Title：The two sided Arnoldi algorithm to get left and right eigenvectors of a nonsymmetric matrix

Speaker：Prof. Axel Ruhe （Kungliga Tekniska högskolan，KTH）

Date：16:00-17:00 pm, November 3, 2014

Place：Conference Room A304, Department of Mathematical Sciences

Abstract：Numerical Analysis group, Department of Mathematics, School of Science,

KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden At the Matrix Pencil Conference in Pitea 1982, I described a two sided Arnoldi algorithm to compute approximative left and right eigenvectors, y0 and x, of a large nonsymmetric matrix B, [5]. After that, I moved to Goteborg and forgot all about left eigenvectors, until I did send the paper [6] by Wu, Wei, Jia, Ling and Zhang to print for BIT, and noted that it contained a perturbation analysis for the same nonsymmetric eigenproblem, in the spirit of the Kahan, Parlett and Jiang paper [2]. In the talk, I will show how to use a two sided Arnoldi to get approximations, whose accuracy can be assessed with the theory in [6]. The Arnoldi algorithm is standard software, see [3]. The crucial step is a Gram Schmidt biorthogonalization, described in [5], applied after running Arnoldi on first B and then B. We will compare this to using the two sided nonsymmetric Lanczos algorithm as described in [4, 1] . I hope to report results on both some test matrices and a large scale computation on a matrix from a fluid dynamics application.

Profile of speaker：Axel Ruhe is a professor emeritus of Numerical Analysis at the department of Mathematics at KTH, the Royal Institute of Technology, Stockholm, Sweden. He was born in Jonkoping, Sweden, 1942, graduated with a Ph D from Lund University 1970, was appointed full professor at Umea University during 1975-1983. From 1983 to 2001 he was professor of Numerical Analysis in the School of Mathematical and Computing Sciences at the Chalmers University of Technology and the University of Goteborg. He acted as a dean of that school 1987-1993.

He is the editor in chief of BIT, Numerical Mathematics, since 2003. He has served as

editor for several journals since 1969. His works deal primarily with numerical computation of eigenvalues for matrices, first numerical treatment of matrices with very ill- conditioned eigenproblems and composite Jordan normal form, then large symmetric pencils arising from mechanical vibration problems, the Spectral Transformation Lanczos algorithm. His current works concern Rational

Krylov algorithms, and the use of eigenvalue methods to compute reduced order models of electrical and mechanical linear dynamic systems. He has also developed algorithms to predict singularities and bifurcations of large nonlinear systems. Some works deal with theoretical aspects on matrix eigenvalues, finding the closest normal matrix, and algorithms for nonlinear parameter estimation, fitting a positive sum of exponentials to a series of data. A recent interest is Krylov sequence algorithms for Information Retrieval.

Contact：Zhongxiao Jia