K. V. Fernando and B. N. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of widely differing magnitudes) than by tridiagonalization based methods. We briefly discuss the algorithmic considerations in Section 4 and then present numerical results in Section 5. Though, of course, one could take the Cholesky factor of the tridiagonal and then apply the dqds algorithm [5] to obtain the eigenvalues of the tridiagonal to high relative accuracy. Let Mm;n denote the space of m Theta n real matrices , and let Mn = M n;n . For a symmetric matrix H we will let 1 (H) 2 (H) Delta Delta Delta n (H) denote its eigenvalues, ordered in decreasing order. For X 2 Mm we ....

....errors the singular values of the bidiagonal matrix. It remains to determine whether the implicit Cholesky algorithm with shifts computes the singular values to high relative accuracy. It does in the case p = 2 as in this case it is, if suitably implemented, just the dqds algorithm discussed in [5]. Right handed Jacobi s method (without accumulating transformations) can exploit both bandedness and gradedness and compute the singular values (the squares of which are the eigenvalues of H) and the left singular vectors (which are the eigenvectors of H) in O(np ) flops. If one uses the ....

K. V. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Math., 67:191--229, 1994.

....his opera tionally minimal implementation. The dqds algorithm was rediscovered independently by Fernando and Parlett in 1992 and they showed that the extra multiplication, compared to Rutishauser s qd.s, allowed dqds to compute all the eigenvalues, however small, to high relative accuracy. See [3]. Here ends the historical commen tary. Here is the transform applied to a segment of Z, Z(iO: nO) with shift r. d:q(iO) r temp = q(i 1) i) 40 d = d, temp r For contrast we present Rutishauser s qd transform with shift qds: iO) q(iO) e(iO) r for i: i0, n0 1 do (i) ....

....ql C1 q p forj = 1, n 1 do qj l ozj l ej q p An alternative, careful, expression for qj l is qj l : max(ozj l, p) ej) q min(ozj l, p) One must remember to subtract p from the eigenvalues computed by the algorithm in order to recover those of T. 1. 1 Overflow In [3] it was shown that dqd.s preserves eigenvalues to high relative accuracy in the absence of overflow and underflow. In this section we identify and eliminate those exceptions that are unnecessary . The concerns of this subsection arise almost exclusively in single precision where the exponent ....

[Article contains additional citation context not shown here]

K. V. Fernando and B. N. Parlett, 'Accurate Singular Values and Differ- ential qd Algorithms'. Numerische Mathematik, vol. 67, (March 1994), no. 2, pp. 191-229.

....computes the singular values of a bidiagonal matrix to high relative accuracy; this algorithm obtains the singular vectors to high accuracy, too [34] Such strong statements do not hold for the previously standard version of the QR algorithm used, for example, in LINPACK. Fernando and Parlett [60], 109] have developed a quotient difference (qd) algorithm for computing the singular values of a bidiagonal matrix. This algorithm, which goes back to much earlier work of Rutishauser, is more accurate than the algorithm of Demmel and Kahan and, because it allows the incorporation of shifts, ....

K. Vince Fernando and Beresford N. Parlett. Accurate singular values and differential qd algorithms. Numer. Math., 67:191--229, 1994.

....the basic part of the algorithm, without the iterative refinement step. All calculations were carried out in MATLAB on a Silicon Graphics Indigo workstation with IEEE floating point standard. For computing the singular values of the computed bidiagonal we use the method due to Fernando and Parlett [5]. a. Implicit versus explicit. Let us consider the following products : A 1 [n; m] T m n 15 where T n is a n Theta n symmetric Toeplitz matrix whose first column is [2; Gamma1; 0; 0; Delta Delta Delta ; 0] singular values and singular vectors of such matrices are known [8] Since it ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms. Numer. Math. 67, pp 191-229, 1994.

....the classical perturbation theory would indicate. A relative perturbation theory is then called for to exploit the situations for better bounds on the relative differences between e and . The development of such a theory goes back to Kahan [20, 1966] and is becoming a very active research area [1, 6, 7, 8, 9, 11, 12, 14, 16, 10, 28, 34]. In this paper, we develop a theory by a unifying treatment that sharpens some existing bounds and covers many previously studied cases. We shall deal with perturbations that have multiplicative structures, namely perturbations to unperturbed matrices are realized by multiplying the unperturbed ....

