Tridiagonal matrix algorithm

Abstract: The main results of a componentwise error analysis for a parallel partitioning algorithm [4] in the cases of banded and tridiagonal linear systems are presented. It is shown that for some special classes of matrices, i.e. diagonally dominant (row or column), symmetric positive definite, and M-matrices, the algorithm is numericaly stable.

Abstract: The alternating-direction-implicit finite-difference time-domain method (ADI-FDTD) is considered as a very efficient algorithm. The key problem of the implementation of the ADI-FDTD method is to solve the tridiagonal matrix system. Two numerical algorithms for the tridiagonal matrix system are analyzed in detail in this paper. The interpretations for the stability of the algorithms are also given clearly

Abstract: The orthogonal qd algorithm with shifts (oqds algorithm), proposed by von Matt, is an algorithm for computing the singular values of bidiagonal matrices. This algorithm is accurate in terms of relative error, and it is also applicable to general triangular matrices. In particular, for lower tridiagonal matrices, BLAS Level 2.5 routines are available in preprocessing stage for this algorithm. BLAS Level 2.5 routines are faster than BLAS Level 2 routines widely used in preprocessing for bidiagonalization. Generally, it takes O(n3) operations to reduce a full n-by-n matrix to a band matrix such as bidiagonal or lower tridiagonal matrix. On the other hand, computing the singular values of a bidiagonal or lower tridiagonal matrices takes only O(n2) operations. Consequently, if we have an algorithm for computing the singular values of lower tridiagonal matrices, we can expect that the total computation time including preprocessing to obtain the singular values is reduced. In this paper, we consider the oqds algorithm for lower tridiagonal matrices. We propose a shift strategy for lower tridiagonal matrices to accelerate convergence and derive criteria for deflation or splitting. (This paper is submitted to PDPTA’13)