M. Gu and S. C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Mat. Anal. Appl., 16(1):79--92, January 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the pseudospectrum domain in order to reduce the number of points where oe min needs to be computed [3,7,8,14] Matrix based: Methods that attempt to reduce C oe min at each point. These include dense and sparse matrix techniques for the sequential or parallel evaluation of singular triplets [6,9,16,17,20,22,23,25]. See also [26] for a nice collection of on line resources related to this topic. In this paper we are primarily in domain based techniques for the following version of the pseudospectrum problem: Problem PSe: Given A 2 C and an ffl 0, compute the corresponding ffl pseudospectrum curve ....

M. Gu and S.C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Matrix Anal. Appl., 16(1):79--92, 1995.

....aim is to orthogonalize the columns of the matrix, after which the SVD is readily obtained. Relevant references include Hari and Veseli c [78] and de Rijk [33] Divide and conquer algorithms for finding the SVD of a bidiagonal matrix are developed by Jessup and Sorensen [93] and Gu and Eisenstat [74]; they are related to the divide and conquer algorithms for the symmetric eigenproblem. A new algorithm for computing the SVD of a dense matrix that first reduces to bidiagonal form and then applies divide and conquer is described by Gu, Demmel and Dhillon [72] Their method incorporates ....

Ming Gu and Stanley C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Matrix Anal. Appl., 16(1):79--92, 1995.

....couplings with the LAPACK [14] routines DSTEGR, DBDSQR and DBDSDC on a SUN 16 Ultra Sparc 60. The routine DSTEGR performs an eigendecomposition of a symmetric tridiagonal matrix using the RRR algorithm. DBDSQR and DBDSDC solve the bSVD using the QR method [1] and a divide and conquer approach [13] respectively. As test matrices we choose T = B T B with prescribed eigenvalues or prescribed entries (ffl = machine precision) 1. Geometric distribution: i = ffl (n Gammai) n Gamma1) i = 1 : n 2. Arithmetic distribution i = ffl (1 Gamma ffl) i Gamma 1) n Gamma 1) i = 1 : n 3. ....

M. Gu and S. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Matrix Anal. Appl., 16:79--92, 1995.

....to find the singular values of a bidiagonal matrix is an important part of the overall process of finding the singular values of a real matrix A. Parallel processing can be used to reduce the execution time. Different parallel methods have been used to solve the SVD problem of bidiagonal matrices [2, 3, 7, 9], and space limitations do not allow us to review them here. Many algorithms use an approach that first finds the eigenvalues of a symmetric tridiagonal (ST) matrix that is related to the bidiagonal matrix B. Li et al. [10] recently proposed a new SVD algorithm that combines two eigenvalue ....

M. Gu and S. C. Eisenstat, A divide-and-conquer algorithm for the bidiagonal SVD, Tech. Rep. YALEU/DCS/RR-933, Yale University, December 1992.

....for the evaluation of Numerische Mathematik Electronic Edition page numbers may differ from the printed version page 285 of Numer. Math. 69: 283 301 (1995) T.Y. Li et al. singular values of B with absolute accuracy, various versions of the divide andconquer algorithm were reported in [2, 3, 13, 14, 15]. 2. The split merge algorithm We briefly summarize the split merge algorithm, developed by Li and Zeng [19] for the symmetric tridiagonal eigenvalue problem. Let S be a symmetric tridiagonal matrix S = 2 6 6 6 6 6 6 6 6 6 4 ff 1 fi 1 fi 1 ff 2 fi 2 0 . 0 fi ....

Gu, M., Eisenstat, S.C. (1992): A divide-and-conquer algorithm for the bidiagonal SVD. Research Report YALEU/DCS/RR-933

No context found.

M. Gu and S. C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Mat. Anal. Appl., 16(1):79--92, January 1995. 44

No context found.

M. Gu and S. C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Mat. Anal. Appl., 16(1):79--92, January 1995.

....but U 0 1 , W w) and Q can be very different from U 0 1 , W w) and Q, respectively. Thus one is usually content with backward stable algorithms for computing the eigendecompositions of A 0 and M . We compute a numerical SVD of M of the form (1. 5) by using the techniques in [8, 9, 10] (see Section 2) We compute the right singular vector matrix as V Q. If the left singular vector matrix is updated, we compute it according to (1.4) with (W w) replaced by ( W w) It takes O(n 2 ) floating point operations to compute a numerical SVD of M (see Section 2) It takes O(mn) ....

.... U 0 0 1 W ; V 0 1 = V 1 v)Q and (v V 0 2 ) V 2 H ; with v 2 R n being the first column of V 2 H. Since M 1 is not related to U we can update the singular values and right singular vectors of A 0 without it. We compute a numerical SVD of M 1 (cf. 1. 5) by using the techniques in [8, 9, 10] (see Section 2) We stably compute the right and left singular vector matrices of A 0 by using the fast multipole method in O(mn log 2 2 ffl) and O(m 2 log 2 2 ffl) floating point operations, respectively. Similar to the previous case, the singular values of A 0 and M 1 are always ....

[Article contains additional citation context not shown here]

M. Gu and S. C. Eisenstat, A divide-and-conquer algorithm for the bidiagonal SVD, Research Report YALEU/DCS/RR-933, Department of Computer Science, Yale University, December 1992. To appear in SIMAX.

....this computation requires O(mn 2 ) floating point operations, much cheaper than O(m 3 ) for m AE n. Equation (2.1) provides an efficient way to compute oe min ( M j C] and hence E (x M ) as well. In fact, equation (2. 1) is similar to the secular equations solved in Gu and Eisenstat [10] and Li [13] and their methods can be easily modified to compute oe min ( M j C] Both computing the SVD of M and solving equation (2.1) can be done reliably. Recently, Karlson and Wald en [12] discussed a method for estimating E (x M ) that requires O(mn) floating point operations. While their ....

M. Gu and S. C. Eisenstat, A divide-and-conquer algorithm for the bidiagonal SVD, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 79--92.

....of G, independent of any additional errors introduced by the algorithms. To explain the higher accuracy to which we aspire to compute the SVD, we will contrast it with the accuracy provided by conventional SVD algorithms, such as QR iteration, bisection and inverse iteration, or divide and conquer [12, 28, 31]. Their model of uncertainty asserts that ffi G is bounded in norm, and that kffiGk=kGk 1 (k Delta k is the two norm) This model of uncertainty is appropriate because roundoff error in these algorithms means that ffiG typically satisfies kffiG 0 k=kGk = Omega Gamma ) i.e. at least order ....

M. Gu and S. C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Mat. Anal. Appl., 16(1):79--92, January 1995.

No context found.

M. Gu and S. C. Eisenstat. A divide-and-conquer algorithm for the bidiagonal SVD. SIAM J. Mat. Anal. Appl., 16(1):79--92, January 1995.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC