Abstract

A new and efficient split-step polynomial preconditioning algorithm is discussed. In this algorithm, basic polynomial preconditioning iterations are repeated several times, each time with an improved matrix. With minor modifications this algorithm can be used for matrix inversion and condition number estimation. The split-step algorithms consist of two nested iterative loops involving matrix multiplications and thus are attractive for parallel optical processors. The details of realization of splitstep algorithms on three different optical matrix multiplier architectures are outlined. The condition number after m outer loop iterations is given by1/{1−[1−1/c(a)]^pm }, where p is the number of inner loop iterations and C(A) is the initial condition number.¹ This strategy reduces the condition number approximately at the rate O(p^−m), much faster compared to other preconditioning algorithms. The results of numerical experiments on the split-step algorithm applied to a case study of an optimum phased array antenna design problem show that this new preconditioning algorithm is a viable tool for improving the performance of analog optical processors. Preliminary analyses of the split-step algorithm demonstrate its robustness with regard to the spatial errors and noise present in most optical processors. Using these split-step algorithms an interconnected optical processor capable of computing solutions with a specified accuracy can be designed.

© 1989 Optical Society of America