V. Hari and K. Veseli c, On Jacobi methods for singular value decompositions, SIAM Journal on Scienti c and Statistical Computing, 8 (1987), pp. 741-754.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Either twosided or one sided transformations can be used; in the former case, diagonal form is approached directly, whereas with one sided 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 ....

Vjeran Hari and Kresimar Veseli'c. On Jacobi methods for singular value decompositions. SIAM J. Sci. Stat. Comput., 8(5):741--754, 1987.

....[20] which can be considered as an extension of Jacobi s method for the computation of the eigenvalue decomposition of THE MAX ALGEBRAIC QRD AND THE MAX ALGEBRAIC SVD 15 a real symmetric matrix. We now state the main properties of this algorithm. The explanation below is mainly based on [4] and [17]. A Givens matrix is a square matrix of the form 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 0 0 0 0 0 1 0 0 0 0 . 0 0 cos( sin( 0 0 . 0 0 sin( cos( ....

....= n 1)n 2 iterations. Sweeps are repeated until S k becomes diagonal. If we have an upper triangular matrix at the beginning of a sweep then we shall have a lower triangular matrix after the sweep and vice versa. For triangular matrices the row cyclic Kogbetliantz algorithm is globally convergent [11, 17]. Furthermore, for triangular matrices the convergence of this algorithm is quadratic if k is large enough [2, 3, 15, 16, 31] 9K 2 N such that 8k K : kS k N k o kS k k 2 o (31) for some constant that does not depend on k, under the assumption that diagonal entries that correspond to ....

[Article contains additional citation context not shown here]

V. Hari and K. Veseli c, On Jacobi methods for singular value decompositions, SIAM Journal on Scientic and Statistical Computing, 8 (1987), pp. 741-754.

.... LR algorithm which usually has some nontrivial diagonalizing effect (the pivoting cares for the proper ordering) This effect will be more pronounced with growing (H 0 ) Quite analogous effects are present if a Jacobi SVD algorithm is preceded by the QR decomposition with column pivoting [10]. Our third result, which we state rather informally, is that the larger the range of numbers on the diagonal D 2 of H , the smaller is (A 1 ) this effect was also observed in [20] We argue as follows. Let L = DLA be the factor obtained from complete pivoting. Here, LA has rows of unit norm. ....

V. Hari and K. Veseli'c. On Jacobi methods for singular value decompositions. SIAM J. Sci. Stat. Comp., 8:741--754, 1987.

....these algorithms are slower than the Golub Reinsch algorithm on a serial computer. But the Jacobi like algorithms have been found very well suited for vectorized or parallel implementations, like vector and multi processors, systolic and other mesh connected computational (VLSI) arrays [33, 3, 23, 24, 35, 14, 36, 8, 1, 7]. These algorithms have, however, been proposed independently and most often with emphasis devoted to implementation on e.g. pure logical systolic arrays or a specific vector processor. Hence similarities and differences of the various algorithms are not clear. Also, due to the lack of any ....

V. Hari and K. Veselic. On Jacobi Methods for Singular Value Decomposition. SIAM Journal on Scientific and Statistical Computing, 8(5), 1987.

....[20] which can be considered as an extension of Jacobi s method for the computation of the eigenvalue decomposition of THE MAX ALGEBRAIC QRD AND THE MAX ALGEBRAIC SVD 15 a real symmetric matrix. We now state the main properties of this algorithm. The explanation below is mainly based on [4] and [17]. A Givens matrix is a square matrix of the form 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 Delta Delta Delta 0 Delta Delta Delta 0 Delta Delta Delta 0 0 0 1 Delta Delta Delta 0 Delta Delta Delta 0 Delta Delta Delta 0 0 . 0 0 Delta Delta ....

....1)n 2 iterations. Sweeps are repeated until S k becomes diagonal. If we have an upper triangular matrix at the beginning of a sweep then we shall have a lower triangular matrix after the sweep and vice versa. For triangular matrices the row cyclic Kogbetliantz algorithm is globally convergent [11, 17]. Furthermore, for triangular matrices the convergence of this algorithm is quadratic if k is large enough [2, 3, 15, 16, 31] 9K 2 N such that 8k K : kS k N k off fl kS k k 2 off (31) for some constant fl that does not depend on k, under the assumption that diagonal entries that correspond to ....

[Article contains additional citation context not shown here]

V. Hari and K. Veseli' c, On Jacobi methods for singular value decompositions, SIAM Journal on Scientific and Statistical Computing, 8 (1987), pp. 741--754.

