Accurate Symmetric Indefinite Linear Equation Solvers

Ashcraft, Cleve; Grimes, Roger G.; Lewis, John G.

doi:10.1137/s0895479896296921

Cited by 120 publications

(163 citation statements)

References 21 publications

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“…The potential instability stems from replacing a symmetric rank-2 update by two rank-1 updates, via the use of an eigendecomposition, a change that is enough to upset the rather delicate stability of the partial pivoting strategy. This instability has not been detected by the LAPACK test software and so far has been observed only on examples specially constructed by Ashcraft et al (1995). The problem is corrected in LAPACK 3.0.…”

Section: Lapack's Symmetric Indefinite Factorizationmentioning

confidence: 94%

“…LAPACK includes an implementation of the symmetric indefinite factorization with the partial pivoting strategy of Bunch & Kaufman (1977) (routine xSYTRF). An unusual feature of the partial pivoting strategy, whose implications for stability were first pointed out by Ashcraft, Grimes & Lewis (1995) and Higham (1995a), is that L ∞ / A ∞ can be arbitrarily large, even though the factorization itself is backward stable.…”

Section: Lapack's Symmetric Indefinite Factorizationmentioning

confidence: 99%

“…The implementation of the symmetric indefinite factorization with partial pivoting in LAPACK 2.0 can be unstable when L ∞ is large, as pointed out and explained by Ashcraft et al (1995). The potential instability stems from replacing a symmetric rank-2 update by two rank-1 updates, via the use of an eigendecomposition, a change that is enough to upset the rather delicate stability of the partial pivoting strategy.…”

Section: Lapack's Symmetric Indefinite Factorizationmentioning

confidence: 99%

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Testing linear algebra software

Higham

1997

Quality of Numerical Software

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How can we test the correctness of a computer implementation of an algorithm such as Gaussian elimination, or the QR algorithm for the eigenproblem? This is an important question for program libraries such as LAPACK, that are designed to run on a wide range of systems. We discuss testing based on verifying known backward or forward error properties of the algorithms, with particular reference to the test software in LAPACK. Issues considered include the choice of bound to verify, computation of the backward error, and choice of test matrices. Some examples of bugs in widely used linear algebra software are described.

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Section: Lapack's Symmetric Indefinite Factorizationmentioning

confidence: 94%

Section: Lapack's Symmetric Indefinite Factorizationmentioning

confidence: 99%

Section: Lapack's Symmetric Indefinite Factorizationmentioning

confidence: 99%

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Testing linear algebra software

Higham

1997

Quality of Numerical Software

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“…In the case that pivoting is required to ensure existence of an LDL decomposition and/or numerical stability the state-of-the-art is considered to be the use of the Bunch-Kaufmann algorithm. The latter is a pivoting strategy working with 1 × 1 and 2 × 2 blocks which retains the symmetry of the problem -see elsewhere [3,4,7] for details and further references. Note, however, that for the real version of (2.27) pivoting is not required to ensure existence, since the matrix is quasidefinite [4,42].…”

Section: Efficient Solution Of the Discretised Piezoelectric Pdesmentioning

confidence: 99%

PDE based determination of piezoelectric material tensors

Kaltenbacher¹,

Lahmer²,

Mohr³

et al. 2006

Eur. J. Appl. Math

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The exact numerical simulation of piezoelectric transducers requires the knowledge of all material tensors that occur in the piezoelectric constitutive relations. To account for mechanical, dielectric and piezoelectric losses, the material parameters are assumed to be complex. The issue of material tensor identification is formulated as an inverse problem: As input measured impedance values for different frequency points are used, the searched-for output is the complete set of material parameters. Hence, the forward operator F mapping from the set of parameters to the set of measurements, involves solutions of the system of partial differential equations arising from application of Newton's and Gauss' law to the piezoelectric constitutive relations. This, via two or three dimensional finite element discretisation, leads to an indefinite system of equations for solving the forward problem. Well-posedness of the infinite dimensional forward problem is proven and efficient solution strategies for its discretized version are presented. Since unique solvability of the inverse problem may hardly be verified, the system of equations we have to solve for recovering the material tensor entries can be rank deficient and therefore requires application of appropriate regularisation strategies. Consequently, inversion of the (nonlinear) parameter-to-measurement map F is performed using regularised versions of Newton's method. Numerical results for different piezoelectric specimens conclude this paper.

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“…The well-known partial pivoting method, based on the Bunch-Kaufman algorithm [8], is implemented in LAPACK [1] and requires at each step of the factorization the exploration of two columns, resulting in a total of O(n 2 ) comparisons. This algorithm has good stability properties [14, p. 219] but in certain cases L may be unbounded, which is a cause for possible instability [3], leading to a modified algorithm referred to as rook pivoting or bounded Bunch-Kaufman pivoting. The latter involves between O(n 2 ) and O(n 3 ) comparisons depending on the number of 2 × 2 pivots.…”

Section: Introductionmentioning

confidence: 99%

Reducing the Amount of Pivoting in Symmetric Indefinite Systems

Becker

Baboulin

Dongarra

2012

Parallel Processing and Applied Mathematics

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Abstract. This paper illustrates how the communication due to pivoting in the solution of symmetric indefinite linear systems can be reduced by considering innovative approaches that are different from pivoting strategies implemented in current linear algebra libraries. First a tiled algorithm where pivoting is performed within a tile is described and then an alternative to pivoting is proposed. The latter considers a symmetric randomization of the original matrix using the so-called recursive butterfly matrices. In numerical experiments, the accuracy of tile-wise pivoting and of the randomization approach is compared with the accuracy of the Bunch-Kaufman algorithm.

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Accurate Symmetric Indefinite Linear Equation Solvers

Cited by 120 publications

References 21 publications

Testing linear algebra software

Testing linear algebra software

PDE based determination of piezoelectric material tensors

Reducing the Amount of Pivoting in Symmetric Indefinite Systems

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