# Ivan Slapničar (Univerza v Splitu): Forward stable eigenvalue decomposition of real symmetric arrowhead matrices and rank-one modifications of diagonal matrices

Date: 14. 4. 2017

Source: Numerical analysis seminar

Source: Numerical analysis seminar

Sreda, 19.4.2017, od 10h do 12h, soba 3.06 na Jadranski 21

Arrowhead matrices and rank-one modifications of diagonal matrices are closely related in terms of inverses. We present forward stable algorithms for solving an eigenvalue problem for real symmetric matrices of both types. The algorithms compute each eigenvalue and all components of the corresponding eigenvector in a forward stable manner to almost full accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes algorithms suitable for parallel computing and for cases when only part of the spectrum is required. We extend the results to singular value decomposition of half arrowhead matrices, Hermitian case, and computing zeros of polynomials with only real distinct roots. We also present several techniques for deflation. The software, written in Julia, is available at

https://github.com/ivanslapnicar/Arrowhead.jl.

This is joint work with Nevena Jakovčević Stor, University of Split, and Jesse L.\ Barlow, the Pennsylvania State University.