On Lanczos’ Algorithm for Tridiagonalizing Matrices

TL;DR: Several new algorithms are defined that substantially mitigate the problem of breakdown in the Lanczos method, andumerical comparisons of the new algorithms and the standard algorithms are given.

Abstract: The Lanczos or biconjugate gradient method is often an effective means for solving nonsymmetric systems of linear equations. However, the method sometimes experiences breakdown, a near division by zero which may hinder or preclude convergence. In this paper we present some theoretical results on the nature and likelihood of the phenomenon of breakdown. We also define several new algorithms that substantially mitigate the problem of breakdown. Numerical comparisons of the new algorithms and the standard algorithms are given.

TL;DR: It is concluded that the tridiagonal reduction method used in FEER would serve as a valuable addition to NASTRAN for highly increased efficiency in obtaining structural vibration modes.

TL;DR: It will be shown that there exists a vector x such that the algorithm starting from xι = jι = x can be continued so that "unlucky" case may not occur, and one of the initial vectors can be chosen arbitrarily to avoid this case.

TL;DR: In this article, a review of the essential points of Lanczos's orthogonalization procedure is presented, which is of great importance for the determination of the eigenvalues of a real, but otherwise general matrix.

TL;DR: In this article, the eigenvalues of an arbitrary matrix with complex elements were found in two steps: first, the matrix A was reduced to a tri-diagonal (i.e., Jacobi) matrix J; and second, the coefficients of J were computed.