....in this paper covers many previously studied cases and yields bounds that are at least as sharp as existing ones. Our results are applicable to the computations of sharp error bounds in the Demmel KahanQR [8, 1990] algorithm and Fernando Parlett s implementation of the Rutishauser QD algorithm [14, 1994]; see Li [23] Previous approaches to building a relative perturbation theory are more or less along the lines of using the min max principle for Hermitian matrix eigenvalue problems. Our approach in this paper, however, is through deriving the perturbation equations (5.2) and (5.3) A major ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms, Numerische Mathematik, 67 (1994), pp. 191--229.

....to e B = D 1 BD 2 are also presented. 1 Introduction Eigenvalue and singular value computations to high relative accuracy have been attracting lots of attention over the last 10 years or so. Tremendous progress has been made both in theoretical understanding and numerical algorithms; see [1, 4, 7, 8, 9, 10, 12, 13, 14, 11, 18, 25, 26, 27, 28] and references therein. On the algorithmic side there are DemmelKahan QR method for bidiagonal singular value computations [8] two sided) Jacobi methods for the eigenvalue problems of positive definite matrices and for the singular value computations [9, 25, 28] Bisection method for scaled ....

.... [8] two sided) Jacobi methods for the eigenvalue problems of positive definite matrices and for the singular value computations [9, 25, 28] Bisection method for scaled diagonally dominant matrices [1] and for matrices with acyclic graphs [7, 17] new implementations of the qd method [14, 27], Department of Mathematics, University of Kentucky, Lexington, KY 40506 (rcli ms.uky.edu. This work was supported in part by the National Science Foundation under Grant No. ACI 9721388 and by the National Science Foundation CAREER award under Grant No. CCR 9875201. 1 Ren Cang Li: Relative ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms, Numerische Mathematik, 67 (1994), pp. 191--229.

....by the QR iteration does not necessarily compute the eigenvalues of a graded matrix to high relative accuracy. There are algorithms that will compute the eigenvalues of a positive definite tridiagonal matrix T to a relative accuracy of ffl(S T TST ) for example bisection or the qd algorithm in [7]. If one has a graded positive definite matrix H and reduces it to a tridiagonal T by Givens rotations (possibly fast rotations) one can monitor the possible loss of accuracy by computing the value of ff for each of the transformations applied. If the maximum of these ff s is not large then the ....

K. V. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Technical Report PAM-544, Center for Pure and Applied Mathematics, University of Calfornia, Berkeley, 1992. to appear in Numerische Mathematik.

.... 2I The execution times in Table 2 demonstrate that the favorable complexity of the RRR based approaches leads to a superior performance. Note that the RRR methods can be further optimized using preconditioning techniques or by minimizing the support of the eigenvectors. Both use the qd algorithm [9] to determine the initial eigen singular value approximations. Matrix type n DSTEGR RRR coup DBDSDC DBDSQR geometric 500 0.13 0.31 0.30 8.13 distribution 1000 0.44 1.34 1.64 108.85 2000 1.70 5.51 7.19 arithmetic 500 0.28 0.35 1.37 17.1 distribution 1000 1.11 1.54 9.20 227.1 2000 4.48 6.44 61.05 ....

K. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numer. Math., 67(2):191--229, 1994. 19

.... the backward error in linear fractional recurrences evaluated by parallel prefix, parallel computation of triangular factorizations of tridiagonal matrices (the QR factorization is discussed in detail) and the parallel implementation of the differential qd algorithm of Fernando and Parlett [6]. Department of Mathematics, College of William Mary, Williamsburg, VA 23187. e mail: na.mathias nanet. ornl.gov. This research was supported in part by National Science Foundation grant DMS 9201586 and much of it was done while the author was visiting the Institute for Mathematics and its ....

....the sum of terms of the form kS 1 S 2 Delta Delta Delta S i j k 2 kS i j 1 Delta Delta Delta S j k 2 kS j 1 Delta Delta Delta S k j k 2 kS k j 1 Delta Delta Delta S n k 2 which is potentially much smaller, though harder to analyze. 5.3. qd Algorithms. Recently Fernando and Parlett [6] revived the qd algorithms of Rutishauser and showed that when they are suitably implemented (in serial) they deliver the singular values of a bidiagonal matrix to high relative accuracy. Their algorithm is also simpler to analyze and cheaper to implement that the 0 shift QR step of Demmel and ....

