T. F. Chan, Rank Revealing QR Factorizations , Linear Algebra and Its Applications, 88/89 (1987), pp. 67--83.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....zero, however, there is an increasingly great gap between the next to last and last singular values, while the ratio of the corresponding R values remains near one. This example has inspired researchers to look for other pivoting strategies under the rubric of rank revealing QR decompositions [2, 3, 4, 5, 7, 8]. There are, however, certain limitations to any pivoted QR decomposition. For example, the first R value is the norm of the first column of A Pi R . We hope this number will approximate oe 1 , which, however, is the spectral norm of the entire matrix A. Thus r 11 will in general underestimate oe ....

T. F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67--82, 1987.

....identification [19] communication [1] In practice, where the problems are usually time varying, it is important to be able to track these subspaces. Therefore, in recent years various subspace tracking algorithms have been proposed. These algorithms are based on rank revealing decompositions [4], 23] the Lanczos algorithm [5] or the SVD updating algorithm [20] This work was done while with Rice University, Houston, Texas supported by the Alexander von Humbodt Foundation and Texas Advanced Technology Program The SVD updating algorithm incorporates a new data vector by a matrix ....

T.F. Chan. Rank Revealing QR Factorization. Linear Algebra and Its Applications, 89:67--82, 1987.

....O(n 3 ) flops for a matrix of order n even for a simple update such as adding a new row. To circumvent these drawbacks any decomposition that estimates the rank and the orthogonal spaces can be used in the place of the SVD. The Rank Revealing QR (RRQR) decomposition could be one possible choice [3]. RRQR algorithms can track the rank, and the computational complexity of this approach is O(n 2 ) However, it has been shown [4] that the quality of the approximation depends on the gap between the singular values, and approximations to the right singular subspaces are not directly available. ....

....[1] S.Qiao, Computing the ULLV Decomposition , CRL Report 278, Communications Research Laboratory, McMaster Uni. Hamilton, Canada, pp. 1 13, January, 1994) 2] G.W.Stewart, An Updating Algorithm for Subspace Tracking , IEEE Trans. on SP, Vol. 40, No. 6, pp. 1535 1541, June, 1992) [3] G.W.Stewart, Updating a Rank Revealing ULV Decomposition , SIAM Jour. on Matrix Analysis and Applications, Vol. 14, No. 2, pp. 494 499, April, 1993) Revision Peter S. K. Hansen, IMM, DTU, Denmark, April 10, 1997. Peter S. K. Hansen 18 Chapter 2. Toolbox Reference ullv dwa Purpose ....

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T. F. Chan. Rank Revealing QR Factorizations. Linear Algebra and Its Applications, 88:67--82, 1987.

....from zero, however, there is an increasingly great gap between the next to last and last singular values, while the ratio of the corresponding R values remains near one. This example has inspired researchers to look for other pivoting strategies under the rubric of rank revealing QR decompositions [2, 3, 4, 5, 7, 8]. There are, however, certain limitations to any pivoted QR decomposition. For example, the first R value is the norm of the first column of A Pi R . We hope this number will approximate oe 1 , which, however, is the spectral norm of the entire matrix A. Thus r 11 will in general underestimate oe ....

T. F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67--82, 1987.

....updating algorithm for the QR decomposition. The name QR decomposition is from Francis s QR algorithm [33] which uses the decomposition. Although pivoting for column size while computing the QR decomposition has long been regarded as a reliable way of determining rank (e.g. see [39, 62] Chan [13] was the first to give bounds for a rank revealing decomposition (the descriptive phrase rank revealing was coined by him) Unfortunately, the bounds were exponential in the defect p Gamma k in the rank. In fact, only recently have Hong and Pan [49] established the existence of a rank ....

T. F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67--82, 1987.

....state space system identification [13] In practice, where the problems are usually time varying, it is important to be able to track these subspaces. Therefore, in recent years various subspace tracking algorithms have been proposed. These algorithms are based on rank revealing decompositions [3, 17], the Lanczos algorithm [4] or the SVD updating algorithm [14] The SVD updating algorithm incorporates a new data vector by a matrix vector multiplication, a QRD updating step and a fraction of a sweep of Kogbetliantz s SVD algorithm [14, 6] These types of computations are well suited for ....

T.F. Chan. Rank Revealing QR Factorization. Linear Algebra and Its Applications, 89:67--82, 1987.

.... demanding and difficult to update for either dense [5] or sparse matrices [1,19] This can be a drawback for recursive procedures which require simple matrix updates (e.g. appending or deleting a row or column) Alternatively, rank revealing QR (RRQR) algorithms such as those by Foster [15] Chan [6], and modifications [4] can be used to obtain subspace information from matrices [7] 8] RRQR decompositions, however, yield subspaces whose accuracies depend on the gap in the singular values [13] in the sense that a large gap is required to produce good approximations to the singular ....

T. F. Chan. Rank Revealing QR Factorizations. Linear Algebra and Its Applications, 88:67--82, 1987.

....element in the factor R. It is necessary to divide the analysis because the condition estimator SPICE will fail in the presence of a zero diagonal element. We first consider the full structural rank case where R has no zero diagonal elements. In this case our analysis is almost identical to Chan [8], except that we require only approximations to the singular vectors. We only establish enough in the second case, with zero diagonal entries, to reduce it to the first case. In both cases the result of the analysis is a bound 22 D. J. PIERCE and J. G. LEWIS 1 3 7 9 R 1 ; S 1 x x x x Gamma ....

