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We present a new transform, triple dqds, to help to compute the eigenvaluesof a real tridiagonal matrix C using real arithmetic. The algorithm uses thereal dqds transform to shift by a real number and triple dqds to shift by acomplex conjugate pair. We present what seems to be a new criteria forsplitting the current pair L,U. The algorithm rejects any transform whichsuffers from excessive element growth and then tries a new transform. Ournumerical tests show that the algorithm is about 100 times faster than theEhrlich-Aberth method of D. A. Bini, L. Gemignani and F. Tisseur. Our code iscomparable in performance to a complex dqds code and is sometimes 3 timesfaster.

Real dqds for the nonsymmetric tridiagonal eigenvalue problem