Tridiagonal matrix algorithm

Abstract: It has been shown that a nonsingular symmetric tridiagonal linear system of the form Tx = b can be solved in a backward-stable manner using diagonal pivoting meth- ods, where the LBL T decomposition of T is computed, i.e., T = LBL T , where L is unit lower triangular and B is block diagonal with 1 1 and 2 2 blocks. In this paper, we generalize these methods for solving unsymmetric tridiagonal matrices. We present three algorithms that compute the LBM T factorization, where L and M are unit lower triangular and B is block diago- nal with 1 1 and 2 2 blocks. These algorithms are normwise backward stable and reduce to the LBL T factorization when applied to symmetric matrices. We demonstrate the robustness of the algorithms for the unsymmetric case using a wide range of well-conditioned and ill-conditioned linear systems. Numerical results suggest that these algorithms are comparable to Gaussian elimination with partial pivoting (GEPP). However, unlike GEPP, these algorithms do not require row interchanges, and thus, may be used in applications where row interchanges are not possible. In addition, substantial computational savings can be achieved by carefully managing the nonzero elements of the factors L, B, and M.

Abstract: The present invention provides an eigenvalue decomposition apparatus that can perform processing in parallel at high speed and high accuracy. The eigenvalue decomposition apparatus comprises a matrix dividing portion 14 that repeatedly divides a symmetric tridiagonal matrix T into two symmetric tridiagonal matrices, an eigenvalue decomposition portion 15 that performs eigenvalue decomposition on the symmetric tridiagonal matrix after the division, an eigenvalue computing portion 17 that repeatedly computes eigenvalues of the symmetric tridiagonal matrix that is the division origin and matrix elements of the symmetric tridiagonal matrix that is the division origin, based on eigenvalues and matrix elements obtained by eigenvalue decomposition performed by the eigenvalue decomposition portion 15, the matrix elements being part of elements of orthogonal matrices constituted by eigenvectors, until an eigenvalue of the symmetric tridiagonal matrix T is computed, and an eigenvector computing portion 19 that computes an eigenvector of the symmetric tridiagonal matrix T based on the symmetric tridiagonal matrix T and the eigenvalue thereof using twisted factorization.