Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations

Bunch, James R.; Parlett, Beresford Ν.

doi:10.1137/0708060

Cited by 359 publications

(219 citation statements)

References 7 publications

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“…Suppose an (n + m) × (n + m) permutation matrix P π is defined as in (6) and F = P T π AP π , where A is (n + m) × (n + m) transformed symmetric indefinite matrix in (5). Then there exists (n + m) × (n + m) nonsingular block lower triangular matrix L b with blocks of order 1 and 2 such that…”

Section: Existence Uniqueness and Sparsity The Blockmentioning

confidence: 99%

“…Suppose a saddle point matrixÅ ∈ R n×n is symmetrically transformed into A as in (5) such that the lower trapezoidal form B = [B 1 B 2 ] satisfies the condition: |b kk | ≥ |b ki |, 1 ≤ i ≤ n for each k = 1, . .…”

Section: Numerical Stabilitymentioning

confidence: 99%

“…The permutation matrix P is introduced for (i) pivoting dynamically and (ii) reducing the fill-ins in L ifÅ is sparse. The block diagonal pivoting strategies are mainly due to Bunch-Kaufman [4], Bunch-Parlett [5] and Bunch-Kaufman-Parlett (BKP) [6].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparse block factorization of saddle point matrices

Lungten

Schilders

Maubach

2016

Linear Algebra and its Applications

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The factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices. A variant of Gaussian elimination equivalent to the Cholesky's factorization is suggested and implemented for factorizing the saddle point matrices block-wise with small blocks of order 1 and 2. The Gaussian elimination applied to these small blocks on block level also induces a block 3 × 3 structured factorization of which the blocks have special properties. We compare the new block factorization with the Schilders' factorization in terms of the sparsity of their factors and computational efficiency. The factorization can be used as a direct method, and also anticipate for preconditioning techniques.

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Section: Existence Uniqueness and Sparsity The Blockmentioning

confidence: 99%

Section: Numerical Stabilitymentioning

confidence: 99%

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Sparse block factorization of saddle point matrices

Lungten

Schilders

Maubach

2016

Linear Algebra and its Applications

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show abstract

“…These methods involve different numbers of comparisons to find the pivot and have various stability properties. As for the LU factorization, the complete pivoting method (also called Bunch-Parlett algorithm [9]) is the most stable pivoting strategy. It guarantees a satisfying growth factor bound [14, p. 216] but also requires up to O(n 3 ) comparisons.…”

Section: Introductionmentioning

confidence: 99%

“…To ensure stability in solving such linear systems, the classical method used is called the diagonal pivoting method [9] where a block-LDL T factorization 5 is obtained such as P AP T = LDL T…”

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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1997

Interior Point Algorithms

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Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations

Cited by 359 publications

References 7 publications

Sparse block factorization of saddle point matrices

Sparse block factorization of saddle point matrices

Reducing the Amount of Pivoting in Symmetric Indefinite Systems

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