chapter presents some basic and new results for solving large sets of linear systems on multiprocessors. Asynchronous numerical methods are introduced that are suitable for multiprocessor systems. In particular, the numerical solution of partial differential equations that is required for the solution of many engineering problems is accomplished by replacing the equations by a banded system of linear equations whose solution yields an approximation of the exact solution in the form of a set of values generated by a function that approximates the true analytic solution. Direct methods solve the system of equations in a known number of arithmetic operations, and errors in the solution arise entirely from rounding errors introduced in the computation. The main motivation for defining chaotic relaxation is that when iterative methods are implemented on a multiprocessor system by using a chaotic relaxation scheme the communication and synchronization between the cooperating processes are significantly reduced.
