E. Jessup and D. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. Mathematics and Computer Science Division Report ANL/MCS-TM-102, Argonne National Laboratory, Argonne, IL, December 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... special cases (see [10] 3 Parallelization Issues Divide and conquer algorithms have been successfully implemented on shared memory multiprocessors for solving the symmetric tridiagonal eigenvalue problem and for the computation of the singular value decomposition of bidiagonal matrices [14] [24]. By contrast, the implementation of these algorithms on distributed memory machines poses difficulties. Several issues need to be addressed and several implementations are possible. The first issue is how to split the work among the processors. As shown in Figure 3.1, the recursive matrix ....

E. R. Jessup and D. C. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. SIAM J. Matrix Anal. Appl., 15(2):530-- 548, 1994.

....transformations the 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 ....

E. R. Jessup and D. C. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. SIAM J. Matrix Anal. Appl., 15 (2):530--548, 1994.

....throughout the remainder of this section assumes the matrix K is square, without any loss of generality. The singular values and singular vectors of M are related to the eigenvalues and eigenvectors of M T M = D zz T . For work on the rank one modification of the symmetric eigenproblem, see [12, 13, 23, 27, 38, 48, 66, 95]. The following result characterizes the singular values and singular vectors of M . It is assumed that the singular values of K are distinct and z i 6= 0 for i = 1; nq. Deflation may be used to reduce the original problem to a problem where these assumptions hold and is briefly discussed ....

....that the eigenvectors are guaranteed to be orthogonal if the secular equation is evaluated using precision which doubles the working precision. In addition, they present extra precision primitives which can be used on machines which do not provide the required extra precision. Jessup and Sorensen [66] extend this approach to the svd. The disadvantage is that this approach yields machine dependent software. Gu and Eisenstat [48, 49] propose an alternative approach which is machine independent, but requires calculating all of the singular values of M . For applications requiring the complete ....

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E. R. Jessup and D. C. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. SIAM J. Matrix Anal. Appl., 15:530-- 548, 1994.

....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 ....

E. R. Jessup and D. C. Sorensen, A parallel algorithm for computing the singular value decomposition of a matrix, Tech. Rep. ANL-MCS-TM-102, Argonne National Laboratory, July 1991.

....that the algorithm takes only O(n 2.3 ) flops on average and the cost can even be as low as O(n 2 ) for some special cases (see [9] 3. Parallelization issues and implementation details. Divide and conquer algorithms have been successfully implemented on shared memory multiprocessors [14] [23] but di#culties have been encountered on distributed memory machines [22] Several issues need to be addressed. A more detailed discussion of the issues discussed below can be found in [30] 3.1. Data distribution. The first issue, and perhaps the most critical step when writing a parallel ....

E. R. Jessup and D. C. Sorensen, A parallel algorithm for computing the singular value decomposition of a matrix, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 530--548.

....can also evaluate singular values to a high relative accuracy. The study of parallel features of this algorithm is to be reported in [9] As another approach for the evaluation of singular values of B with absolute accuracy, various versions of the divide and conquer 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 6 6 6 4 ff 1 fi 1 fi 1 ff 2 fi 2 0 . ....

E. R. Jessup and D. C. Sorensen, A parallel algorithm for computing the singular value decomposition of a matrix: a revision of Argonne National Labortory Tech. Report ANL/MCS-TM-102, Technical Report CU-CS-623-92, University of Colorado at Boulder, Department of Computer Science, Oct. 1992.

....procedures described in [8, 9] and a simple permutation. The scheme for finding the SVD of M 1 (see (1.6) appears in [8, 9] The techniques for both problems are very similar. The following lemma characterizes the singular values and singular vectors of M . Lemma 2. 1 (Jessup and Sorensen [12]) Let W Omega 0 Q T be the SVD of M with W = w 1 ; wn ; wn 1 ) Omega = diag( 1 ; n ) and Q = q 1 ; q n ) where 0 1 : n . Then M T M = D 2 zz T = Q Omega 2 Q T : is the eigendecomposition of M T M . The singular values f ....