[Article contains additional citation context not shown here]

K. V. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Math., 67:191--229, 1994.

....of widely differing magnitudes) than by tridiagonalization based methods. We briefly discuss the algorithmic considerations in Section 4 and then present numerical results in Section 5. 2 Though, of course, one could take the Cholesky factor of the tridiagonal and then apply the dqds algorithm [5] to obtain the eigenvalues of the tridiagonal to high relative accuracy. 4 roy mathias Let Mm;n denote the space of m Theta n real matrices , and let Mn = M n;n . For a symmetric matrix H we will let 1 (H) 2 (H) Delta Delta Delta n (H) denote its eigenvalues, ordered in decreasing order. ....

....errors the singular values of the bidiagonal matrix. It remains to determine whether the implicit Cholesky algorithm with shifts computes the singular values to high relative accuracy. It does in the case p = 2 as in this case it is, if suitably implemented, just the dqds algorithm discussed in [5]. Right handed Jacobi s method (without accumulating transformations) can exploit both bandedness and gradedness and compute the singular values (the squares of which are the eigenvalues of H) and the left singular vectors (which are the eigenvectors of H) in O(np 2 ) flops. If one uses the ....

K. V. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Math., 67:191--229, 1994.

....THE QD ALGORITHM TO TACKLE MULTIVARIATE PROBLEMS Annie Cuyt 1 and Brigitte Verdonk 2 Abstract. A lot has been said and done about the qd algorithm [13, 9, 7, 5, 14]. Our main interest is to analyze how the original algorithm and its various improvements can be generalized for use in several multivariate applications. The present paper recalls the known univariate results in the sections 2.1 and 3, and discusses their multivariate generalization in the ....

....eigenvalues of a tridiagonal matrix and the singular values and vectors of a bidiagonal matrix. We refer among others to the differential qd algorithm [13, pp. 505 506] the qd algorithm with shift [13, pp. 472 474] the stationary qd algorithm [13, pp. 508 513] and the orthogonal qd algorithm [9, 14]. The main aim of these variants is either to avoid the pitfalls of a floating point implementation or to accelerate the convergence of the algorithm or both. These improved schemes, however, rely on the fact that the matrix elements satisfy certain properties (such as positivity) Such properties ....

K. Vince Fernando and Beresford N. Parlett. Accurate singular values and differential qd-algorithms. Numer. Math., 67:191--229, 1994. Extending the QD-Algorithm 25

....matrix is tridiagonal. In the last few years, it has gradually become clear that the standard representation of a tridiagonal matrix via its entries is an unfortunate one, and that it is better, both for accuracy and efficiency, to represent the matrix as a product of bidiagonals; see, e.g. [8, 21, 22]. This paper has demonstrated that the same is true for the Lanczos matrix associated with the symmetric band Lanczos process. Instead of representing that matrix via its entries, it is preferable to present it via the entries of an LDL T factorization. Acknowledgment. We would like to thank ....

B. N. Parlett and K. Fernando, Accurate singular values and differential QD algorithms, Numer. Math., 67 (1994), pp. 191--229.

....We begin by introducing the family of GR algorithms in Section 2. These are iterative methods that move a matrix toward upper triangular form via similarity transformations. We discuss the convergence 2 Amazingly the quotient difference algorithm has had a recent revival. Fernando and Parlett [28], 42] introduced new versions for finding singular values of bidiagonal matrices and eigenvalues of symmetric, tridiagonal matrices. QR LIKE ALGORITHMS 3 of GR algorithms briefly. In Section 3 we show how to implement GR algorithms economically as bulge chasing procedures on Hessenberg matrices. ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191--229.

....to e B = D 1 BD 2 are also presented. 1 Introduction Eigenvalue and singular value computations to high relative accuracy have been attracting lots of attention over the last 10 years or so. Tremendous progress has been made both in theoretical understanding and numerical algorithms; see [1, 4, 7, 8, 9, 10, 12, 13, 14, 11, 18, 25, 26, 27, 28] and references therein. On the algorithmic side there are DemmelKahan QR method for bidiagonal singular value computations [8] two sided) Jacobi methods for the eigenvalue problems of positive definite matrices and for the singular value computations [9, 25, 28] Bisection method for scaled ....

