Implementation of the Orthogonal QD Algorithm for Lower Tridiagonal Matrices
نویسندگان
چکیده
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)
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