Least Squares Computations by Givens Transformations Without Square Roots

Gentleman, W. Morven

doi:10.1093/imamat/12.3.329

Cited by 436 publications

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“…Therefore this method may turn out to be more competitive on modern computers where the time taken for floating-point additions and multidimensional array indexings are not negligible compared to that for multiplications. Also, the use of fast Givens [4] may result in substantial improvement in efficiency.…”

Section: S O M E C O M P U T a T I O N A L Detailsmentioning

confidence: 99%

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An Improved Algorithm for Computing the Singular Value Decomposition

Chan

1982

ACM Trans. Math. Softw.

209

102

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The most well-known and widely used algorithm for computing the Singular Value Decomposition (SVD) A ---U ~V T of an m x n rectangular matrix A is the Golub-Reinsch algorithm (GR-SVD). In this paper, an improved version of the original GR-SVD algorithm is presented. The new algorithm works best for matrices with m >> n, but is more efficient even when m is only slightly greater than n (usually when m ~ 2n) and in some cases can achieve as much as 50 percent savings. If the matrix U ~s exphcltly desired, then n 2 extra storage locations are required, but otherwise no extra storage is needed. The two main modifications are: (1) first triangularizing A by Householder transformations before bldmgonahzing it (thin idea seems to be widely known among some researchers in the field, but as far as can be determined, neither a detailed analysis nor an lmplementatmn has been published before), and (2) accumulating the left Givens transformations in GR-SVD on an n x n array instead of on an m x n array. A PFORT-verified FORTRAN Implementation m included. Comparisons with the EISPACK SVD routine are given.

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Section: S O M E C O M P U T a T I O N A L Detailsmentioning

confidence: 99%

“…It is assumed that slow Givens is used throughout the calculation. If fast Givens [4] had been used, then the entries would become approximately 2mn 2 and 2n 3, instead.…”

Section: Mod-svdmentioning

confidence: 99%

An Improved Algorithm for Computing the Singular Value Decomposition

Chan

1982

ACM Trans. Math. Softw.

209

102

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“…An implementation described in Section 5 uses aggregated rotations and further reduces both the memory traffic and the flop count. For real shifts, we have (5/4)n 2 + O(n) flops with double rotations, (7/6)n 2 + O(n) flops with triple rotations, and fast rotations [Gentleman 1973;Hammarling 1974], run with only n 2 + O(n) flops. In the case of complex shifts, we have (5/2)n 2 + O(n) flops for double rotations, and (7/3)n 2 + O(n) for triple rotations.…”

Section: The New Algorithmmentioning

confidence: 99%

A note on shifted Hessenberg systems and frequency response computation

Beattie

Drmavč

Gugercin

2011

ACM Trans. Math. Softw.

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“…An error analysis of the QR decomposition by HRs and GRs are given by Gentleman (1975) and Higham (2000), respectively.…”

Section: Algorithm 3 Qmentioning

confidence: 99%

Numerical Linear Algebra

Gentle

2011

Handbook of Computational Statistics

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Least Squares Computations by Givens Transformations Without Square Roots

Cited by 436 publications

References 0 publications

An Improved Algorithm for Computing the Singular Value Decomposition

An Improved Algorithm for Computing the Singular Value Decomposition

A note on shifted Hessenberg systems and frequency response computation

Numerical Linear Algebra

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