The multishift QR algorithm.: part I:: Maintaining well-focused shifts and level 3 performance

This paper presents a small-bulge multishift variation of the multishift QR algorithm that avoids the phenomenon of shift blurring, which retards convergence and limits the number of simultaneous shifts. It replaces the large diagonal bulge in the multishift QR sweep with a chain of many small bulges. The small-bulge multishift QR sweep admits nearly any number of simultaneous shifts even hundreds without adverse effects on the convergence rate. With enough simultaneous shifts, the small-bulge multishift QR algorithm takes advantage of the level 3 BLAS, which is a special advantage for computers with advanced architectures.