I'm doing a Lanczos diagonalization of a large sparse matrix (~2 million elements). Almost all of the steps in the Lanzcos algorithm are done in parallel on the GPU, except for diagonalizing the Lanczos matrix to check for convergence. For that, I've been using the TQLI algorithm from Numerical Recipes. Are there methods for finding the eigensystem of a tridiagonal matrix that are parallel or easily parallelizable? Does a parallel version of TQLI exist?
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3 Answers
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I suggest using a library like SLEPc, which includes interfaces to many different methods for solving eigensystems in serial or parallel. The user manual includes references to several different methods for solving eigenvalue problems.
answered Mar 22, 2012 at 20:03
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$\begingroup$ Actually, no existing sparse eigensolvers use parallel linear algebra for the Rayleigh quotient. I wrote such an eigensolver this summer, but it is unfortunately closed source. $\endgroup$ Commented Aug 12, 2012 at 21:22
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TQL cannot be parallelized.
The standard parallel algorithm is that of Cuppen:
JJM Cuppen,
A divide and conquer method for the symmetric tridiagonal eigenproblem, 1980.
https://doi.org/10.1007/BF01396757
see also:
F. Tisseur,
A parallel divide and conquer algorithm for the symmetric eigenvalue problem on distributed memory architectures, 1999
http://eprints.ma.man.ac.uk/981/01/covered/MIMS_ep2007_225.pdf
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.4109&rep=rep1&type=pdf
http://www14.in.tum.de/konferenzen/Jass09/courses/2/Kleine_Albers_paper.pdf
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I recommend using Parallel Multiple Relatively Robust Representations (PMRRR) for the parallel tridiagonal eigensolve. It can compute all of the eigenpairs of the tridiagonal matrix in $O(n^2)$ work in parallel. You can find an overview of the method here. There is also the implementation in ScaLAPACK, which is discussed here.
