Relative Perturbation Techniques for Eigenvalue and Singular Value Decompositions

这个项目发展了摄动技术，用来推导奇异值和特征值的相对误差的界，以及关于相对间隙的奇异向量和特征向量的误差角的界。基于这些边界的收缩和收敛准则有可能导致计算特征值、奇异值和URV分解的高精度和高效率的QR型和Jacobi算法。在矩阵B的奇异值分解的背景下，研究了一类表示所有可能的不改变B的秩值的扰动。特别地，它包括双对角矩阵和双循环矩阵的分量相对扰动，以及消灭块三角矩阵中的外对角块的扰动。到目前为止，已经从这类扰动的一般结果中得到了双对角阵的奇异值和向量的许多现有的相对扰动界。同时，证明了双圈、三角和移位三角矩阵的奇异值和向量摄动的新结果。这些结果被推广到广义特征值问题，以及结构和非厄米特矩阵的特征值问题。

This project develops perturbation techniques for deriving bounds on the relative error of singular values and eigenvalues, as well as bounds on the error angles of singular vectors and eigenvectors in terms of a relative gap. Deflation and convergence criteria based on these bounds have the potential to lead to highly accurate and efficient QR-type and Jacobi algorithms for computing eigenvalue, singular value and URV decompositions. In the context of the singular value decomposition of a matrix B, a class of perturbations is investigated which represents all possible that do not change the rank of B. In particular, it includes componentwise relative perturbations of bidiagonal and biacyclic matrices, and perturbations that annihilate an offdiagonal block in a block triangular matrix. So far, many existing relative perturbations bounds have been derived for singular values and vectors of bidiagonal matrices from general results for this class of perturbations. Also, new perturbation results have been proved for singular values and vectors of biacyclic, triangular and shifted triangular matrices. These results are extended to the generalized eigenvalue problem, and to the eigenvalue problems for structured and non-Hermitian matrices.