On Jacobi and Jacobi-I ike Algorithms for a Parallel Computer

Many existing algorithms for obtaining the eigenvalues and eigenvectors of matrices would make poor use of such a powerful parallel computer as the ILLIAC IV. In this paper, Jacobi's algorithm for real symmetric or complex Hermitian matrices, and a Jacobi-like algorithm for real nonsymmetric matrices developed by P. J. Eberlein, are modified so as to achieve maximum efficiency for the parallel computations. 1. Introduction. With the advent of parallel computers, the study of compu- tationally massive problems became economically possible. Such problems include, for example, solution of sets of partial differential equations over sizable grids, and multiplication, inversion, or determination of eigenvalues and eigenvectors of large matrices. An example of a parallel computer is the ILLIAC IV.* This computer is es- sentially an array of coupled arithmetic units driven by instructions from a common control unit. Each of the arithmetic units, called processing elements (PE's), have 2048 words of 64-bit memory with an access time under 420 nanoseconds. Each PE is capable of 64-bit floating-point multiplication in about 550 nanoseconds. Two 32-bit floating-point operations may be performed in each PE in approximately the same times. The PE instruction set is similar to that of conventional machines with two exceptions. First, the PE's are capable of communicating data to four neigh- boring PE's by means of routing instructions. Second, the PE's are able to set their own mode registers to effectively disable or enable themselves. For a more detailed