M. Gu and S. Eisenstat. An efficient algorithm for computing a strong rank--revealing QR factorization. SIAM J. Sci. Comput., 17(4):848 -- 869, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rank deficiency. Both methods need a condition estimator, to produce approximate null vectors of certain triangular matrices. Chandrasekaran and Ipsen [26] give a systematic treatment of algorithms for computing rank revealing QR factorizations. In an important recent development, Gu and Eisenstat [75] derive an algorithm that is guaranteed to compute a strong form of rank revealing QR factorization whose properties include that it stably provides an approximation to the null space; the algorithm has the same complexity as the column pivoting strategy, though is up to 50 more expensive in ....

Ming Gu and Stanley C. Eisenstat. Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput., 17(4): 848--869, 1996.

....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 ....

M. Gu and S. C. Eisenstat. An efficient algorithm for computing a strong rankrevealing QR-factorization. SIAM Journal on Scientific Computing, 17:848--869, 1996.

.... 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 ....

M. Gu and S. Eisenstat, An efficient algorithm for computing a strong rank-revealing factorization, Tech. Report YALEU/DCS/RR-967, Yale University, Department of Computer Science, 1994.

....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 ....

M. Gu and S. C. Eisenstat. An efficient algorithm for computing a strong rankrevealing QR-factorization. SIAM Journal on Scientific Computing, 17:848--869, 1996.

.... 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 ....

M. Gu and S. Eisenstat, An efficient algorithm for computing a strong rank-revealing factorization, Tech. Report YALEU/DCS/RR-967, Yale University, Department of Computer Science, 1994.

.... 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 ....

M. Gu and S. Eisenstat, An efficient algorithm for computing a strong rank-revealing factorization, Tech. Report YALEU/DCS/RR-967, Yale University, Department of Computer Science, 1994.

....in section 4.1 below) The factor (R 0 ) in the error bound depends on how well the pivoting during the QR decomposition of XD reveals the rank of XD. The bound O(2 n ) comes from the standard column pivoting algorithm [15] and choosing D 0 ii = R ii , but better alternatives are available [30, 1, 5, 6, 16, 20, 28]. For example, M. Gu has a pivoting scheme that reduces O(2 n ) to O(n 1 (1=4)log 2 n ) analogous to the pivot growth bound for GECP. See also [27] We can eliminate the factor (R 0 ) by using the following more expensive algorithm: Algorithm 2. Compute a high accuracy SVD of G = XDY T . ....

M. Gu and S. Eisenstat. An efficient algorithm for computing a strong rank--revealing QR factorization. SIAM J. Sci. Comput., 17(4):848 -- 869, 1996.

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M. Gu and S. Eisenstat. An efficient algorithm for computing a strong rank--revealing QR factorization. SIAM J. Sci. Comput., 17(4):848 -- 869, 1996.

No context found.

M. Gu and S. Eisenstat. An efficient algorithm for computing a strong rank--revealing QR factorization. SIAM J. Sci. Comput., 17(4):848 -- 869, 1996.

....related issue is to study the impact of the difference choices of i j s on the accuracy of the numerical solution. One way to reduce the upper bound on ( b L) is to perform a rank revealing LU (RRLU) factorization on C instead of using GEPP GECP (see, for example, Chan [10] Gu and Eisenstat [29] and Hwang, Lin, and Yang [33] It would be interesting to see if fast RRLU factorization algorithms can be developed to guarantee that ( b L) is always modest. Acknowledgements. The author is grateful to Profs. A. Bjorck and S. Chandrasekaran for helpful discussions, Prof. Golub for pointing ....

M. Gu and S. C. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848--869.

....several digits of relative accuracy, though the largest eigenvalue was 10 70 times larger [26] X and Y are perfectly conditioned) but there are many others, depending on how large a condition number for X and Y one will tolerate. For example the factorizations provided by rank revealing QR [51, 4, 13, 14, 30, 33, 42] and Gaussian elimination with complete pivoting (GECP) or other pivoting [41, 44] GECP (usually) provides an RRD since the unit lower and upper triangular factors L and U in G = P r LDUP c have offdiagonal entries bounded by 1 in absolute value (here P r and P c are permutations 2 ) We then ....

....(Y ) Remark. The factor (R 0 ) in the error bound depends on how well the pivoting during the QR decomposition of XD reveals the rank of XD. The bound O(2 n ) comes from the standard column pivoting algorithm [28] and choosing D 0 ii = R ii , but better alternatives are available [51, 4, 13, 14, 30, 33, 42]. For example, Gu has a pivoting scheme that reduces O(2 n ) to O(n 1 (1=4)log 2 n ) analogous to the pivot growth bound for GECP. See also [41] Proof: We proceed through the algorithm line by line, showing that the backward error introduced by every step but (3) is of the form (I E)G(I ....

M. Gu and S. Eisenstat. An efficient algorithm for computing a strong rank--revealing QR factorization. SIAM J. Sci. Comput., 17(4):848 -- 869, 1996.

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M. Gu and S. Eisenstat, An efficient algorithm for computing a strong rank-revealing factorization, Tech. Report YALEU/DCS/RR-967, Yale University, Department of Computer Science, 1994.

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M. Gu AND S. ESENSTAT, An efficient algorithm for computing a strong rank-revealing factorization, Tech. Report YALEU/DCS/RR-967, Yale University, Department of Computer Science, 1994.

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M. Gu and S. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848--69.

No context found.

M. Gu and S. Eisenstat, An efficient algorithm for computing a strong rank-revealing factorization, Tech. Report YALEU/DCS/RR-967, Yale University, Department of Computer Science, 1994.

No context found.

M. Gu and S. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848--69.

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