....u t 1 n X i=1 (d i z i ) 2 Gamma d 2 i Gamma 2 j Delta 2 and q j = z 1 d 2 1 Gamma 2 j ; z n d 2 n Gamma 2 j T ,v u u t n X i=1 z 2 i Gamma d 2 i Gamma 2 j Delta 2 (i.e. replace j by j in equations (2.3) and (2. 5) as in [1, 12]) For even if j is close to j , the approximate ratios z i = d 2 i Gamma 2 j ) and d i z i = d 2 i Gamma 2 j ) can still be very different from the exact ratios z i = d 2 i Gamma 2 j ) and d i z i = d 2 i Gamma 2 j ) resulting in singular vectors very different ....

E. R. Jessup and D. C. Sorensen, A parallel algorithm for computing the singular value decomposition of a matrix. Revision of Tech. Report ANL-MCS-TM-102, Argonne National Laboratory, 1991.

....in O(log 2 k) steps whenever possible, and collapsing a chain of k nodes into a single node via parallel prefix in O(log 2 k) steps whenever possible. If we could understand the numerical stability of parallel prefix, we could probably analyze this more general scheme as well. Divide and conquer [7, 10, 18, 12] has been widely used for the tridiagonal eigenproblem and bidiagonal singular value decomposition. This can be straightforwardly extended to the acyclic case. In terms of the tree, just remove the root by a rank two tearing , solve the independent child subtrees recursively and in parallel, and ....

E. Jessup and D Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. Mathematics and Computer Science Division Report ANL/MCSTM -102, Argonne National Laboratory, Argonne, IL, December 1987.

.... special cases (see [10] 3 Parallelization Issues Divide and conquer algorithms have been successfully implemented on shared memory multiprocessors for solving the symmetric tridiagonal eigenvalue problem and for the computation of the singular value decomposition of bidiagonal matrices [14] [24]. By contrast, the implementation of these algorithms on distributed memory machines poses difficulties. Several issues need to be addressed and several implementations are possible. The first issue is how to split the work among the processors. As shown in Figure 3.1, the recursive matrix ....

E. R. Jessup and D. C. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. SIAM J. Matrix Anal. Appl., 15(2):530-- 548, 1994.

....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 ....

Jessup, E.R., Sorensen, D.C. (1992): A parallel algorithm for computing the singular value decomposition of a matrix: a revision of Argonne National Labortory Tech. Report ANL/MCS-TM-102. Technical Report CU-CS-623-92, University of Colorado at Boulder, Department of Computer Science

....where is a small multiple of ffl specified in [17] Any matrix of the form (2.7) can be reduced to one that satisfies these conditions by the deflation procedure described in [17] The following lemma characterizes the singular values and singular vectors of M . Lemma 1 (Jessup and Sorensen [18]) Let S SigmaG T be the SVD of M with S = s 1 ; s n ) Sigma = diag(oe 1 ; oe n ) and G = g 1 ; g n ) where 0 oe 1 : oe n : Then the singular values foe i g n i=1 satisfy the interlacing property 0 = d 1 oe 1 d 2 : d n oe n d n jjzjj ....

E. Jessup and D. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. Mathematics and Computer Science Division Report ANL/MCS-TM-102, Argonne National Laboratory, Argonne, IL, December 1987.

.... special cases (see [10] 3 Parallelization Issues Divide and conquer algorithms have been successfully implemented on shared memory multiprocessors for solving the symmetric tridiagonal eigenvalue problem and for the computation of the singular value decomposition of bidiagonal matrices [14] [24]. By contrast, the implementation of these algorithms on distributed memory machines poses difficulties. Several issues need to be addressed and several implementations are possible. The first issue is how to split the work among the processors. As shown in Figure 3.1, the recursive matrix ....

E. R. Jessup and D. C. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. SIAM J. Matrix Anal. Appl., 15(2):530--548, 1994.

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E. Jessup and D. Sorensen. A parallel algorithm for computing the singular value decomposition of a matrix. Mathematics and Computer Science Division Report ANL/MCS-TM-102, Argonne National Laboratory, Argonne, IL, December 1987.

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