C. Paige. Some aspects of generalized QR factorization. In M. Cox and S. Hammarling, editors, Reliable Numerical Computations. Clarendon Press, Oxford, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 lower triangular. ....

C. C. Paige. Some aspects of generalized QR factorizations. In Reliable Numerical Computation, M. G. Cox and S. J. Hammarling, editors, Oxford University Press, 1990, pages 73--91.

....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 0 ; 4.14) ....

C. C. Paige. Some aspects of generalized QR factorizations. In Reliable Numerical Computation, M. G. Cox and S. J. Hammarling, editors, Oxford University Press, 1990, pages 73--91.

....reveal an interesting and potentially useful feature of A and B. Sensitivity in rank decisions is fundamental to any generalized URV or generalized QR algorithm that starts with an estimate of the range space of A and proceeds with a rank decision for P # A B, including the methods described in [9, 1]. The algorithms in [5, 6] are somewhat di#erent in that they make rank decisions on a matrix with singular values equal to the generalized singular values of A and B. Since generalized singular values can be sensitive to perturbations [8] the rank decisions in these methods can also be di#cult. ....

....costly than computing URV decompositions for A and B separately. Depending on the ranks involved, it is often not much more costly than updating the URV decomposition of a single matrix with 4i columns. The generalized URV decomposition is similar in spirit to the generalized QR factorization of [9]. However, a generalized QR factorization does not lend itself to updating. In terms of computational complexity, the use of the URV updating method is justifiable only when updates are needed. The method is not competitive for finding the subspaces associated with the generalized SVD of a single ....

C. C. Paige, Some aspects of generalized QR factorizations, in Reliable Numerical Computation, M. G. Cox and S. J. Hamarling, eds., Clarendon Press, Oxford, 1990, pp. 71--91. Cited in A. Bjorck, available via anonymous ftp from math.liu.se in pub/references.

....structure. To get (2) the r p (n r a r b r p ) matrix B (3) 34 can be compressed into r p nonzero columns using a further transformation Q (4) Clearly the singular values of the product A 22 B T 32 are the nonzero singular values of A B T . Related decompositions may be found in [3, 1]. The above discussion shows that r p can be found from the SVD of B (1) 1 . It can also be found directly from the singular values of A B T . However, without access to the unperturbed A and B neither of these approaches is entirely satisfactory. For the latter this can be seen ....

C. C. Paige. Some aspects of generalized QR factorizations. In M. G. Cox and S. J. Hammarling, editors, Reliable Numerical Computation, pages 71-91, Oxford, 1990. Clarendon Press.

....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, then the GQR ....

C. Paige, Some aspects of generalized QR factorization.

....rank revealing quotient decomposition of the form (3) in x2. While we do not develop an updating algorithm, the decomposition might be considered a quotient generalization of a URV decomposition. It can also be viewed as a complete orthogonal version of a generalized QR factorization as in [9]. Related generalizations have been proposed in [3, 5, 6, 11] The algorithm described in [1] and implemented in LAPACK as a preprocessing step for a full QSVD is essentially the rst part of a quotient generalization of a complete orthogonal decomposition. While less contrived examples are not as ....

C. C. Paige. Some aspects of generalized QR factorizations. In M. G. Cox and S. J. Hammarling, editors, Reliable Numerical Computation, pages 71-91, Oxford, 1990. Clarendon Press.

....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 # ; Q T BV ....

....are much more difficult to interpret than (3.22) and (3.23) respectively because of the complicated expressions. In the next section we compare the bounds on the relative error in x (3.19) and (3.22) and the bounds on the relative error in y (3.21) and (3.23) on some numerical examples. Paige [12] derives a perturbation bound for x also, but the results are derived differently and give bounds in terms of singular values of A and S 22 . Let oe(C) denote the smallest nonzero singular value of C and ffiC i a properly partitioned pertubation of C, then the error bound found in [12] is ....

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C. C. Paige. Some aspects of generalized QR factorizations. In Reliable Numerical Computation, M. G. Cox and S. J. Hammarling, editors, Oxford Unversity Press, 1990, pages 73-91.

.... pivoting to reveal rank, although more sophisticated rank revealing schemes exist [19, 32, 35, 52] Recall that for our purposes, we only need the unitary factor Q and the rank of C 01 D (or D H C 0H ) It turns out that by using the generalized QR (GQR) decomposition technique developed in [45, 3], we can get the desired information without computing C 01 or C 0H . In fact, in order to compute the QR decomposition with pivoting of C 01 D, we first compute the QR decomposition with pivoting of the matrix D: D = Q 1 R 1 5; 2:3) and then we compute the RQ factorization of the matrix Q ....

C. Paige. Some aspects of generalized QR factorization. In M. Cox and S. Hammarling, editors, Reliable Numerical Computations. Clarendon Press, Oxford, 1990.

....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 problem, ....

....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 p, then the ....

C. Paige, Some aspects of generalized QR factorization.

....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 0 n Gammap L ....

C. C. Paige. Some aspects of generalized QR factorizations. In Reliable Numerical Computation, M. G. Cox and S. J. Hammarling, editors, Oxford University Press, 1990, pages 73--91.

....B T . 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 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 ....

C. C. Paige. Some aspects of generalized QR factorizations. In Reliable Numerical Computation, M. G. Cox and S. J. Hammarling, editors, Oxford University Press, 1990, pages 73--91.

....inverse (A p B p ) Gamma1 and subsequent products. This will yield the ultimate inverse free algorithm. Recall that for our purposes, we only need the unitary factor Q and the rank of (A p B p ) Gamma1 A p . It turns out that by using the generalized QR decomposition technique developed in [45, 2], we can get the desired information without computing (A p B p ) Gamma1 . In fact, in order to compute the QR decomposition with pivoting of (A p B p ) Gamma1 A p , we first compute the QR decomposition with pivoting of the matrix A p : A p = Q 1 R 1 Pi; 2.7) and then we compute the RQ ....

C. Paige. Some aspects of generalized QR factorization. In M. Cox and S. Hammarling, editors, Reliable Numerical Computations. Clarendon Press, Oxford, 1990.

....simultaneously. The URV decomposition, 4] is an easily updated decomposition which, in some applications, may be used to replace the SVD. The fact that an intersection of the range spaces of two matrices is required suggests that a generalization of the URV decomposition along the lines of [3] might be helpful. Such a decomposition was introduced in [5] along with an O(n 2 ) updating algorithm. This paper will give a brief description of the decomposition and show how it can be used as part of an on line identification algorithm. Given a sequence of m Theta 1 input vectors, u(k) ....

....T 2 . In fact, it can be shown that if the E and F blocks are zero, then the first columns, U 1 , of U corresponding to the number of columns in R 14 are a basis for the intersection of the range space of W j T T 1 and W j T T 2 . Details concerning decompositions of this type can be found in [3]. If we partition V 2 in a manner which matches the decomposition V 2 = h V 23 V 24 V 25 i and assume that the E and F blocks are zero then U 1 R 14 = W j T T 2 V 24 and the full rank property of R 14 imply that W j T T 2 V 24 is also a basis for the intersection. This fact makes it ....

C. C. Paige. Some Aspects of Generalized QR Factorizations. In : M. G. Cox and S. Hammarling (Eds.), Reliable Numerical Computation, Oxford Univ. Press, pp. 73-91, 1990.

....GSVDs was further analysed in [10] The dimensions of the blocks that occur in any GSVD can be expressed as ranks of the matrices involved and certain products and concatenations of these. We will present a summary of the results below. As for generalizations of the QRD, it is mainly Paige in [25] who pointed out the importance of generalized QRDs for two matrices as a basic conceptual and mathematical tool. The motivation is that in some applications, one needs the QRD of a product of two matrices AB where A 2 m Thetan and B 2 n Thetap . For general matrices A and B such a ....

....motivation is that in some applications, one needs the QRD of a product of two matrices AB where A 2 m Thetan and B 2 n Thetap . For general matrices A and B such a computation avoids forming the product explicitly and transforms A and B separately to obtain the desired results. Paige [25] refers to such a factorization as a product QR factorization. Similarly, in some applications one needs the QR factorization of AB Gamma1 where B is square and nonsingular. A general numerically robust algorithm would not compute the inverse of B nor the product explicitly, but would ....

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C. C. PAIGE, Some aspects of generalized QR factorizations, Reliable Numerical Computation, 1990, pp. 73-91, M. Cox and S. Hammarling (eds.), Oxford University Press.

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C. Paige. Some aspects of generalized QR factorization. In M. Cox and S. Hammarling, editors, Reliable Numerical Computations. Clarendon Press, Oxford, 1990.

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Paige, C. C., Some Aspects of Generalized QR Factorizations, in Reliable Numerical Computation, Eds. Cox and Hammarling, pp. 73-91, Oxford Univ. Press, Oxford, 1990.

No context found.

C. Paige. Some aspects of generalized QR factorization. In M. Cox and S. Hammarling, editors, Reliable Numerical Computations. Clarendon Press, Oxford, 1990.

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