J. Bunch, P. Nielsen, and D. Sorensen. Rank-one modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....table 2 to table 4. 4 in [6] we note that our algorithm requires fewer flops than one power method iteration in [7] if M 6 ( 4M Gamma 1:5)M versus 24M O(M iterations) Note that in step 11, we may further reduce computation to 8:5M by updating only one of the noise eigenvectors [1] [2] however simulations show that this would lead to an inaccurate array response estimate due to the fact that the noise subspace is not perfectly spherical. Flops per weight update per frame Step # # of flops 1 3 (4M 2M 1) 5M 4M 4 0:5M 2:5M 7 5 14 36 (4M Gamma ....

J. Bunch, C. Nielsen, and D. Sorensen. Rank-one modification of the symmetric eigenproblem. Numer. Math., 3:111--129, 1978.

....subject classifications. 65F15, 68C25 1 Introduction The divide and conquer algorithm is an important recent development for solving the tridiagonal symmetric eigenvalue problem. The algorithm was first developed by Cuppen [8] based on previous ideas of Golub [17] and Bunch, Nielson and Sorensen [5] for the solution of the secular equation and made popular as a practical parallel method by Dongarra and Sorensen [14] This simple and attractive algorithm was considered unstable for a while because of a lack of orthogonality in the computed eigenvectors. It was thought that extended precision ....

....A recursive application of the strategy described above on the two tridiagonal matrices in (2.1) leads to the divide and conquer method for the symmetric tridiagonal eigenvalue problem. 2. 1 Computing the Spectral Decomposition of a RankOne Perturbed Matrix An updating technique as described in [5], 8] 17] can be used to compute the spectral decomposition of a rank one perturbed matrix (2.5) where D = diag(d 1 ; d 2 ; d n ) z = z 1 ; z 2 ; z n ) and ae is a nonzero scalar. By setting the characteristic polynomial of D aezz equal to zero, we find that the ....

James R. Bunch, Christopher P. Nielsen, and Danny C. Sorensen. Rank-one modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.

....of T is given by T = U e U ; U = diag(Q 1 ; Q 2 ) e Q: The formation of U is a matrix multiplication and dominates the operation count. This algorithm was originally suggested by Cuppen [30] and how to solve the secular equation efficiently was shown by Bunch, Nielson 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 ....

James R. Bunch, Christopher P. Nielsen, and Danny C. Sorensen. Rankone modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.

....of these methods can be grouped into three families. In the first one, classical batch ED SVD methods like the QR algorithm, Jacobi rotation and the power iteration have been modified for use in adaptive processing [20 23] In the second family, variations of Bunch s rank one updating algorithm [24] have been proposed. The third class of algorithms considers the ED SVD as a constrained or unconstrained optimization problem. In [25] the signal subspace has been shown to be the solution of an unconstrained minimization problem and the projection approximation subspace tracking (PASTd) ....

R. Bunch, C. P. Nielsen and D. Sorenson. "Rank-one modification of the symmetric eigenproblem." Numerische Mathematik, vol 31, pp. 31-48, 1978.

....iteration (b Gamma Ax ) Gamma 1 kx ; 3.2) A A Gamma I) b: 3.3) This iteration will converge monotonically at a rate that is asymptotically quadratic. The convergence of this method can be improved by using a rational interpolation similar to that in [6] to solve the secular equation. However, in any case, will converge to oe n 1 and x to the TLS solution only if the initial approximation satisfies 2 (oe n ) 3.4) In general it is hard to verify this assumption. For the special case of a Toeplitz TLS problem Kamm and Nagy [14] use ....

J. R. Bunch, C. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31 (1978), pp. 31--48.

....either case, the observations used to construct the eigenspace model are the training observations;thatis, they are assumed to be instances from some class. This model may then used to decide whether further observations belong to the class. Incremental eigenanalysis has been studied previously [1, 2, 3, 4, 7, 13], but surprisingly these authors either have ignored the fact that a change in data changes the mean, or else have handled it in an ad hoc way. Only our previous work allows for a change of mean [9] where we allowed for the inclusion of only a single new datum. In contrast, our algorithms here ....