.... [8] two sided) Jacobi methods for the eigenvalue problems of positive definite matrices and for the singular value computations [9, 25, 28] Bisection method for scaled diagonally dominant matrices [1] and for matrices with acyclic graphs [7, 17] new implementations of the qd method [14, 27], and Demmel s algorithms for structured matrices [6] and more recently [10] showed how to compute singular value decompositions to high relative accuracy for matrices that Department of Mathematics, University of Kentucky, Lexington, KY 40506 (rcli ms.uky.edu. This work was supported in part ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms, Numerische Mathematik, 67 (1994), pp. 191--229.

....accuracy as the elements of the matrix. Demmel and Kahan [23] use this result, together with rounding error analysis, to show that a zero shift version of the QR algorithm computes the singular values 20 NICHOLAS J. HIGHAM of a bidiagonal matrix to high relative accuracy. Fernando and Parlett [28] have developed a shifted quotient difference algorithm that satisfies even smaller error bounds than the algorithm of Demmel and Kahan, and is at least as fast. Barlow and Demmel [4] were interested in finding classes of matrices other than those in Theorem 6.6 or Corollary 6.7 that determine ....

K. Vince Fernando and Beresford N. Parlett, Accurate singular values and differential qd algorithms, Numer. Math. 67 (1994), 191--229.

.... the iterative application of Cholesky decomposition [21] The divide and conquer approach can also be combined with it [4, 5] More recently, the Cholesky decomposition, or one of its variants, has been used in connection with the accurate computation of the singular values of bidiagonal matrices [11, 15], and of the eigenvalues of specially structured symmetric tridiagonal matrices [9] Moreover, it has been shown that Francis QR algorithm (see [16, 17] can be implemented using a band Cholesky decomposition [3] Cholesky decomposition, followed by the parallel solution of the respective ....

K. V. Fernando and B. Parlett, Accurate singular values and differential qd algorithms, Numerische Mathematik, 67 (1994), pp. 191-- 229.

....ffl, the bound is not tight. 2 There are algorithms whose relative error bounds do not depend on the eigenvalues. These algorithms compute all eigenvalues or singular values to high relative accuracy, even those of small magnitude: the dqds algorithm for singular values of bidiagonal matrices (Fernando and Parlett 1994, Parlett 1995) for instance, as well as Jacobi methods for eigenvalues of symmetric positive definite matrices and for singular values (Demmel 1997, x5.4.3) Mathias 1995a) Absolute perturbation bounds cannot account for this phenomenon. Absolute error bounds are well suited for describing ....

.... ff n 1 C C C A : Bidiagonal matrices can also arise when one computes the vibrational frequencies of a linear mass spring system (Demmel et al. 1997, x12.1) There are several algorithms for computing singular values of a bidiagonal matrix to high relative accuracy (Demmel and Kahan 1990, Fernando and Parlett 1994). Because such algorithms apply a sequence of transformations to reduce B to diagonal form, they need to decide when an off diagonal element fi j is small enough to be neglected without severely harming the singular values. Suppose we are contemplating the removal of a single off diagonal element. ....

[Article contains additional citation context not shown here]

K. Fernando and B. Parlett (1994), `Accurate singular values and differential qd algorithms', Numer. Math. 67(2), 191--229.

....bound may not be good enough, because there are algorithms that compute all eigenvalues to high relative accuracy, even those of small magnitude. Among such algorithms are Jacobi methods for Hermitian positive definite matrices [4, 13] and the dqds algorithm for certain tridiagonal matrices [7]. These algorithms have genuine relative error bounds that do not depend on the eigenvalues. Our original motivation was to determine under which circumstances we can find genuine relative perturbation bounds that do not depend on the eigenvalues. In particular, does the existence of such a ....

K. Fernando and B. Parlett, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191--229.

....1 diagonal and off diagonal elements of T 1 . Since T 1 is positive definite we can compute its bidiagonal Cholesky factor L 1 . The singular values, oe j , of L 1 may now be computed to high relative accuracy using either bisection or the much faster and more elegant dqds algorithm given in [56] (remember that in exact arithmetic the eigenvalues of T 1 are the squares of the singular values of L 1 and the eigenvectors of T 1 are the left singular vectors of L 1 ) Recall that the singular values of L 1 are such that oe 2 1 = 2 O( 2 ) oe 2 2 = O( 2 ) and oe 2 3 = ....

