Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. SIAM Journal on Scientific and Statistical Computing, 12(6):1332--1350, November 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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 subspaces. In LSI applications ....

C. Bischof and P. Hansen. Structure-preserving and rank-revealing qr factorizations. SIAM Journal on Scientific and Statistical Computing, 12(1):1332--1350, 1991.

....one should not make too much of this, since empirical studies [11, 9] have shown approximate condition estimators to be quite good in practice. The condition calculator is most effective for dense matrices many more rows than columns. For sparse matrices the technique of Bischof and Hansen [2], which combines restricted forward selection with a backward rank revealing pass [3] Other combinations will be suggested by the application at hand. ....

C. H. Bischof and P. C. Hansen (1989). "Structure-Preserving and RankRevealing QR-Factorizations." Preprint MCS-P100-0989, Mathematics and Computer Science Division, Argonne National Laboratory.

....of these methods is quite good; however, the existence of parallel implementations is unclear at this point. It might be possible to derive a deflated Schur type method as well. Acknowledgement. The authors are grateful to P.C. Hansen for providing us with an implementation of the RRQR, based on [2, 3, 13]. We are also grateful for the valuable comments of the reviewers. ....

C. H. Bischof and P. C. Hansen, "Structure preserving and rank-revealing QR-factorizations," in SISSC, vol. 12, pp. 1332--1350, 1991.

....computationally inefficient. These issues are discussed in more detail in [12] The situation is even more complicated if one wishes to exploit parallelism. Considerable progress has been made in the implementation of linear algebra kernels, such as orthogonal factorizations, on parallel machines [2, 3, 21, 23]. With respect to the computation of derivative information, approaches to computing finitedifference approximations in parallel using graph coloring approaches have been successful [7, 19, 22] but again accuracy may be lost. We, in turn, suggest the use of automatic differentiation to compute ....

Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. Technical Report MCS-P100-0989, Argonne National Laboratory, Mathematics and Computer Sciences Division, September 1989.

.... 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 ....

C. H. Bischof and P. C. Hansen, Structure-preserving and rank-revealing QR factorizations, SIAM Journal on Scientific and Statistical Computing, 12 (1991), pp. 1332--1350.

....the regularity assumption, the active space of Q is R(Y ) A nonregular basis kernel transformation can easily be transformed into a regular one, as follows. Suppose Y is rank deficient. Let Y P = Y R, with R = R 11 R 12 0 0 , be a rank revealing QR decomposition of Y (see, for example, [2,3]) that is, R 11 is nonsingular and rank(R 11 ) rank(Y ) Then, Q(Y; S) Q( Y ; S) with S = RP T SPR T . Thus, we can assume without loss of generality that Y is of full rank. Now suppose S is singular. We know from the proof of Lemma 2 that S = U SU T for some U and S of ....

Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. SIAM Journal on Scientific and Statistical Computing, 12(6):1332--1350, November 1991.

.... 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 ....

C. H. Bischof and P. C. Hansen, Structure-preserving and rank-revealing QR factorizations, SIAM Journal on Scientific and Statistical Computing, 12 (1991), pp. 1332--1350.

....Incremental condition estimation [3, 5] is an O(n) scheme to arrive at an estimate for the condition number of R when R = R w fl ; that is, R is R augmented by a column. This estimator is well suited for restricting column exchanges in rank revealing orthogonal factorizations [1, 2, 4, 6]. Adaptive condition estimation schemes address the issue of rank one updates of a triangular matrix R. Pierce and Plemmons [17, 16] suggest an O(n) scheme and Ferng, Golub, and Plemmons [11] an O(n 2 ) scheme for the situation where R T R = R T R uu T : These schemes are designed for ....

Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. Preprint MCS-P100-0989, Argonne National Laboratory, Mathematics and Computer Science Division, September 1989.

.... 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 ....

C. H. Bischof and P. C. Hansen, Structure-preserving and rankrevealing QR factorizations, SIAM Journal on Scientific and Statistical Computing, 12 (1991), pp. 1332--1350.

....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 ....

Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. SIAM Journal on Scientific and Statistical Computing, 12(6):1332--1350, November 1991.

.... Incremental condition estimation [5, 7] is an O(n) scheme to arrive at an estimate for the condition number of R when R = R w fl ; that is, R is R augmented by a column. This estimator is well suited for restricting column exchanges in rank revealing orthogonal factorizations [3, 4, 6, 8]. Adaptive condition estimation schemes address the issue of rank one updates of a triangular matrix R. Pierce and Plemmons [25, 24] suggest an O(n) scheme and Ferng, Golub, and Plemmons [14] an O(n 2 ) scheme for the situation where R T R = R T R uu T : These schemes are designed ....

Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. Preprint MCS-P100-0989, Argonne National Laboratory, Mathematics and Computer Science Division, September 1989.

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Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. SIAM Journal on Scientific and Statistical Computing, 12(6):1332--1350, November 1991.

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C. H. Bischof and P. C. Hansen, Structure-preserving and rankrevealing QR factorizations, SIAM Journal on Scientific and Statistical Computing, 12 (1991), pp. 1332--1350.

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C.H. BISCHOF AND P.C. HANSEN, Structure-preserving and rank-revealing QR factorizations. SIAM J. Sci. Stat. Comput. 12 (1991), pp. 1332-1350.

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Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR factorizations. SIAM Journal on Scientific and Statistical Computing, 12(6):1332--1350, November 1991.

No context found.

C. H. Bischof and P. C. Hansen, Structure-preserving and rankrevealing QR factorizations, SIAM Journal on Scientific and Statistical Computing, 12 (1991), pp. 1332--1350.

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