J. R. Bunch, C. P. Nielsen, and D. C. Sorenson. Rank-one modification of the symmetric eigenproblem. Numerische Mathematik, 31:31--48, 1978.

....of these methods can be grouped into three families. In the first one, classical batch ED SVD methods like the QR algorithm, Jacobi rotation and the power iteration have been modified for use in adaptive processing [20 23] In the second family, variations of Bunch s rank one updating algorithm [24] have been proposed. The third class of algorithms considers the ED SVD as a constrained or unconstrained optimization problem. In [25] the signal subspace has been shown to be the solution of an unconstrained minimization problem and the projection approximation subspace tracking (PASTd) ....

R. Bunch, C. P. Nielsen and D. Sorenson. "Rank-one modification of the symmetric eigenproblem." Numerische Mathematik, vol 31, pp. 31-48, 1978.

....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 ....

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J. R. Bunch, C. P. Nielsen, and D. C. Sorensen. Rank-one modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.

....memory architecture AMS subject classifications. 65F15, 68C25 PII. S1064827598336951 1. Introduction. The divide and conquer algorithm for the symmetric tridiagonal eigenvalue problem was first developed by Cuppen [8] based on previous ideas of Golub [16] and Bunch, Nielsen, and Sorensen [5] for the solution of the secular equation. The algorithm was popularized as a practical parallel method by Dongarra and Sorensen [14] who implemented it on a shared memory machine. They concluded that divide and conquer algorithms, when properly implemented, can be many times faster than ....

J. R. Bunch, C. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31 (1978), pp. 31--48.

.... case, the singular values of A 0 and M 1 are always well conditioned with respect to a perturbation, the singular vectors can be extremely sensitive to such perturbations [5] In both cases, the problem of updating the SVD has been considered by Bunch and Nielsen [1] using results from [2, 4]. Their scheme for finding the SVD of M and M 1 can be unstable [1, 2] And their algorithm takes about 2n min 2 (m; n) and 2mmin 2 (m; n) floating point operations to update the right and the left singular vector matrices, respectively. The lack of a fast algorithm for updating the SVD is ....

.... with respect to a perturbation, the singular vectors can be extremely sensitive to such perturbations [5] In both cases, the problem of updating the SVD has been considered by Bunch and Nielsen [1] using results from [2, 4] Their scheme for finding the SVD of M and M 1 can be unstable [1, 2]. And their algorithm takes about 2n min 2 (m; n) and 2mmin 2 (m; n) floating point operations to update the right and the left singular vector matrices, respectively. The lack of a fast algorithm for updating the SVD is one of the reasons for the recent development of URV and ULV ....

J. R. Bunch, C. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31 (1978), pp. 31--48. 32

.... : z p ) 2 R p , are the zeros of the secular equation f(i) j 1 p X j=1 z 2 j fl j Gamma i = 0 : 20) The secular equation (20) appears in the context of symmetric eigenvalue problems [10] and can be solved through a fast, quadratically convergent procedure, as described in [2, 4]. Based on the above formulation, we solve r secular equations to find the set of SVs ffi ij g i=1; r; j=1; p of the matrices B i = B i Gamma1 i z i z i ; i = 1; r; 21) where B 0 = Gamma and B i Gamma1 = diag(fi i Gamma1;1 ; fi i Gamma1;p ) for i 1. ....

....the singular values, but neglect the updating of the singular vectors. The second term of Bhattacharyya distance (2) is then given by d 2 B2 1 2 ln Q p j=1 fi j 2 p q jC q j jC d j : 23) The number of flops necessary to find all sets ffi ij g is not significant, having the order of rp [4]. Therefore, in the case of r p the arithmetic complexity of the approximate computation of the complete Bhattacharyya distance remains 2rp 2 2p 2 flops. Each entry in the logical database should contain the mean vector d , the matrix V r , the SVs 1 ; r , and the value of the ....

