2016
DOI: 10.1016/j.laa.2015.07.042
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Sparse block factorization of saddle point matrices

Abstract: 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 bl… Show more

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Cited by 4 publications
(8 citation statements)
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“…Combining the outer matrices here with the outer matrices in Equation yields the alternative, but equivalent, LTL T factorization scriptA=[]arrayarrayIarray0arrayLAarrayB2TB1TarrayL2arrayKarray0array0arrayB1L2[]arrayarrayDAarray0arrayIarray0arrayIarray0arrayIarray0array0T2[]arrayarrayIarrayB11B2array0array0arrayL2Tarray0arrayLATarrayKTarrayB1TL2, where K=X+B2TB1TLA. This factorization was recently proposed both for use as a direct method and as the basis of a preconditioner for an iterative method by Lungten et al We refer to it as Factorization 2 or the LSM factorization.…”
Section: Null‐space Methods As a Factorizationmentioning
confidence: 99%
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“…Combining the outer matrices here with the outer matrices in Equation yields the alternative, but equivalent, LTL T factorization scriptA=[]arrayarrayIarray0arrayLAarrayB2TB1TarrayL2arrayKarray0array0arrayB1L2[]arrayarrayDAarray0arrayIarray0arrayIarray0arrayIarray0array0T2[]arrayarrayIarrayB11B2array0array0arrayL2Tarray0arrayLATarrayKTarrayB1TL2, where K=X+B2TB1TLA. This factorization was recently proposed both for use as a direct method and as the basis of a preconditioner for an iterative method by Lungten et al We refer to it as Factorization 2 or the LSM factorization.…”
Section: Null‐space Methods As a Factorizationmentioning
confidence: 99%
“…Lungten et al focus on problems for which the nonsingular matrix B 1 is also upper triangular (or it is easy to transform the problem into this form). In this case, if Factorization 2 is formed via an modified Cholesky algorithm, they show that, for the dense case, it takes 13(n3m3)+12(n27m2)16(5n+m)+nm(nm+4) flops to factorize the saddle point matrix scriptA.…”
Section: Null‐space Methods As a Factorizationmentioning
confidence: 99%
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“…There are exactly m 2 × 2 and ( n − m ) 1 × 1 diagonal blocks K i i that form “a priori” pivots. Lungten et al showed that in the important and special case that B 1 is nonsingular and upper triangular and C =0, using this pivot sequence results in a stable factorization provided that the entries of B 1 satisfy the following condition: false|bkkfalse|false{false|bkjfalse|,3ptj=k+1,,nfalse},2emfor2ptk=1,,m. …”
Section: Bamd Orderingmentioning
confidence: 99%