SML000847

SML000847 - c 1998 Society for Industrial and Applied...

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ON HYPERBOLIC TRIANGULARIZATION: STABILITY AND PIVOTING * MICHAEL STEWART AND G. W. STEWART SIAM J. M ATRIX A NAL. A PPL . c ± 1998 Society for Industrial and Applied Mathematics Vol. 19, No. 4, pp. 847–860, October 1998 001 Abstract. This paper treats the problem of triangularizing a matrix by hyperbolic Householder transformations. The stability of this method, which finds application in block updating and fast algorithms for Toeplitz-like matrices, has been analyzed only in special cases. Here we give a gen- eral analysis which shows that two distinct implementations of the individual transformations are relationally stable. The analysis also shows that pivoting is required for the entire triangularization algorithm to be stable. Key words. hyperbolic transformation, triangularization, relational stability, pivoting AMS subject classifications. 15A23, 65F, 65G05 PII. S0895479897319581 1. Introduction. Let A be a positive definite matrix of order p and let R T R be its Cholesky factorization. Given an m × p matrix X , the Cholesky updating problem is to compute the Cholesky factorization ˆ R T ˆ R = ˆ A A + X T X from that of A . Since X T X is positive semidefinite, ˆ A is positive definite and always has a Cholesky factor. It is well known that the Cholesky updating problem can be solved by orthogonal triangularization. Specifically, there is an orthogonal matrix Q such that Q T ± R X ² = ± ˆ R 0 ² , where ˆ R is upper triangular. From the orthogonality of Q , it follows that ˆ R T ˆ R = ± R X ² T QQ T ± R X ² = A + X T X, so that ˆ R T ˆ R is the required Cholesky factorization. The matrix Q is usually generated as a product of Householder transformations or plane rotations. For details see, e.g., [11]. Now let Y be an n × p matrix. The Cholesky downdating problem is to calculate the Cholesky factor ˆ R of ˆ A = A - Y T Y from that of A . The downdating problem is known to be difficult. An obvious problem is that ˆ A can be indefinite, in which case the problem has no (real) solution. A more subtle problem is that information present * Received by the editors April 4, 1997; accepted for publication (in revised form) November 26, 1997; published electronically May 7, 1998. http://www.siam.org/journals/simax/19-4/31958.html Computer Sciences Laboratory, Research School of Information Sciences and Engineering, Aus- tralian National University, Canberra ACT 0200, Australia. The work of this author was supported in part by the Department of the Air Force under grant F496220-95-1-0525-P00001. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 ([email protected]). The work of this author was supported by National Science Foundation grant CCR-95-03126. 847
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848 MICHAEL STEWART AND G. W. STEWART in the original problem may be represented only imperfectly in the Cholesky factor.
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SML000847 - c 1998 Society for Industrial and Applied...

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