....transformation given by Rutishauser, the differences are not significant enough to warrant inventing new terminology. More recently, Fernando and Parlett have developed another qd algorithm that gives a fast way of computing the singular values of a bidiagonal matrix to high relative accuracy [56]. The term stationary is used for (4.4.17) since it represents an identity transformation when = 0. Rutishauser used the term progressive instead for the formation of U Gamma D Gamma U T Gamma from LDL T . In the rest of this chapter, we will denote L (i 1; i) by L (i) U Gamma ....

[Article contains additional citation context not shown here]

K. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

....by reversing the order of Q k and R k and carrying out the multiplication. It is proved that except for contrived examples X k converges to a diagonal matrix with the eigenvalues of A T A ordered along the diagonal, i.e. X1 = 6 2 . A square root version of this algorithm has been derived in [6, 9, 13] (see also [5] for a related result) With A = QARA , one has X 0 = R T A z lower 1 RA z upper def = R T 0 1 R 0 The QR factorization of X 0 (cfr. first iteration) is obtained from the QR factorization of R T 0 X 0 = R T 0 z Q 0] R 0] 1R 0 = Q 0] z Q 0 ....

V. Fernando, B. Parlett, `Accurate singular values and differential QD algorithms', Report No. UCB/PAM-554, Center for Pure and Applied Mathematics, Univ. of California, Berkeley, July 1992.,

.... which can be solved this way include tridiagonal Gaussian elimination (a three term recurrence) solving bidiagonal linear systems (two terms) Sturm sequence evaluation for the symmetric tridiagonal eigenproblem (three terms) and the bidiagonal dqds algorithm for singular values (three terms) [63]. The numerical stability of these algorithms is not completely understood. For some applications, it is easy to see the error bounds are rather worse than the those of the sequential implementation [20] For others, such as Sturm sequence evaluation [76] empirical evidence suggests it is stable ....

....as described in section 2.2 The stability is unproven. Experiments with bisection [76] are encouraging, but the only published analysis [20] is very pessimistic. Initial results on the dqds algorithm for the bidiagonal SVD, on the other hand, indicate stability may be preserved in some cases [63]. On the other hand, bisection can easily be parallelized by having different processors refine disjoint intervals, evaluating the Sturm sequence in the standard serial way. This involves much less communication, and is preferable in most circumstances, unless there is special support for parallel ....

B. Parlett and V. Fernando. Accurate singular values and differential QD algorithms. Math Department PAM-554, University of California, Berkeley, CA, July 1992.

.... the iterative application of Cholesky decomposition [21] The divide and conquer approach can also be combined with it [6, 7, 8] More recently, the Cholesky decomposition, or a variation of it, has been used in connection with the accurate computation of the singular values of bidiagonal matrices [12, 15], and of the eigenvalues of specially structured symmetric tridiagonal matrices [10] Moreover, it has been shown that Francis QR algorithm (see [16, 17] can be implemented using a band Cholesky decomposition [3, 4, 5] We also point out that, despite the amount of work devoted to the parallel ....

K. V. Fernando and B. Parlett, Accurate singular values and differential qd algorithms, Numeriche Mathematics, 67 (1994), pp. 191--229.

....The whole field had its genesis with Rutishauser s quotient difference algorithm [28] 29] which Rutishauser then generalized to the LR algorithm [30] The QR algorithm followed shortly thereafter. Amazingly the quotient difference algorithm has had a recent revival. Fernando and Parlett [17], 27] introduced new versions for finding singular values of bidiagonal matrices and eigenvalues of symmetric, tridiagonal matrices. Other examples are the HR [6] 8] and SR [10] 11] algorithms, The H stands for hyperbolic and the S for symplectic. Other examples will come up later. Now let us ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms, Numer. Math. 67 (1994), 191--229.

....which is used as a preprocessing step for the implicit Jacobi method in the case when the matrix G has much more rows than columns, and analyzed numerical aspects of accurate computing as overflow underflow and the accuracy of various implementations of plane rotations. Fernando and Parlett [19] showed that various variants of the differential QD algorithm compute the singular values of bidiagonal matrices to high relative accuracy, and that this algorithm has some better properties than the zero shift QR algorithm. Gu and Eisenstat [22] further extended the relative perturbation theory ....

