Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

Abstract

New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration.An implementation of the new algorithms is presented and experimental data for runtime behaviour and numerical accuracy based on a wide range of test cases are summarized. Compared with competing state-of-the-art tridiagonalization-based methods, the algorithms proposed here show strong reductions in runtime, especially for very large matrices and/or small bandwidths. The residuals of the computed eigenvectors are in general comparable with state-of-the-art methods. In some cases, especially for strongly clustered eigenvalues, a loss in orthogonality of some eigenvectors is observed. This is not surprising, and future work will focus on investigating ways for improving these cases.

Comparison of eigensolvers for symmetric band matrices

“…Five different methods and their implementations are compared in this paper: (i) the Lapack routine L/DSBEVD, (ii) the routine L/DSBEVR which is based on the MRRR algorithm [9] and composed of building blocks from Lapack , (iii) the routine P/DSBEVD which is an adaptation of the Plasma routine P/DSYEVD for band matrices and composed of building blocks from Plasma , (iv) the BD&C algorithm [3,4] , and (v) the BTF method [10] . In the following sections, these five methods and their theoretical properties are briefly reviewed.…”

“…From now on, we will denote methods or routines from Lapack [5] or composed of building blocks from Lapack with the prefix “L/”, and methods or routines composed of building blocks from Plasma [6] with the prefix “P/”. More specifically, we compare L/DSBEVD from Lapack , which first tridiagonalizes the given symmetric band matrix, then applies the tridiagonal divide-and-conquer method [7,8] , and finally transforms the eigenvectors back; L/DSBEVR, which also first tridiagonalizes the given symmetric band matrix, then applies the MRRR algorithm [9] , and finally also transforms the eigenvectors back; P/DSBEVD, which consists of the same basic algorithmic steps as L/DSBEVD, but uses Plasma routines as building blocks; the block divide-and-conquer (BD&C) method [3,4] , which directly computes eigenvalues and eigenvectors of a symmetric band matrix without tridiagonalizing it as a whole; and the BTF method [10] , which tridiagonalizes the given symmetric band matrix only for computing the eigenvalues, but computes the eigenvectors directly from the band matrix and thus avoids constructing or applying the corresponding similarity transformation. …”

“…the BTF method [10] , which tridiagonalizes the given symmetric band matrix only for computing the eigenvalues, but computes the eigenvectors directly from the band matrix and thus avoids constructing or applying the corresponding similarity transformation.…”

“…Every band matrix can be represented as a block tridiagonal matrix with upper and lower triangular off-diagonal blocks. The BTF method [10] tridiagonalizes the band matrix only for computing its eigenvalues, and it factorizes the shifted band (or block tridiagonal) matrix in order to compute its eigenvectors by an inverse iteration process. In [10] is has been shown how a very good starting vector can be determined from the block twisted factorizations of the shifted band matrix, and in general only very few iterations are needed for an accurate eigenvector approximation.…”

Numerical study on thermal buckling of CFRP–Al honeycomb sandwich composites based on homogenization–localization analysis

“…(20). Subsequently, the corresponding buckling shapes, represented by the eigenvector δD, are calculated by employing inverse iteration method [30,31] as follows…”