Some linear algebraic algorithms and their performance on CRAY-1

This report describes new and relevant features of the CRAY-1 hardware, and defines and characterizes the scalar, vector, and super vector performance levels inherent in this computer. Such fundamental applications as polynomial evaluation, matrix multiplication, and the solution of linear equations can be programed to achieve super vector speeds. Methods for achieving optimal performance are described and compared with more conventional methods. The various methods used to solve the following types of linear systems are analyzed: general linear systems by Gaussian elimination and Gauss--Jordan methods, symmetric positive-definite systems by the root-free Cholesky method, matrix inversion by the Gauss--Jordan method, tridiagonal systems, least squares systems by the Householder transformation method, and both general banded systems and symmetric positive-definite banded systems. In addition to estimating CRAY-1 performances, times are given for the CRAY-1, CDC-STAR, and CDC-7600 computers. 3 figures, 6 tables.