Ming Gu and Stanley Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmmetric eigenproblem. Technical Report YALEU/DCS/RR-916, Yale University, Department of Computer Science, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on a sequential computer. The three library routines are DSTEQR, which implements the QR method, DSTEBZ DSTEIN which implements bisection and inverse iteration respectively and finally DSTEDC an implementation of Cuppen s divideand conquer algorithm with modifications proposed by Gu and Eisenstat [45]. The library routines are from LAPACK version 2.0 [3] All codes except EAR, which is in C, are written in FORTRAN and make use of the BLAS routines. Compilation was done with optimization enabled. The tests were run on a single processor Sun SPARCstation 10, using IEEE double precision ....

....that a direct comparison of these two programs is interesting. Their Cuppen type algorithm uses the rank one extension to formulate the update problem. In contrast to the LAPACK algorithm, for which no parallel implementation currently exists, it does not use the technique from Gu and Eisenstat [45] but uses simulated extended precision in order to maintain the orthogonality among eigenvectors. The implementation of simulated extended precision relies on the IEEE floating point standard. This eigensolver EAR is written in C such that the comparison was made with two A 13 . A 02 ....

M. Gu and C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 15:1266--1276, 1994.

....second issue is to maintain orthogonality between eigenvectors in the presence of close eigenvalues. There are two approaches: the extra precision approach of Sorensen and Tang [30] used by Gates and Arbenz [16] in their implementation, and the Lowner Theorem approach proposed by Gu and Eisenstat [19] and adopted for LAPACK [1] 29] There are trade offs that we shall discuss in Section 4.2 between these two approaches. A ScaLAPACK prototype code is available at http: www.netlib.org scalapack prototype The third issue is the back transformation process, which is of great importance for ....

....However, this approach is hard to implement portably across all the usual architectures. There are many machine dependent tricks to make the implementation of extended precision go faster, but on some machines, such as Crays, these tricks do not help and performance suffers. The Gu and Eisenstat [19] approach based on the Lowner Theorem can easily be implemented portably on IEEE machines and Crays using only working precision arithmetic throughout, with a trivial bit of extra work in one place to compensate for the lack of a guard digit in Cray add subtract. On the other hand, the Lowner ....

[Article contains additional citation context not shown here]

Ming Gu and Stanley C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 15(4):1266--1276, 1994.

....and Sorensen [23] building on work of Golub [68] Until recently, it was thought that extended precision arithmetic was needed in the solution of the secular equation to guarantee that sufficiently orthogonal eigenvectors are produced when there are close eigenvalues. However, Gu and Eisenstat [73] have found a new approach that does not require extended precision. The divide and conquer algorithm has natural parallelism. Even on serial computers it can be many times faster than the QR algorithm, though it needs more workspace, hence LAPACK includes the divide and conquer algorithm. A ....

Ming Gu and Stanley C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 15(4):1266--1276, 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 ....

.... oe 2 i T r P nq j=1 z 2 j (oe 2 j Gamma oe 2 i ) 2 ; i = 1; nq: 4.13) For a proof of Lemma 9, see Section 4.3.1. It is well documented that the orthogonality of the singular vectors, resulting from (4.11) 4.13) after solving the secular equation (4. 10) is an issue [12, 13, 27, 105 48, 49, 95]. Sorensen and Tang [95] in the context of the symmetric eigenproblem, show 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 ....

[Article contains additional citation context not shown here]

M. Gu and S. C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 15:1266--1276, 1994.

....arithmetic by techniques similar to the algorithms in appendix A. If such careful computation cannot be per92 formed, perhaps because the properties upon which it depends are not possessed by the computer we wish to use, then we must resort to a clever approach suggested by Gu and Eisenstat [25]. They avoid the use of multiple precision computation by evaluating the relevant function to working precision, yielding a possibly inaccurate result for the given data, then perturbing the original problem to a new problem whose solution is precisely the computed solution. Gu and Eisenstat s ....

Gu, M., and S. Eisenstat, A Stable and Efficient Algorithm for the Rank-one Modification of the Symmetric Eigenproblem, Yale University Research Report YALEU/DCS/RR-916, 1992.

....unification principle, which says that running a task 1 algorithm on the rows of the inverse of the matrix yields a task 2 algorithm. They suggest hybrid algorithms that alternate between task 1 and task 2 steps to refine the separation of the singular values of R. Pan and Tang [1992] and Gu and Eisenstat [1992] presented different classes of algorithms for achieving (8) and (9) addressing the possibility of nontermination of the algorithms because of floating point inaccuracies. The goal of our work was to develop an efficient and reliable RRQR algorithm and implementation suitable for inclusion in a ....

.... based on the work by Bischof [1989] and efficient implementations of RRQR algorithms well suited for triangular matrices, based on the work by Chandrasekaran and Ipsen [1994] and Pan and Tang [1992] These algorithms seemed better suited for triangular matrices than those suggested by Gu and Eisenstat [1992]. We find that Delta 5 1. P = I ; 2. foreach i 2 f1; ng do res i = ka( i)k 2 end do 3. for i = 1 to min(m; n) do 4. Let i pvt n be such that respvt is maximal 5. P ( i) P ( pvt) a( i) a( pvt) respvt : res i ; 6. u i ; a(i : m ; i) genhh(a(i : m ; i) 7. ....

Gu, M. and Eisenstat, S. 1992 . A stable and efficient algorithm for the rank-one modification of the symmmetric eigenproblem. Technical Report YALEU/DCS/RR-916, Yale University, Department of Computer Science.

