Lapack 3.1 xHSEQR: Tuning and Implementation Notes on the Small Bulge Multi-shift QR Algorithm with Aggressive Early Deflation

This note documents implementation details of the small bulge, multi-shift QR algorithm with aggressive early deflation that appears as Lapack version 3.1 programs CHSEQR, DHSEQR, SHSEQR and ZHSEQR and the subroutines they call. These codes calculate eigenvalues and optionally a Schur factorization of a Hessenberg matrix. They do the bulk of the work required to calculate eigenvalues and optionally eigenvectors of a general non-symmetric matrix. This report is intended to provide some guidance for setting the machine dependent tuning parameters, to help maintainers to identify and correct problems, and to help developers improve upon this implementation. ∗Portions of this work were accomplished while this author was on sabbatical leave at and partially supported by the Lawrence Berkeley National Laboratory. This work was also partially supported by by the National Science Foundation under awards 0098150 and 0112375.