Efficiency measurements for a proposed algorithm to find all eigenvalues for dense, nonsymmetric, real matrices are presented. The proposed algorithm combines the well-known QR and LR algorithms; the former preserves numerical stability, and the latter has higher speed. Technical details of the hybrid algorithm are described in another paper [1].

The performance measurements for the algorithm are the normalized error and the time to find eigenvalues. The main reference is to the HQR code of the EISPACK software package. The computers involved range from PCs to RISC machines and the Cray X-MP. The authors found a speed improvement of up to 50 percent in some cases with almost no degradation in error or stability. The results depend on the hardware (better for PC, worse for Cray). An open problem is why this hybrid algorithm does not calculate the eigenvectors so well. It is clear that the authors have tested a large number of cases, including matrices with diverse dimensions, including some ill-conditioned defective companion matrices. The matrices were generally random generated. The code can be obtained from the second author via electronic mail. The code is written in FORTRAN.

This paper is valuable for computer engineers, researchers looking for higher-speed algorithms, and students. It has good references and is presented well.