β Scribed by Petko Yanev; Erricos John Kontoghiorghes
No coin nor oath required. For personal study only.
Computationally efficient serial and parallel algorithms for estimating the general linear model are proposed. The sequential block-recursive algorithm is an adaptation of a known Givens strategy that has as a main component the Generalized QR decomposition. The proposed algorithm is based on orthogonal transformations and exploits the triangular structure of the Cholesky QRD factor of the variance-covariance matrix. Specifically, it computes the estimator of the general linear model by solving recursively a series of smaller and smaller generalized linear least squares problems. The new algorithm is found to outperform significantly the corresponding LAPACK routine. A parallel version of the new sequential algorithm which utilizes an efficient distribution of the matrices over the processors and has low inter-processor communication is developed. The theoretical computational complexity of the parallel algorithms is derived and analyzed. Experimental results are presented which confirm the theoretical analysis. The parallel strategy is found to be scalable and highly efficient for estimating large-scale general linear estimation problems.
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This paper is primarily concerned with extending the results of Brandwein and Strawderman in the usual canonical setting of a general linear model when sampling from a spherically symmetric distribution. When the location parameter belongs to a proper linear subspace of the sampling space, we give a