Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

LAPACK Working Note 163: How the MRRR Algorithm Can Fail on Tight Eigenvalue Clusters

Beresford N. Parlett and Christof Voemel

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-04-1367
March 2005

http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/CSD-04-1367.pdf

During the last ten years, Dhillon and Parlett devised a new algorithm (Multiple Relatively Robust Representations, MRRR) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with O( n^2) cost. It has been incorporated into LAPACK version 3.0 as routine STEGR.

We have discovered that the MRRR algorithm can fail in extreme cases. Sometimes, eigenvalues agree to working accuracy and MRRR cannot compute orthogonal eigenvectors for them. In this paper, we describe and analyze these failures and various remedies. (Revised version)


BibTeX citation:

@techreport{Parlett:CSD-04-1367,
    Author = {Parlett, Beresford N. and Voemel, Christof},
    Title = {LAPACK Working Note 163: How the MRRR Algorithm Can Fail on Tight Eigenvalue Clusters},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {2005},
    Month = {Mar},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/6206.html},
    Number = {UCB/CSD-04-1367},
    Abstract = {During the last ten years, Dhillon and Parlett devised a new algorithm (Multiple Relatively Robust Representations, MRRR) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix <i>T</i> with <i>O</i>(<i>n</i>^2) cost. It has been incorporated into LAPACK version 3.0 as routine STEGR. <p> We have discovered that the MRRR algorithm can fail in extreme cases. Sometimes, eigenvalues agree to working accuracy and MRRR cannot compute orthogonal eigenvectors for them. In this paper, we describe and analyze these failures and various remedies. (Revised version)}
}

EndNote citation:

%0 Report
%A Parlett, Beresford N.
%A Voemel, Christof
%T LAPACK Working Note 163: How the MRRR Algorithm Can Fail on Tight Eigenvalue Clusters
%I EECS Department, University of California, Berkeley
%D 2005
%@ UCB/CSD-04-1367
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/6206.html
%F Parlett:CSD-04-1367