J.R. Bunch, C.P. Nielsen, D.C. Sorensen, "Rank-One Modification of the Symmetric Eigenproblem," Numerische Matematik, vol. 31, pp. 31--48, 1978.

....efficient for parallel computation [25, 46] It is quite remarkable that this method can also be faster than other implementations on a serial computer The matrix T may be expressed as a modification of a direct sum of two smaller tridiagonal matrices. This modification may be a rank one update [15], or may be obtained by crossing out a row and column of T [72] The eigenproblem of T can then be solved in terms of the eigenproblems of the smaller tridiagonal matrices, and this may be done recursively. For several years after its inception, it was not known how to guarantee numerically ....

J. Bunch, P. Nielsen, and D. Sorensen. Rank-one modification of the symmetric eigenproblem. Num. Math., 31:31--48, 1978.

....of the matrix, thereby obscuring key properties of the equation. It is used there to compute the smallest even eigenvalue and it too uses polynomial approximations. The idea of a rational approximation for secular equations is not new. In a different context, it was already used in, e.g. [5] and many other references, the most relevant to this work being [19] However, apparently because of the somewhat complicated nature of their analysis, it seems that these rational approximations are rarely considered beyond the first order. We consider a different approach that enables us to ....

Bunch, J.R., Nielsen, C.P., Sorensen, D.C. (1978): Rank-one modification of the symmetric eigenproblem. Numer. Math. 31, pp. 31--48.

.... eigenvalues can be e#ectively approximated by the known methods (bisection, divide and conquer, or QR) In particular, BP91] and [BP,b] show how this can be done in linear arithmetic time (up SOLVING A POLYNOMIAL EQUATION 213 to a polylog factor) based on the divide and conquer approach (see [BNS78], C81] and [DS87] on some preceding works) and [BP92] shows a practical modification of the algorithm of [BP91] Would a similar approach work for an arbitrary polynomial p(x) or at least for a large class of polynomials p(x) with complex zeros The idea is to start by computing a complex ....

J. R. BUNCH,C .P.NIELSEN,D.C.SORENSEN, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31 (1978), pp. 31--48.

....are known as secular equations and functions like OE( OE 0 ( are sometimes called secular functions. The term seems to come from celestial mechanics (see [1] Secular equations appear in many contexts such as in the solution of certain eigenvalue problems, see for example [1] 3] 23] [7], 49] 51] 50] and [52] These works are concerned with computing all the solutions of a secular equation and they usually assume that fl i 6= 0; i = 1; 2; n, therefore excluding the possibility of the hard case. This fact prevents the use of such methods for solving the general TRS. ....

J.R. Bunch, C.P. Nielsen, and D.C. Sorensen. Rank--one modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.

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J. Bunch, P. Nielsen, and D. Sorensen. Rank-one modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.

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J. R. Bunch, Ch. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31(1978), 31--48.

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J. Bunch, P. Nielsen, and D. Sorensen. Rank-one modification of the symmetric eigenproblem. Numer. Math., 31:31--48, 1978.

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J. R. Bunch, C. P. Nielsen, and D. C. Sorenson. Rank-one modification of the symmetric eigenproblem. Numerische Mathematik, 31:31--48, 1978.

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James R. Bunch, Christopher P. Nielsen, and Danny C. Sorenson. Rank-one modification of the symmetric eigenproblem. Numerische Mathematik, 31:31-- 48, 1978.

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J. R. Bunch, Christopher P. Nielsen, and Danny C. Sorensen. Rank-One Modification of the Symmetric Eigenproblem. Numer. Math., 31:31--48, 1978.

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Bunch J. R., Nielsen C. P., Sorensen D. C., Rank-one modification of the symmetric eigenproblem, Numer. Math. 31, pp. 31--48, (1978).

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J. R. Bunch, C. P. Nielsen, D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math. 31, 31--48, (1978).

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Bunch J. R., Nielsen C. P., Sorensen D. C., Rank-one modification of the symmetric eigenproblem, Numer. Math. 31, pp. 31--48, (1978).

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J. R. Bunch, C. P. Nielsen, and D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., 31 (1978), pp. 31--48.

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