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# Tridiagonalization of a Symmetric Band Matrix

• H. R. Schwarz
Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)

## Abstract

The well known method proposed by Givens  reduces a full symmetric matrix A = (a ik ) of order n by a sequence of appropriately chosen elementary orthogonal transformations (in the following called Jacobi rotations) to triput diagonal form. This is achieved by (n - 1)(n - 2)/2 Jacobi rotations, each of which annihilates one of the elements a ik with |i - k|>1. If this process is applied in one of its usual ways to a symmetric band matrix A = (a ik ) of order n and with the band width m>1, i.e. with
$${a_{ik}} = 0{\rm{ for all }}i{\rm{ and }}k{\rm{ with |}}i - k{\rm{| >}}m,$$
(1)
it would of course produce a tridiagonal matrix, too. But the rotations generate immediately nonvanishing elements outside the original band that show the tendency to fill out the matrix. Thus it seems that little profit with respect to computational and storage requirements may be taken from the property of the given matrix A to be of band type.

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## References

1. 1.
Givens, W.: A method for computing eigenvalues and eigenvectors suggested by classical results on symmetric matrices. Nat. Bur. Standards Appl. Math. Ser. 29, H7–122 (1953).
2. 2.
Rutishauser, H.: On Jacobi rotation patterns. Proceedings of Symposia in Applied Mathematics, Vol. 15 Experimental Arithmetic, High Speed Computing and Mathematics, 1963, 219–239.Google Scholar
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Schwarz, H. R.: Die Reduktion einer symmetrischen Bandmatrix auf tridiagonale Form. Z. Angew. Math. Mech. (Sonderheft) 45, T76–T77 (1965).Google Scholar
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Schwarz, H. R.: Reduction of a symmetric bandmatrix to triple diagonal form. Comm. ACM 6, 315–316 (1963).
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Barth, W., R. S.Martin, and J.H.Wilkinson. Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection. Numer. Math. 9, 386–393 (1967). Cf. II/5.
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Reinsch, C, and F. L. Bauer. Rational QR transformation with Newton shift for symmetric tridiagonal matrices. Numer. Math. 11, 264 -272 (1968). Cf. II/6.
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Rutishauser, H.: The Jacobi method for real symmetric matrices. Numer. Math. 9, 1–10 (1966). Cf. II/l.

## Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

## Authors and Affiliations

• H. R. Schwarz

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