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Computing Eigenvalues Rapidly and/or Accurately
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{\bf James Demmel }
%Collaborators:
Mathematics and Computer Science
University of California at Berkeley
demmel@cs.berkeley.edu
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We discuss recent algorithms for computing eigenvalues
of symmetric matrices and singular values of general matrices
with high speed and/or high accuracy. Unfortunately both goals
are not yet achievable simultaneously in all cases.
We begin with recent work, joint with M. Gu and H. Ren,
on a divide-and-conquer algorithm for the SVD. This is
currently the fastest serial algorithm available for
current superscalar architectures and memory hierarchies.
Second, we describe a parallel algorithm, due to B. Parlett
and I. Dhillon, for the symmetric tridiagonal eigenproblem.
While still under development, this algorithm, which is similar to
inverse iteration, promises to be highly accurate, guaranteed $O(n^2) $
in cost, and embarrassingly parallel. It could become the algorithm of
choice on serial as well as parallel machines.
Third, we discuss a family of high accuracy algorithms for the
singular value decomposition, which greatly extend the class
of matrices whose SVDs can be computed to high relative accuracy,
i.e. with the smallest singular values computed with as many
correct digits as the largest. This is joint work with M. Gu,
S. Eisenstat, I. Slapnicar, K. Veselic, and Z. Drmac.
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