....we would now perform a few sweeps of Kogbetliantz s SVD algorithm (without any QR updates ) and then again check the norm of the cross terms. Classical convergence results for Kogbetliantz s algorithm turn out to be useless in this respect. Linear convergence bounds are extremely conservative [5, 6, 11, 13], while ultimate quadratic convergence results do not apply to the initial convergence, where the off diagonal elements in Rs [k] can be very large [3, 19, 24] Jacobi type algorithms are considered extremely fast, but for the general case no estimates are available whatsoever as for the speed of ....

HARI V., VESELI ' C K., 1987. On Jacobi methods for singular value decompositions. SIAM. J. Sci. Stat. Comp., Vol. 8, No. 5, pp. 741-754.

....QR methods. Therefore, parallel implementations of these algorithms have been presented by several authors [1, 3, 16, 4] For the KA it is advantageous to execute a QR decomposition as a preparatory step and applying the KA to the resulting upper triangular matrix (called TKA in the sequel) [15, 14]. The complexity of the parallel implementations is mainly determined by the complexity of the rotation evaluations. Therefore, different strategies for modifying the rotation evaluations have been presented: M1 approximate rotations [16, 3] M2 factorized rotations [7, 18, 12] M3 CORDIC ....

....factorized approximate rotations [10, 9] M1 M3 CORDIC based approximate rotations [11] Which of these modified schemes yields the most efficient parallel implementation depends on the particular parallel processor. The global and the ultimate quadratic convergence of the JA [6, 19] and the TKA [14, 13] have been proved. In this paper we give the theorems and the outline of new proofs for the global and the ultimate quadratic convergence which hold for the JA as well as for the TKA and include the case of using approximate rotations (the proofs of the exact methods are obtained as special ....

[Article contains additional citation context not shown here]

V. Hari and K. Veselic. On Jacobi Methods for Singular Value Decomposition. SIAM J. Sci. Stat. Comput., 8:741-- 754, 1987.

....be of importance in practice. Veseli c [29] proved that the J Gammaorthogonal Jacobi method is globally convergent for the optimal strategy, threshold strategies, row cyclic strategy, and all other strategies which are equivalent to the row cyclic one (for example, the modulus parallel strategy [18]) He also proved a very interesting fact that all J Gammaorthogonal matrices V which satisfy (3.1.2) have the same condition number. Moreover, if V 1 and V 2 are two such matrices, then V 2 = V 1 U; U = U 1 0 0 U 2 # ; where U 1 , U 2 are othogonal matrices of order m, n Gamma m, ....

V. Hari, K. Veseli'c, On Jacobi methods for singular value decompositions, SIAM J. Sci. Stat. Comp., Vol. 8, (741--754) 1987.

....6,2 5,8 4,7 Processor 4: 4,5 3,4 2,3 8,2 7,8 6,7 5,6 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 This is clearly easiest to apply when we are only applying Jacobi rotations to columns of the matrix, rather than both rows and columns. Such a one sided Jacobi is natural when computing the SVD [98], but requires some preprocessing for the symmetric eigenproblem [51, 184] for example, in the symmetric positive definite case one can perform Cholesky on A to get A = LL T , apply one sided Jacobi on L or L T to get its (partial) SVD, and then square the singular values to get the ....

V. Hari and K. Veseli'c. On Jacobi methods for singular value decompositions. SIAM J. Sci. Stat. Comput., 8:741--754, 1987.

....is the eigenvalue decomposition (EVD) of A. The methods of choice for the fast parallel computation of these decompositions are Kogbetliantz s (SVD) and Jacobi s (EVD) algorithm [3, 23] since they exhibit a significantly higher degree of parallelism than the QR algorithm. It is well known [6, 20, 8, 5, 21], that it is advantageous to apply the Kogbetliantz algorithm (KA) to the triangular matrix R obtained from A by a preparatory QRdecomposition A = QR. This triangular Kogbetliantz algorithm (TKA) as well as the Jacobi algorithm (JA) only have to work with triangular matrices, since the upper ....

....and t Psi (3. 8) one has to distinguish between using case 1 if a (k) qq a (k) pp and using case 2 if a (k) pp a (k) qq in order to guarantee jd K j 1 (for the exact scheme using one case is sufficient, but the test a (k) qq a (k) pp is also necessary for a stable computation [20, 6]) Obviously, 3.5) is the same formula for computing t Phi with K1 as (2.5) for computing t ex with J (the same holds for (3.8) and K2 ) Therefore, all the approximations of Table 2.1 can also be applied to the TKA (i.e. to (3.5) and (3.8) For (3.6) and (3.9) no further approximations ....

[Article contains additional citation context not shown here]

V. Hari and K. Veseli' c, On Jacobi methods for singular value decomposition, SIAM J. Sci. Stat. Comput., 5 (1987), pp. 741--754.

No context found.

V. Hari and K. Veseli c, On Jacobi methods for singular value decompositions, SIAM Journal on Scienti c and Statistical Computing, 8 (1987), pp. 741-754.

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