报告人：陈小山教授 华南师范大学
会议时间：2022年10月31日， 15:30-17:30 (GMT+08:00) 中国标准时间 - 北京

主持人：夏强教授

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摘  要：We introduce the two-sided Rayleigh quotient shift for the QR algorithm for non-Hermitian matrices. For the singly shifted case, the two-sided Rayleigh quotient iteration is incorporated in the QR iteration in an economic manner. A modified version of the method is proposed, where the two-sided Rayleigh quotient is computed directly from current upper Hessenberg matrix. Based on the observation that the Francis double-shift QR iteration is related to a 2D Grassmann-Rayleigh quotient iteration, we propose the doubly shifted QR algorithm with the two-sided 2D Grassmann-Rayleigh quotient double-shift. A modified version of the method is also proposed. For semi-simple eigenvalues, we claim that the QR algorithm with either the two-sided Rayleigh quotient shift or the 2D Grassmann-Rayleigh quotient double-shift has a cubic local convergence rate.Numerical examples are presented to support the claim. The proposed algorithms may be used for computing the Schur form of submatrices needed in the multishift QR algorithm with aggressive early deflation.

报告人简介：陈小山，华南师范大学数学科学学院教授，博导。主要从事数值线性代数及其应用方面的研究工作，主持一项国家自然科学基金和二项省自然科学基金。2012年获广东省自然科学技术奖二等奖（排名第三）。在《Numersche Mathematik》,《SIAM J. Matrix Anal. Appl.》，《Numer. Linear Albebra Appl.》,《BIT Numer. Math.》和《Linear Algebra Appl.》等国际知名期刊上发表论文40多篇。

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