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Partial Factorization of a Dense Symmetric Indefinite Matrix

机译:密集对称不定矩阵的部分分解

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At the heart of a frontal or multifrontal solver for the solution of sparse symmetric sets of linear equations, there is the need to partially factorize dense matrices (the frontal matrices) and to be able to use their factorizations in subsequent forward and backward substitutions. For a large problem, packing (holding only the lower or upper triangular part) is important to save memory. It has long been recognized that blocking is the key to efficiency and this has become particularly relevant on modern hardware. For stability in the indefinite case, the use of interchanges and 2x2 pivots as well as 1 x 1 pivots is equally well established. In this article, the challenge of using these three ideas (packing, blocking, and pivoting) together is addressed to achieve stable factorizations of large real-world symmetric indefinite problems with good execution speed. The ideas are not restricted to frontal and multifrontal solvers and are applicable whenever partial or complete factorizations of dense symmetric indefinite matrices are needed.
机译:在求解稀疏对称方程组的正面或多面求解器的核心,需要部分分解密集矩阵(正面矩阵),并能够在后续的前向和后向替换中使用它们的因式分解。对于大问题,打包(仅保留下部或上部三角形部分)对于节省内存很重要。长期以来,人们一直认识到阻塞是提高效率的关键,而这在现代硬件上已变得尤为重要。为了在不确定的情况下保持稳定,同样可以确定使用互换和2x2枢轴以及1 x 1枢轴。在本文中,解决了将这三个概念(打包,阻塞和旋转)一起使用所面临的挑战,以实现具有良好执行速度的大型现实世界对称不定问题的稳定分解。这些思想不仅限于正面求解器和多正面求解器,并且在需要密集对称不定矩阵的部分或全部分解时适用。

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