Implicit standard Jacobi gives high relative accuracy

Dopico, Froilán M.; Koev, Plamen; Molera, Juan M.

doi:10.1007/s00211-009-0240-8

Cited by 28 publications

(40 citation statements)

References 44 publications

(120 reference statements)

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“…In this section, we will use an approach completely different to the one in [46] to show that a relative perturbation bound similar to (4.1) holds for the eigenvalues of symmetric indefinite diagonally dominant matrices. This approach is inspired in the algorithm presented in [15] for computing, with high relative accuracy, the eigenvalues of any symmetric matrix expressed as a rank-revealing decomposition, and by its error analysis. The key point is that in the case of symmetric diagonally dominant matrices the strong perturbation bounds given in Theorem 2.4 for the LDL T factorization can be expressed as a small multiplicative perturbation of the original matrix, which allows us to apply the eigenvalue relative perturbation results developed in [17].…”

Section: Bounds For Eigenvalues Of Symmetric Matricesmentioning

confidence: 99%

“…Furthermore, this algorithm can be combined with the algorithms presented in [11] to compute the singular values with relative errors in the order of machine precision. In fact, the algorithm for the LDU factorization in [45] can be combined also with the algorithms in [7,15,16] to compute, with high relative accuracy, solutions to linear systems and solutions to least squares problems involving diagonally dominant matrices, and eigenvalues of symmetric diagonally dominant matrices.…”

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confidence: 99%

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Relative Perturbation Theory for Diagonally Dominant Matrices

Dailey¹,

Dopico²,

Ye³

2014

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

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Section: Bounds For Eigenvalues Of Symmetric Matricesmentioning

confidence: 99%

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confidence: 99%

Relative Perturbation Theory for Diagonally Dominant Matrices

Dailey¹,

Dopico²,

Ye³

2014

SIAM J. Matrix Anal. & Appl.

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“…Throughout this proof L, D and U denote the exact (1) ). Note first that according to Lemma 8, one can consider that A is obtained from A by applying a sequence of n + 1 perturbations that can be: (1) of type (15) with δ = u (recall (19) and (20)); or, (2) perturbations that are reversals of type (15) …”

Section: Lemma 9 Use the Same Notation And Assumptions As In Lemma 8mentioning

confidence: 99%

“…The fundamental point in (3) is that the error bound is governed by the condition numbers of the well conditioned factors X and Y , and not by κ(A) which may be extremely large. Symmetric RRDs have been also used to compute accurate eigenvalues and eigenvectors of symmetric matrices [21,18,19]. Very recently [20], it has been shown that computing an accurate RRD, in the sense of (1)- (2), of an n × n matrix A also leads to very important benefits in the accuracy of the numerical solution of the system Ax = b.…”

Section: Introductionmentioning

confidence: 99%

Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

Dopico

Koev

2011

Numer. Math.

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We present a structured perturbation theory for the LDU factorization of (row) diagonally dominant matrices and we use this theory to prove that a recent algorithm of Ye (Math Comp 77(264):2195-2230, 2008 computes the L , D and U factors of these matrices with relative errors less than 14n 3 u, where u is the unit roundoff and n × n is the size of the matrix. The relative errors for D are componentwise and for L and U are normwise with respect the "max norm" A M = max i j |a i j |. These error bounds guarantee that for any diagonally dominant matrix A we can compute accurately its singular value decomposition and the solution of the linear system Ax = b for most vectors b, independently of the magnitude of the traditional condition number of A and in O(n 3 ) flops. Mathematics Subject Classification (2000)

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“…Here, U is a unitary matrix of order m, while V is J-unitary, (i.e., V * JV = J) of order n. The HSVD in (2.10) can be computed by orthogonalization of, either the of columns G * by trigonometric rotations [12], or the columns of G by hyperbolic rotations [41].…”

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A Hierarchically Blocked Jacobi SVD Algorithm for Single and Multiple Graphics Processing Units

Novaković¹

2015

SIAM J. Sci. Comput.

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Abstract. We present a hierarchically blocked one-sided Jacobi algorithm for the singular value decomposition (SVD), targeting both single and multiple graphics processing units (GPUs). The blocking structure reflects the levels of GPU's memory hierarchy. The algorithm may outperform MAGMA's dgesvd, while retaining high relative accuracy. To this end, we developed a family of parallel pivot strategies on GPU's shared address space, but applicable also to inter-GPU communication. Unlike common hybrid approaches, our algorithm in a single GPU setting needs a CPU for the controlling purposes only, while utilizing GPU's resources to the fullest extent permitted by the hardware. When required by the problem size, the algorithm, in principle, scales to an arbitrary number of GPU nodes. The scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a Tesla S2050 system with four GPUs vs. a single Fermi card. Key words. Jacobi (H)SVD, parallel pivot strategies, graphics processing units AMS subject classifications. 65Y05, 65Y10, 65F151. Introduction. Graphics processing units have become a widely accepted tool of parallel scientific computing, but many of the established algorithms still need to be redesigned with massive parallelism in mind. Instead of multiple CPU cores, which are fully capable of simultaneously processing different operations, GPUs are essentially limited to many concurrent instructions of the same kind-a paradigm known as SIMT (single-instruction, multiple-threads) parallelism.SIMT type of parallelism is not the only reason for the redesign. Modern CPU algorithms rely on (mostly automatic) multi-level cache management for speedup. GPUs instead offer a complex memory hierarchy, with different access speeds and patterns, and both automatically and programmatically managed caches. Even more so than in the CPU world, a (less) careful hardware-adapted blocking of a GPU algorithm is the key technique by which considerable speedups are gained (or lost).After the introductory paper [29], here we present a family of the full block [21] and the block-oriented [20] one-sided Jacobi-type algorithm variants for the ordinary (SVD) and the hyperbolic singular value decomposition (HSVD) of a matrix, targeting both a single and the multiple GPUs. The blocking of our algorithm follows the levels of the GPU memory hierarchy; namely, the innermost level of blocking tries to maximize the amount of computation done inside the fastest (and smallest) memory of the registers and manual caches. The GPU's global RAM and caches are considered by the mid-level, while inter-GPU communication and synchronization are among the issues addressed by the outermost level of blocking.At each blocking level an instance of either the block-oriented or the full block Jacobi (H)SVD is run, orthogonalizing pivot columns or block-columns by conceptually the same algorithm at the lower level. Thus, the overall structure of the algorithm is hierarchical (or recursive) in nature, and ready to fit not only the cur...

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Implicit standard Jacobi gives high relative accuracy

Cited by 28 publications

References 44 publications

Relative Perturbation Theory for Diagonally Dominant Matrices

Relative Perturbation Theory for Diagonally Dominant Matrices

Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

A Hierarchically Blocked Jacobi SVD Algorithm for Single and Multiple Graphics Processing Units

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