Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Anthony P. Austin, Lloyd N. Trefethen
2015 SIAM Journal on Scientific Computing  
Powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals; best known are the works of Sakurai and coauthors and Polizzi and coauthors. Even if the matrices are real symmetric, most such methods rely on complex arithmetic, leading to expensive linear systems to solve. An appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation
more » ... nal interpolation problems, for which there is no need to leave the real axis. Investigation shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective. This article presents a technique of this kind that is related to previous work published in Japanese by Murakami.
doi:10.1137/140984129 fatcat:3wzaiwq7ibeuzfm7vheokfgfby