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Computation of Eigenvectors on Concurrent Processors
Author(s) -
CHANG H. Y.,
UTKU S.
Publication year - 1989
Publication title -
computer‐aided civil and infrastructure engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.773
H-Index - 82
eISSN - 1467-8667
pISSN - 1093-9687
DOI - 10.1111/j.1467-8667.1989.tb00018.x
Subject(s) - tridiagonal matrix , eigenvalues and eigenvectors , matrix (chemical analysis) , computation , mathematics , tridiagonal matrix algorithm , transformation (genetics) , symmetric matrix , diagonal matrix , qr decomposition , diagonal , algorithm , geometry , physics , chemistry , chromatography , quantum mechanics , gene , biochemistry
Suppose a real symmetric general matrix A is reduced into a real symmetric tridiagonal matrix C by either Householder's method or Givens method. Eigenvalues of the tridiagonal matrix are the eigenvalues of the original matrix since they are similar. After eigenvalues are calculated, the eigenvectors of the original matrix may be obtained by the eigenvectors of the tridiagonal matrix and the transformation matrices used in either Householder's or Givens tridiagonalization methods [1]. In these methods, however, transformation matrices usually are not stored due to memory considerations. Only critical information such as rotation angles in Givens method is saved. Generally, the recovery of all transformation matrices involves more computation than the tridiagonalization procedure. Therefore such a scheme is rarely used. In this paper, a serial eigenvector solution procedure, assuming eigenvalues are known, is first described. This procedure applies Gauss elimination with diagonal pivoting on the characteristic equations. A parallel procedure is then developed by using scattered column decomposition. The efficiency and speedup are given in terms of problem and machine parameters. The efficiency of the parallel algorithm increases as the number of scattered columns assigned to each processor increases, and decreases as communication cost between processors increases.