.... 4 6 7 9 R 4 ; S 4 x x x x I 5 6 8 9 R 5 ; S 5 x x x x P P P P P P P P P Pi 6 7 8 9 2 x x x x x x x x x x x x x x x F 6 x x x x x x x 6 6 7 8 9 2 x x x x x R 6 ; S 6 x x x x x x x x x T 6 x Figure 7: Modified Factor after Pivoting Column SPARSE RRQR 23 on kTk 2 similar to that in [8], where exact singular vectors were required. We assume that we have computed a triangular factor R for the first r columns of A. We denote this submatrix as A 1:r , so that we have A 1:r = QR: Further assume that the condition estimate has exceeded a tolerance of 1= signaling an ....

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T.F. Chan, Rank revealing QR factorizations, Linear Algebra and its Applications, 88/89, 1987, pp. 67--82.

....problem so we can safely compute the Cholesky factor. Problems in computing the Cholesky factorization arise when some of A s columns are only weakly independent of the others. This case can be dealt with through a rank revealing Cholesky factorization. As with rank revealing QR factorizations [5,2,3], there is a permutation matrix P such that P T (A T A)P = C T C; C = C 11 C 12 C 22 ; 27) where C 11 2 R r Thetar is wellconditioned, r is the numerical rank of A, and kC 22 k 2 is small. If C 22 is numerically negligible, then the last k Gamma r columns of AP can be ....

Tony F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67--82, 1987.

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T. F. Chan, Rank Revealing QR Factorizations , Linear Algebra and Its Applications, 88/89 (1987), pp. 67--83.

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Tony F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67--82, 1987.

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T. F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67-82, 1987.

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T. F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67-82, 1987.

....So although in practice the diagonal elements of R are rough approximations to the singular values, this need not always be the case. Indeed, the Kahan matrix provides a well known counterexample. Kah66] This is one consideration that motivates the rank revealing QR factorization (RRQR) [Fos86, Cha87]. If R is partitioned as A ; an RRQR algorithm tries to maximize the smallest singular value of R 11 and or minimize the largest singular value of R 22 [CI94] This essentially means making, respectively, kR 11 k as large as possible and kR 22 k as small as possible. From the interlacing ....

....out, of course. We need only determine the permutation which puts the correct column of R into the n th column. Implemented correctly, determining the permutation and getting the correct column into the n th position with the new R of course retriangulated requires O(n ) operations [Cha87]. The second item gives QLP quality information about n when it works. Once again recall that Mathias and Stewart [MS93] show that if 1; then (1 : n 1; n)k kR (1 : n 1; n)k: 63 Now we know that the norm of the n th column of R is just jr nn j, that is ....

T. F. Chan. \Rank Revealing QR Factorizations." Linear Algebra and Its Applications, 88/89:67-82, 1987.

.... and in the calculation of splines [18] Other applications arise in beam forming [8] spectral estimation [23] regularization [21, 29] and eigenproblems [3] Algorithms for the reliable computation of rank revealing factorizations have recently received considerable attention (see, for example [6, 7, 10, 11, 20, 26, 27]) However, the most common approach to computing such a RRQRF is the column pivoting procedure suggested by Businger and Golub [9] This QR factorization with All authors were partially supported by the Advanced Research Projects Agency, under contract DM28E04120 and P 95006. Quintana also ....

T. F. Chan, Rank revealing QR factorizations, Linear Algebra and Its Applications, 88/89 (1987), pp. 67--82.

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Tony F. Chan. Rank revealing QR factorizations. Linear Algebra and Its Applications, 88/89:67--82, 1987.

.... and the calculation of splines [18] Other applications arise in beam forming [8] spectral estimation [23] regularization [21, 29] and eigenproblems [3] Algorithms for the reliable computation of rank revealing factorizations have recently received considerable attention (see, for example [6, 7, 10, 11, 20, 26, 27]) However, the most common approach to computing such an RRQRF is the columnpivoting procedure suggested by Businger and Golub [9] This QR factorization with column pivoting (QRP) may fail to reveal the numerical rank correctly, but it is widely used because of its simplicity and practical ....

T. F. Chan, Rank revealing QR factorizations, Linear Algebra and Its Applications, 88/89 (1987), pp. 67--82.

.... and in the calculation of splines [18] Other applications arise in beam forming [8] spectral estimation [23] regularization [21,29] and eigenproblems [3] Algorithms for the reliable computation of rank revealing factorizations have recently received considerable attention (see, for example [6, 7, 10, 11, 20, 26, 27]) However, the most common approach to computing such a RRQRF is the column pivoting procedure suggested by Businger and Golub [9] This QR factorization with column pivoting (QRP) may fail to reveal the numerical rank correctly, but it is widely used due to its simplicity and practical ....

T. F. Chan, Rank revealing QR factorizations, Linear Algebra and Its Applications, 88/89 (1987), pp. 67--82.

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T. F. Chan, Rank revealing QR factorizations, Linear Algebra and Its Applications, 88/89 (1987), pp. 67--82.

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T. F. Chan. Rank Revealing QR Factorizations. Linear Algebra and Its Applications, 88:67--82, 1987.

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T. F. CHAN, Rank revealing QR factorizations, Linear Algebra and Its Applications, 88/89 (1987), pp. 67-82.

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T. F. Chan, Rank revealing QR factorizations, Linear Algebra and Its Applications, 88/89 (1987), pp. 67--82.

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