K. V. Fernando, B. N. Parlett, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191--229.

....in numerical analysis is due to Golub and Kahan [38] Until recently reduction to bidiagonal form followed by a variant of the QR algorithm, due to Golub [40] has been the standard way to compute the decomposition. Recently new algorithms for reducing the bidiagonal matrix have been proposed [25, 30]. The idea of first computing the QR decomposition has been exploited by Chan [12, 11] Beltrami [6] first established the existence of the singular value decomposition in 1873 by computing the eigendecomposition of the cross product matrix, and this is still a popular way of doing things in some ....

K. V. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Technical Report PAM--554, Department of Mathematics, University of California, Berkeley, 1992.

....a major improvement over the original SVD subroutine svdc as it has been implemented in the LINPACK linear algebra library [3] On the other hand K. V. Fernando and B. N. Parlett discovered in 1992 a variant of the qd algorithm for obtaining maximal relative accuracy for all the singular values [4]. Their approach is based on the so called differential form of the progressive qd algorithm. In contrast to the work of Demmel and Kahan their algorithm cannot compute the left and right singular vectors simultaneously with the singular values. 4. Progressive Quotient Difference Step. The ....

....in Figure 4, except that the transformations are applied from bottom to top. We will refer to the four transformations, that have been introduced in this section, by the generic term of orthogonal qd steps. Fernando and Parlett also present a root free version of the orthogonal left lustep in [4]. However, such a root free algorithm has the disadvantage that it operates on the squares of the singular values. Consequently, the largest singular value must not exceed the square root of the largest machine representable number. Similarly, singular values smaller than the square root of the ....

[Article contains additional citation context not shown here]

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191--229.

.... T 1 X 1 Gamma X T 2 X 2 = R T DR ; 17) as in (14) Remark 3 For two special cases with X 1 = flI n and X 2 = flI n , respectively, there are some important applications in linear algebra and signal processing: ffl X 2 = flI n : The implicit Cholesky decomposition of Fernando and Parlett [4] and von Matt [14] is actually the presented Schur type method, where fl is the shift for the singular value computation. The presented Schur type method for arbitrary matrices also allows shifts larger than the smallest singular value without causing a breakdown of the algorithm. ffl X 1 = flI ....

K.V. Fernando, B.N. Parlett. Accurate singular values and differential qd algorithm. Numer. Math., 67:191--229, 1994.

....of bidiagonal reduction is currently under intensive investigations [7] We intend to concentrate only on the evaluation of singular values of bidiagonal matrices. Recently K. V. Fernando and B. N. Parlett reported a dqd (the differential form of the progressive quotient difference) algorithm [8] which 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 Numerische Mathematik Electronic Edition page numbers may differ from the printed version page 285 of Numer. ....

Fernando, K.V., Parlett, B.N. (1992): Accurate singular values and differential QD algorithm. preprint PAM-554, University of California, Berkeley

....and nonnegative definite Hermitian eigenvalue problems. 1 Introduction Relative perturbation theory for eigensystems and singular systems has been becoming a hot topic in the last five years and ever since It was first studied by Kahan [18] in 1966, later by [1, 6, 8, 9, 29] and most recently by [7, 10, 11, 13, 15, 25]. 1.1 What to be Covered This paper deals with perturbations of the following kinds: ffl Eigenvalue problems: 1. A and e A = D AD for Hermitian case, where D is nonsingular and close to I or more generally to a unitary matrix; 2. A and e A = D 1 AD 2 for general diagonalizable case, ....

K. V. Fernando and B. N. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

....the order of Q k and R k and carrying out the multiplication. It is proved that except for contrived examples X k converges to a diagonal matrix with the eigenvalues of A T A ordered along the diagonal, i.e. X1 = Sigma 2 . A square root version of this algorithm has been derived in [6, 9, 13] (see also [5] for a related result) With A = QARA , one has X 0 = R T A z lower Delta RA z upper def = R T 0 Delta R 0 The QR factorization of X 0 (cfr. first iteration) is obtained from the QR factorization of R T 0 X 0 = R T 0 z Q 0] R 0] DeltaR 0 = Q ....

