Evaluation of small elements of the eigenvectors of certain symmetric tridiagonal matrices with high relative accuracy

Abstract

Evaluation of the eigenvectors of symmetric tridiagonal matrices is one of the most basic tasks in numerical linear algebra. It is a widely known fact that, in the case of well separated eigenvalues, the eigenvectors can be evaluated with high relative accuracy. Nevertheless, in general, each coordinate of the eigenvector is evaluated with only high absolute accuracy. In particular, those coordinates whose magnitude is below the machine precision are not expected to be evaluated with any accuracy whatsoever.It turns out that, under certain conditions, frequently ecountered in applications, small (e.g. 10 −50 ) coordinates of eigenvectors of symmetric tridiagonal matrices can be evaluated with high relative accuracy. In this paper, we investigate such conditions, carry out the analysis, and describe the resulting numerical schemes. While our schemes can be viewed as a modification of already existing (and well known) numerical algorithms, the related error analysis appears to be new. Our results are illustrated via several numerical examples.

On generalized prolate spheroidal functions

“…For example, when using double-precision arithmetic, a component of ãn of magnitude 10 −100 will be computed in absolute precision to 116 digits. This fact is proved in a more general setting in [14].…”

Fast botnet detection from streaming logs using online lanczos method

“…The more crucial point here for our purpose is that the eigenvalues and eigenvectors of 𝐓 𝑘 can estimate those of 𝐑 . The eigenvalues and eigenvectors of a symmetric tridiagonal matrix is a basic task in numerical linear algebra, [16,21,[29][30][31][32]. They take advantage of the special form of symmetric tridiagonal matrices to run faster and more accurate than algorithms for general matrices.…”

On the Efficient Evaluation of the Azimuthal Fourier Components of the Green's Function for Helmholtz's Equation in Cylindrical Coordinates

“…Note for κ > 10 6 , the resource requirements of prior methods becomes excessive. 5.78×10 −3 secs -1.52×10 −1 secs -10 15 5.72×10 −3 secs -1.52×10 −1 secs -10 18 5.69×10 −3 secs -1.52×10 −1 secs -Table 8: The evaluation of the modal Green's function in quadruple precision for varying κ with small source-to-target distance (β − = 10 −12 ). The error is evaluated by using adaptive Gaussian quadrature as the gold standard.…”

“…Thus, a classical Miller-type algorithm cannot be applied. However, it was recently observed in [18] that if a recurrence relation is represented as a banded matrix, then the inverse power method can be used to find a solution, even when the stability behavior is mixed in the sense just described. We thus apply the inverse power method, as described in [18], to the resulting five-diagonal matrix corresponding to Matviyenko's recurrence relation.…”

“…However, it was recently observed in [18] that if a recurrence relation is represented as a banded matrix, then the inverse power method can be used to find a solution, even when the stability behavior is mixed in the sense just described. We thus apply the inverse power method, as described in [18], to the resulting five-diagonal matrix corresponding to Matviyenko's recurrence relation. In this fashion, we obtain all the eigenvectors corresponding to the zero eigenvalue; only one vector in this eigenspace corresponds to the vector of modal Green's functions.…”