Accurate Factorization and Eigenvalue Algorithms for Symmetric DSTU and TSC Matrices
Accurate Factorization and Eigenvalue Algorithms for Symmetric DSTU and TSC Matrices
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DOI：
10.1137/050631537
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发表时间：
2006-12
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影响因子：
--
通讯作者：
M. J. Peláez;J. Moro
中科院分区：
文献类型：
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作者：
M. J. Peláez;J. Moro
本文提出了两种算法，它们以较小的分量相对误差计算属于两类矩阵的对称矩阵的块$LDL^T$分解：对角标度全幺模（DSTU）矩阵和全符号复合（TSC）矩阵。这种精度是通过利用这种结构的一些特殊性质来实现的，以避免在整个因式分解过程中进行减法。一旦一个准确的块$LDL^T$分解是可用的，它被证明可以很容易地获得一个准确的对称秩揭示分解，这是算法的起点计算具有高相对精度的特征值和特征向量的任意，可能是不定的，对称矩阵。这证明了对称DSTU和TSC矩阵的特征值和特征向量可以以较高的相对精度计算。
Two algorithms are presented which compute, with small componentwise relative error, a block $LDL^T$ factorization of symmetric matrices belonging to two classes of matrices: diagonally scaled totally unimodular (DSTU) and total signed compound (TSC) matrices. This accuracy is achieved by taking advantage of some special properties of such structures in order to avoid subtractions throughout the factorization process. Once an accurate block $LDL^T$ decomposition is available, it is proved that one can easily obtain an accurate symmetric rank-revealing decomposition, which is the starting point for algorithms computing with high relative accuracy the eigenvalues and eigenvectors of arbitrary, possibly indefinite, symmetric matrices. This proves that eigenvalues and eigenvectors of symmetric DSTU and TSC matrices can be computed with high relative accuracy.