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Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations

机译:低阶扰动下拟准Hermitian矩阵的有效特征值计算

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In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N.
机译:在本文中,我们解决了有效计算大N×N Hermitian矩阵的所有特征值的问题,该矩阵由低秩的可能的非Hermitian扰动修正。通过对矩阵进行相似度的初步转换以转换成上Hessenberg形式,可以大大简化先前提出的QR算法的快速适应性。可以通过一小组参数指定转换后的矩阵,这些参数可以在QR过程中轻松更新。生成的结构化QR迭代可以使用线性内存存储在线性时间内执行。此外,它被证明是向后稳定的。数值实验表明,对于较小的N值,该新算法的性能优于Hessenberg QR算法的可用实现。

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