Balancing sparse matrices for computing eigenvalues

Applying a permuted diagonal similarity transform DPAP T D −1 to a matrix A before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed. This is often called balancing. This paper describes several balancing algorithms for sparse matrices and compare...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 309; no. 1; pp. 261 - 287
Main Authors: Chen, Tzu-Yi, Demmel, James W.
Format: Journal Article
Language:English
Published: Elsevier Inc 15.04.2000
Subjects:
Accurate eigenvalues
Balancing
Norm minimization
Sparse matrix algorithms
Accurate eigenvalues
65F15
Balancing
Norm minimization
65F35
Sparse matrix algorithms
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:Applying a permuted diagonal similarity transform DPAP T D −1 to a matrix A before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed. This is often called balancing. This paper describes several balancing algorithms for sparse matrices and compares them against each other and the traditional dense algorithm. We first discuss our sparse implementation of the dense algorithm; our code is faster than the dense algorithm when the density of the matrix is no more than approximately .5, and is much faster for large, sparse matrices. We next describe a set of randomized balancing algorithms for matrices that are not given explicitly, i.e. given a vector x, we can compute only Ax and perhaps A T x. We motivate these Krylov-based algorithms using Perron–Frobenius theory. Results are given comparing the Krylov-based algorithms to each other and to the sparse and dense direct balancing algorithms, looking at norm reduction, running times, and the accuracy of eigenvalues computed after a matrix is balanced. We conclude that sparse balancing algorithms are efficient preconditioners for eigensolvers.
ISSN:0024-3795
1873-1856
DOI:10.1016/S0024-3795(00)00014-8