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The New qd Algorithms

Published online by Cambridge University Press:  07 November 2008

Beresford N. Parlett
Beresford N. Parlett
Affiliation:
Department of MathematicsUniversity of CaliforniaBerkeley CA 94720, USA, E-mail: [email protected]
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Extract

Let us think about ways to find both eigenvalues and eigenvectors of tridiagonal matrices. An important special case is the computation of singular values and singular vectors of bidiagonal matrices. The discussion is addressed both to specialists in matrix computation and to other scientists whose main interests lie elsewhere. The reason for hoping to communicate with two such diverse sets of readers at the same time is that the content of the survey, though of recent origin, is quite elementary and does not demand familiarity with much beyond triangular factorization and the Gram-Schmidt process for orthogonalizing a set of vectors. For some readers the survey will cover familiar territory but from a novel perspective. The justification for presenting these ideas is that they lead to new variations of current methods that run a lot faster while achieving greater accuracy.

Type
Research Article
Information
Acta Numerica , Volume 4 , January 1995 , pp. 459 - 491
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Demmel, J. and Kahan, W. (1990), ‘Accurate singular values of bidiagonal matrices’, SIAM J. Sci. Sta. Comput. 11, 873–912.CrossRefGoogle Scholar
Fernando, K. V. and Parlett, B. N. (1994), ‘Accurate singular values and differential qd algorithms’, Numerische Mathematik 67, 191–229.CrossRefGoogle Scholar
Henrici, P. (1958), ‘The quotient-difference algorithm’, Nat. Bur. Standards Appl. Math. Series 19, 23–46.Google Scholar
Mathias, R. (1994), ‘The instability of parallel prefix matrix multiplication’, preprint, Dept. of Mathematics, College of William and Mary, Williamsburg, VA 23187.Google Scholar
Parlett, B. N. (1992), ‘Reduction to tridiagonal form and minimal realizations’, SIAM J. on Matrix Analysis and Applications 13, 567–593.CrossRefGoogle Scholar
Parlett, B. N. (1994) ‘Orthogonal eigenvectors for symmetric tridiagonals’, to appear.Google Scholar
Rutishauser, H. (1954), ‘Der Quotienten-Differenzen-Algorithmus’, Z. Angew. Math. Phys. 5, 233–251.CrossRefGoogle Scholar
Rutishauser, H. (1958), ‘Solution of eigenvalue problems with the LR-transformation’, Nat. Bur. Standards Appl. Math. Series 49, 47–81.Google Scholar
Rutishauser, H. (1960), ‘Uber eine kubisch konvergente Variante der LR-Transformation’, Z. Angew. Math. Mech. 11, 49–54.CrossRefGoogle Scholar
Rutishauser, H. (1990), Lectures on Numerical Mathematics, Birkhäuser, Boston.CrossRefGoogle Scholar
Yang, Yao (1994) ‘Backward error analysis for dqds’, submitted for publication.Google Scholar