Sven J. Hammarling. The numerical solution of the general Gauss--Markov linear model. In Mathematics in Signal Processing, T. S. Durrani, J. B. Abbiss, and J. E. Hudson, editors, Oxford University Press, 1987, pages 451--456.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....three versions of the null space method for solving the LSE problem, so called because it employs an orthogonal basis for the null space of the constraint matrix. We begin with a version based on the generalized QR factorization. The generalized QR factorization was introduced by Hammarling [12] and Paige [17] and further analyzed by Anderson, Bai and Dongarra [2] Theorem 2.1 (generalized QR factorization) Let A 2 R B 2 R with m p n p. There are orthogonal matrices Q 2 R U 2 R m Gamman p L 11 0 n Gammap L 21 L 22 ; BQ = p S 0 ; 2.1) where L 22 and S are ....

Sven J. Hammarling. The numerical solution of the general Gauss--Markov linear model. In Mathematics in Signal Processing, T. S. Durrani, J. B. Abbiss, and J. E. Hudson, editors, Oxford University Press, 1987, pages 451--456.

....the other QR factorizes B . We first describe the null space methods, so called because they employ an orthogonal basis for the null space of the constraint matrix. We begin with a version based on the generalized QR factorization. The generalized QR factorization was introduced by Hammarling [27] and Paige [42] and further analyzed by Anderson, Bai and Dongarra [3] and is of interest in its own right. Theorem 4.7 (generalized QR factorization) Let A 2 R and B 2 with m p n p. There are orthogonal matrices Q 2 R and U 2 R m Gamman p L 11 0 n Gammap L 21 L 22 ; BQ = p S ....

Sven J. Hammarling. The numerical solution of the general Gauss--Markov linear model. In Mathematics in Signal Processing, T. S. Durrani, J. B. Abbiss, and J. E. Hudson, editors, Oxford University Press, 1987, pages 451--456.

....is available from public linear algebra library LINPACK[7] Redesigned codes in block algorithm fashion that are better suited for today s high performance architectures can be found in LAPACK. The terminology generali ed R factori ations (GQR factorization) which has been introduced by Hammarling[6] and Paige[9] is to refer to orthogonal transformations that apply to n by m matrix A and n by p matrix B to transform them to triangular forms, respectively, but which corresponds to the QR factorization of B 01 A in the case whenever B is square and nonsingular. For example, if n m, n p, ....

S. Hammarling, The numerical solution of the general Gauss-Markov linear model, NAG Technical Report, TR2/85, 1985.

....that this method of solution is numerically stable and give normwise perturbation theory for the GLM problem. 3.1 Generalized QR method We describe a method for solving the GLM problem based on the generalized QR (GQR) factorization. The generalized QR factorization was introduced by Hammarling [7] and Paige [12] and further analyzed by Anderson, Bai and Dongarra [2] Theorem 3.1 (generalized QR factorization) Let A 2 IR n Thetam and B 2 IR n Thetap with m n m p. There exists orthogonal matrices Q 2 IR n Thetan and V 2 IR p Thetap such that Q T A = m m n Gamma m R 0 ....

S. J. Hammarling. The numerical solution of the general Gauss-Markov linear model. In Mathematics in Signal Processing, T. S. Durrani, J. B. Abbiss, and J. E. Hudson, editors, Oxford University Press, 1987, pages 451-456.

....with or without pivoting of two matrices A and B having the same number of rows, and whenever B is square and nonsingular, the factorization implicitly gives the orthogonal factorization with or without pivoting of B Gamma1 A. The GQR factorization was introduced early by Hammarling[6] and Paige[9] But from the general purpose software development point of view, we proposed the different factorization forms. In addition to the factorization forms and implementation details, we show the applications of GQR factorization in solving the linear equality constraint least square ....

....available from public linear algebra library LINPACK[7] Redesigned codes in block algorithm fashion that are better suited for today s high performance architectures can be found in LAPACK. The terminology generalized QR factorizations (GQR factorization) which has been introduced by Hammarling[6] and Paige[9] is to refer to orthogonal transformations that apply to n by m matrix A and n by p matrix B to transform them to triangular forms, respectively, but which corresponds to the QR factorization of B Gamma1 A in the case whenever B is square and nonsingular. For example, if n m, n ....

S. Hammarling, The numerical solution of the general Gauss-Markov linear model, NAG Technical Report, TR2/85, 1985.

....three versions of the null space method for solving the LSE problem, so called because it employs an orthogonal basis for the null space of the constraint matrix. We begin with a version based on the generalized QR factorization. The generalized QR factorization was introduced by Hammarling [12] and Paige [17] and further analyzed by Anderson, Bai and Dongarra [2] Theorem 2.1 (generalized QR factorization) Let A 2 R m Thetan and B 2 R p Thetan with m p n p. There are orthogonal matrices Q 2 R n Thetan and U 2 R m Thetam such that U T AQ = p n Gammap m Gamman p L 11 ....

Sven J. Hammarling. The numerical solution of the general Gauss--Markov linear model. In Mathematics in Signal Processing, T. S. Durrani, J. B. Abbiss, and J. E. Hudson, editors, Oxford University Press, 1987, pages 451--456.

....where B is square and nonsingular. A general numerically robust algorithm would not compute the inverse of B nor the product explicitly, but would transform A and B separately. Paige in [25] proposed to call such a combined decomposition of two matrices a generalized QR factorization, following [20]. We propose here to reserve the name generalized QRD for the complete set of generalizations of the QR decompositions, which will be developed in this paper. We will also propose a novel nomenclature in a similar way as we have done for the generalizations of the SVD in [6] Apparently, Stoer ....

S. HAMMARLING, The numerical solution of the general Gauss-Markov linear model, NAG Technical Report, TR2/85, Numerical Algorithms Group Limited, Oxford, 1985.

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S. Hammarling. The numerical solution of the general Gauss-Markov linear model. In T. S. Durrani et al., editor, Mathematics in Signal Processing. Clarendon Press, Oxford, 1986.

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