V. Fernando, B. Parlett, `Accurate singular values and differential QD algorithms', Report No. UCB/PAM-554, Center for Pure and Applied Mathematics, Univ. of California, Berkeley, July 1992.,

....C A 3 is an upper bidiagonal matrix. Phase II: Compute the SVD of B: B = Q SigmaW T (1.4) where Q and W are orthogonal matrices. The SVD of A is then computed as A = U 1 Q U 2 ) Sigma 0 (V W ) T Phase II has previously been implemented using QR iteration [11, 12, 8] or qd iteration [22, 24]. This has been the bottleneck of the overall algorithm, taking up to 80 of the total time. We have made three contributions toward overcoming this bottleneck. First, we have implemented a variation of the bidiagonal divide and conquer algorithm (BDC) of Gu and Eisenstat [17] which is based on ....

B. Parlett and V. Fernando. Accurate singular values and differential qd algorithms. Math Department PAM-554, University of California, Berkeley, CA, July 1992. to appear in Num. Math. 19

....guarantee that the two smaller singular values and their singular vectors are accurate to about 5 decimal digits, whereas the absolute error bounds guarantee no correct digits at all. Algorithms capable of computing the SVD of bidiagonal matrices with such high relative accuracy were published in [18, 16, 25]. Our interest in the notion of relative accuracy defined by bounds (7) and (8) arises for two reasons. First, there are a number of physical problems where the smallest singular values (or eigenvalues) are well determined by the physical problem being modeled, and we need to compute them with ....

....SVDs of G and G = G ffiG agree to high relative accuracy, as described by bounds (7) and (8) Many of these papers also provide quite different algorithms that compute the SVD with these bounds, where j is proportional to machine epsilon . These matrix classes include 1. bidiagonal matrices [18, 16, 25] 2. acyclic matrices [17] see below for a definition) 3. scaled diagonally dominant matrices [3] 4. well scalable symmetric positive definite matrices [19] and 5. certain well scalable symmetric indefinite matrices [57, 49, 48] Some of these results depended on the multiplicative perturbation ....

[Article contains additional citation context not shown here]

K. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

.... prefix in (1) Interesting linear algebra problems that can be cast in this way include tridiagonal Gaussian elimination, solving bidiagonal linear systems of equations, Sturm sequence evaluation for the symmetric tridiagonal eigenproblem, and the bidiagonal dqds algorithm for singular values [159]; we discuss some of these below. The numerical stability of these procedures remains open, although it is often good in practice [192] We now turn to the principle of locality. Since this is an issue many algorithms do not take into account, a number of so called shared memory machines have been ....

B. Parlett and V. Fernando. Accurate singular values and differential QD algorithms. Math Department PAM-554, University of California, Berkeley, CA, July 1992.

....without the iterative refinement step. All calculations were carried out in MATLAB on a Silicon Graphics Indigo workstation with IEEE standard (machine accuracy ffl 10 Gamma16 ) For computing the singular values of the computed bidiagonal we use the method due to Fernando and Parlett [6]. a. Implicit versus explicit. Let us consider the following products : A 1 [n; m] T m n 10 40 10 20 10 0 10 20 10 20 10 0 10 20 10 40 singular value a: Accuracy singular values. 10 40 10 20 10 0 10 20 10 20 10 15 10 10 10 5 10 0 singular value b: ....

K. V. Fernando and B. N. Parlett, Accurate singular values and differential qd algorithms. Numerische Mathematik 67, pp 191-229, 1994.

.... has been shown to yield very accurate results despite the fact that the singular values of such product can become very 16 large and very small as N tends to grow [8] The singular values of the computed bidiagonal are then computed to high relative accuracy using an appropriate technique [6]. This decomposition can e.g. be used to find the dominant directions of the state transition map Phi N;1 over a finite time interval [1; N ] In the case that the discrete time system (16) comes from a discretization of a nonlinear continuous time system it is known that the singular values of ....

K.V. Fernando and B.N. Parlett (1994), Accurate singular values and differential qd algorithms, Numerische Mathematik 67, pp. 191--229.

No context found.

K. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

....We achieve Property I by discarding this representation in favor of LDL t = T Gamma I for a suitable shift . Section 3 shows the necessity for this change of representation. Property II is then easily achieved by using bisection or, in the positive definite case, by the dqds algorithm, see [11]. Given a good factorization LDL t , and a highly accurate , we can think of satisfying Property III by using inverse iteration. While traditional inverse iteration often works well in practice, we employ an elegant alternative that uses a rank revealing twisted factorization of T Gamma I. ....