....second issue is to maintain orthogonality between eigenvectors in the presence of close eigenvalues. There are two approaches: the extra precision approach of Sorensen and Tang [30] used by Gates and Arbenz [16] in their implementation, and the Lowner Theorem approach proposed by Gu and Eisenstat [19] and adopted for LAPACK [1] 29] There are trade offs that we shall discuss in Section 4.2 between these two approaches. 1 A ScaLAPACK prototype code is available at http: www.netlib.org scalapack prototype The third issue is the back transformation process, which is of great importance ....

..... However, this approach is hard to implement portably across all the usual architectures. There are many machine dependent tricks to make the implementation of extended precision go faster, but on some machines, such as Crays, these tricks do not help and performance suffers. The Gu and Eisenstat [19] approach based on the Lowner Theorem can easily be implemented portably on IEEE machines and Crays using only working precision arithmetic throughout, with a trivial bit of extra work in one place to compensate for the lack of a guard digit in Cray add subtract. On the other hand, the Lowner ....

[Article contains additional citation context not shown here]

Ming Gu and Stanley C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 15(4):1266--1276, 1994.

....unification principle, which says that running a task 1 algorithm on the rows of the inverse of the matrix yields a task 2 algorithm. They suggest hybrid algorithms that alternate between task 1 and task 2 steps to refine the separation of the singular values of R. Pan and Tang [1992] and Gu and Eisenstat [1992] presented different classes of algorithms for achieving (8) and (9) addressing the possibility of nontermination of the algorithms because of floating point inaccuracies. The goal of our work was to develop an efficient and reliable RRQR algorithm and implementation suitable for inclusion in a ....

.... based on the work by Bischof [1989] and efficient implementations of RRQR algorithms well suited for triangular matrices based on the work by Chandrasekaran and Ipsen [1994] and Pan and Tang [1992] These algorithms seemed better suited for triangular matrices than those suggested by Gu and Eisenstat [1992]. We expect that Delta 5 1. P = I ; 2. foreach i 2 f1; ng do res i = ka( i)k 2 end do 3. for i = 1 to min(m; n) do 4. Let i pvt n be such that respvt is maximal 5. P ( i) P ( pvt) a( i) a( pvt) respvt : res i ; 6. u i ; a(i : m ; i) genhh(a(i : m ; i) 7. ....

Gu, M. and Eisenstat, S. 1992 . A stable and efficient algorithm for the rank-one modification of the symmmetric eigenproblem. Technical Report YALEU/DCS/RR-916, Yale University, Department of Computer Science.

....second issue is to maintain orthogonality between eigenvectors in the presence of close eigenvalues. There are two approaches: the extra precision approach of Sorensen and Tang [30] used by Gates and Arbenz [16] in their implementation, and the Lowner Theorem approach proposed by Gu and Eisenstat [19] and adopted for LAPACK [1] 29] There are trade offs that we shall discuss in Section 4.2 between these two approaches. The third issue is the back transformation process, which is of great importance for the success of divide and conquer algorithms because it reduces the cost of forming the ....

....the extra precision approach is embarrassingly parallel, with each eigenvalue and eigenvector computed without communication. However the extra communication the Lowner approach uses is trivial compared with the communication of eigenvectors elsewhere in the computation. The Lowner approach [19], 26] considers that the computed eigenvalues are the exact eigenvalues of a new rank one modification D ae z z T . By definition we have that det(D ae z z T Gamma I) n Y j=1 ( j Gamma ) 4.1) and also det(D ae z z T Gamma I) 0 1 ae n X j=1 z 2 j d j ....

[Article contains additional citation context not shown here]

Ming Gu and Stanley C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 15(4):1266--1276, 1994.

....= ae n X j=1 i 2 j ffi j Gamma x ; 2) and the eigenvector corresponding to each k is parallel to (D Gamma k I) Gamma1 z: 3) Here I is the n Theta n identity matrix. Numerical difficulties arose when these were not computed accurately enough to be orthogonal, but recent work [11, 9] has overcome these difficulties. Ours is a small contribution to solving the secular equation f(x) 0 as accurately as necessary but faster than before. Certain details necessary for robustness in the code will be discussed too. Inattention to such details can cause accidents like Division by ....

....0 OE k (x) 1: For any approximation y to k , we shall approximate each of k (x) and OE k (x) by a simpler form F or G chosen to match k and OE k in value and derivative at x = y; in other words, we perform osculatory interpolation. So far, nothing we have said differs from past practice [3, 5, 9]. What will be novel will be the way we choose which of F and G should be used with each of k and OE k ; in fact, we shall see that F is best not used at all. 2.1.1 Two Ways to Rationally Interpolate k (x) Let y be a fixed approximation to k somewhere between ffi k and ffi k 1 . Now we have ....

[Article contains additional citation context not shown here]

Ming Gu and S. C. Eisenstadt, A stable and efficient algorithm for the rank--one modification of the symmetric eigenproblems, Research Report YaleU/DCS/RR--916, 1992.

No context found.

Ming Gu and Stanley Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmmetric eigenproblem. Technical Report YALEU/DCS/RR-916, Yale University, Department of Computer Science, 1992.

No context found.

M. Gu and S. Eisenstat, A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1266--1276.

No context found.

M. Gu and S. C. Eisenstat, "A Stable and Efficient Algorithm for the rank-one modification of the symmetric eigenvalue problem," SIAM J. Matrix Anal. Appl., vol. 15, pp. 1266-1276, 1994.

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Ming Gu and Stanley Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmmetric eigenproblem. Technical Report YALEU/DCS/RR-916, Yale University, Department of Computer Science, 1992.

No context found.

M. Gu and S. C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl., 14(4):1266-- 1276, 1994.

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