An improved algorithm for generalized least squares estimation

Xiao-Wen Chang; David Titley-Peloquin

doi:10.3934/naco.2020044

Article Contents

2020, Volume 10, Issue 4: 451-461. Doi: 10.3934/naco.2020044

This issue Previous Article On box-constrained total least squares problem Next Article Convergence of a randomized Douglas-Rachford method for linear system

An improved algorithm for generalized least squares estimation

Xiao-Wen Chang^1, , and
David Titley-Peloquin^2,

1.
School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 0E9
2.
Department of Bioresource Engineering, McGill University, Ste-Anne-de-Bellevue, Quebec, Canada, H9X 3V9

^* Corresponding author
^* Corresponding author

Received: May 2020

Revised: September 2020

Published: September 2020

This work was supported in part by NSERC of Canada Grant RGPIN-2017-0513

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Abstract

The textbook direct method for generalized least squares estimation was developed by Christopher C. Paige about 40 years ago. He proposed two algorithms. Suppose that the noise covariance matrix, rather than its factor, is available. Both of the Paige's algorithms involve three matrix factorizations. The first does not exploit the matrix structure of the problem, but it can be implemented by blocking techniques to reduce data communication time on modern computer processors. The second takes advantage of the matrix structure, but its main part cannot be implemented by blocking techniques. In this paper, we propose an improved algorithm. The new algorithm involves only two matrix factorizations, instead of three, and can be implemented by blocking techniques. We show that, in terms of flop counts, the improved algorithm costs less than Paige's first algorithm in any case and less than his second algorithm in some cases. Numerical tests show that in terms of CPU running time, our improved algorithm is faster than both of the existing algorithms when blocking techniques are used.

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Mathematics Subject Classification: Primary: 62J05, 65F20; Secondary: 15A23; 93E24.

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Figure 6.1. Average CPU running time per solve over $10$ solves for $ \mathbf{A}\in {\mathbb{R}}^{m \times n}$ with $n = 80$, $m = 100:200:3500$

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Algorithm 1:

1: Compute the QL factorization of $ \mathbf{A} $: $ \mathbf{H}_n\cdots \mathbf{H}_1 \mathbf{A}= \begin{bmatrix} {\boldsymbol{0}} \mathbf{L}_{ \mathbf{A}} \end{bmatrix} $
2: Compute $ \bar {\boldsymbol{\Sigma}}= \mathbf{H}_1\cdots \mathbf{H}_n {\boldsymbol{\Sigma}} \mathbf{H}_n\cdots \mathbf{H}_1 $
3: Compute $ \tilde {\mathbf{y}} = \mathbf{H}_1\cdots \mathbf{H}_n {\mathbf{y}} $
4: Compute the Cholesky factorization of $ \bar {\boldsymbol{\Sigma}} $: $ \bar {\boldsymbol{\Sigma}}= \mathbf{L} \mathbf{L}^ {\mkern-1.5mu\mathsf{T}} $
5: Solve $ \mathbf{L}(1\!:\!n,1\!:\!n) {\mathbf{z}}_1 = \tilde {\mathbf{y}}(1\!:\!n) $ for $ {\mathbf{z}}_1 $
6: Solve $ \mathbf{L}_{ \mathbf{A}} {\hat{\mathbf{x}}}=\tilde {\mathbf{y}}(n+1:m)- \mathbf{L}(n+1\!:\!m,1\!:\!n) {\mathbf{z}}_1 $ for $ {\hat{\mathbf{x}}} $

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References

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