....our qdlike transformations in the face of roundoff errors. Before we do so, we emphasize that the particular qd like transformations presented in this section are new. Similar qd recurrences have been studied by Rutishauser [28] 30] and [31] Henrici [16] 17, Chapter 7] Fernando and Parlett [11], and Yao Yang [35] 4.3 Roundoff Error Analysis First, we introduce our model of arithmetic. We assume that the floating point result of a basic arithmetic operation ffi satisfies f l(x ffi y) x ffi y) 1 j) x ffi y) 1 ffi) where j and ffi depend on x, y, ffi, and the arithmetic ....

K. V. Fernando and B. N. Parlett, `Accurate singular values and differential qd algorithms '. Numer. Math., vol. 67, (March 1994), no. 2, pp. 191--229.

.... I is positive definite. 2. Compute the Cholesky Decomposition : T I = LL t : 3. Compute the singular values of L, denoted by 0 oe 1 oe 2 Delta Delta Delta oe n , to high relative accuracy (this may be done by a bisection algorithm or by the dqds algorithm of Fernando and Parlett [10]) 4. Group the eigenvalues of LL t , i = oe 2 i , as : a. isolated m is isolated if min(relgap( m ; m 1 ) relgap( m ; m Gamma1 ) min(10 Gamma3 ; 1=n) b. loose clusters m ; m 1 ; m l form a loose cluster if max(100; n)ffl min(relgap( i ; ....

K. Fernando and B. N. Parlett, Accurate singular values and the differential qd algorithms. Numerische Mathematik, 67:191-229, 1994.

....# (1) which are given by the singular values and the negated singular values of B. Alternatively, it is possible to consider the eigenvalues of BB t or B t B, which are given by the squared singular values of B. The last two approaches lead to many variants of the qd algorithms of Rutishauser [8], 17] 18] Fortunately it is not required to form the numerically undesired matrix products BB t and B t B explicitly to count the eigenvalues. We also show that eigenvalues of a real skew symmetric tridiagonal matrix can be counted by transforming that problem to an equivalent bidiagonal ....

....is a rather economical way to count eigenvalues of a skew symmetric tridiagonal matrix. It is well known that singular values of bidiagonal matrices can be computed to high relative precision if a suitable algorithm is used. See Demmel and Kahan [6] Barlow and Demmel [2] and Fernando and Parlett [8]. Because of this relationship between skew symmetric tridiagonal and bidiagonal matrices, the high relative accuracy is also inherited by skew symmetric tridiagonal matrices. This report is organised as follows. In Section 2, the notation is developed and the transformations between the ....

[Article contains additional citation context not shown here]

K. V. Fernando and B. N. Parlett. Accurate singular values and differential qd algorithms. Numer. Math., 67:191--229, 1994.

....be the first to use the k;k ( in eigenproblems. He introduced them in the context of differential qd algorithms for eigenvalues of tridiagonal matrices. See Section A2.3 of [31] Fernando and Parlett extended the results to differential qd algorithms for singular values of bidiagonal matrices [12] and SVD of triangular matrices [11] via the implicit Cholesky (LR) algorithm where the k;k ( appear as intermediary quantities denoted by d k . We study the nearly homogeneous system of equations defined by (F Gamma I)z( k) k;k ( e k ; z k = 1 where F is any square general matrix, ....

....) k = 1; n is O[n 3 ] for a general dense matrix. However, the complexity is less for structured matrices. As an example, if the matrix is tridiagonal, the cost becomes O[n] See the ICIAM 95 proceedings [10] and the technical report [9] The same count is true for bidiagonal matrices [12] as well as for diagonal matrices with rank one updates. For Toeplitz matrices, using the Trench algorithm, the operation count becomes O[n 2 ] which could be reduced to O[n] by using less numerically stable algorithms) Computational details are not discussed in this report; they will be ....

[Article contains additional citation context not shown here]

K. V. Fernando and B. N. Parlett. Accurate singular values and differential qd algorithms. Numer. Math., 67:191--229, 1994.

No context found.

K. V. Fernando and B. N. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

No context found.

K. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

No context found.

K. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Numerische Mathematik, 67:191--229, 1994.

No context found.

K. V. Fernando and B. Parlett. Accurate singular values and differential qd algorithms. Technical Report PAM-544, Center for Pure and Applied Mathematics, University of Calfornia, Berkeley, 1992.

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