The progressive quotient difference algorithm with shifts (qds) was presented by Rutishauser as early as 1954. It is equivalent to the shifted LR algorithm written in a special notation for tridiagonal matrices. The much more recent differential
qds (dqds) is a sophisticated variant of qds. The triple dqds algorithm consists of three dqds steps performed implicitly and such that real arithmetic is maintained in the presence of complex eigenvalues. The main advantage over the standard Hessenberg QR algorithm is that it preserves the tridiagonal form and thus reduces both storage and time. In this seminar we will describe the triple dqds algorithm and we will present some preliminary numerical results that suggest the robustness of the new algorithm.
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