We consider algorithms for going from a full'' matrix to a condensed band bidiagonal'' form using orthogonal transformations. We use the framework of algorithms by tiles''. Within this framework, we study: (i) the tiled bidiagonalization algorithm , which is a tiled version of the standard scalar bidiagonalization algorithm; and (ii) the R-bidiagonalization algorithm , which is a tiled version of the algorithm which consists in first performing the QR factorization of the initial matrix, then performing the band-bidiagonalization of the R-factor. For both bidiagonalization algorithms and , we use four main types of reduction trees, namely , , , and a newly introduced auto-adaptive tree, . We provide a study of critical path lengths for these tiled algorithms, which shows that (i) has a shorter critical path length than for tall and skinny matrices, and (ii) based schemes are much better than earlier proposed variants with unbounded resources. We provide experiments on a single multicore node, and on a few multicore nodes of a parallel distributed shared-memory system, to show the superiority of the new algorithms on a variety of matrix sizes, matrix shapes and core counts.
BD2VAL step. Much research has been done and is done on this kernel. Much software exists. In , to compute the singular values and optionally the singular vectors of a bidiagonal matrix, the routine xBDSQR uses the Golub-Kahan QR algorithm @cite_1 ; the routine xBDSDC uses the divide-and-conquer algorithm @cite_8 ; and the routine xBDSVX uses bisection and inverse iteration algorithm. Recent research was trying to apply the MRRR (Multiple Relatively Robust Representations) method @cite_18 to the problem.
@cite_18 @cite_1 @cite_8
2058443643 2005423095 2169378749
We describe the design and implementation of a new algorithm for computing the singular value decomposition (SVD) of a real bidiagonal matrix. This algorithm uses ideas developed by Grosser and Lang that extend Parlett’s and Dhillon’s multiple relatively robust representations (MRRR) algorithm for the tridiagonal symmetric eigenproblem. One key feature of our new implementation is that @math singular triplets can be computed using only @math storage units and floating point operations, where @math is the dimension of the matrix. The algorithm will be made available as routine xBDSCR in the upcoming new release of the LAPACK library. A numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form @math , thus exhibiting A’s singular values on @math ’s diagonal. The scheme first transforms A to a bidiagonal matrix J, then diagonalizes J. The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods. Some applications are mentioned, in particular the use of the pseudo-inverse @math to solve least squares problems in a way which dampens spurious oscillation and cancellation. The authors present a stable and efficient divide-and-conquer algorithm for computing the singular value decomposition (SVD) of a lower bidiagonal matrix. Previous divide-and-conquer algorithms all suffer from a potential loss of orthogonality among the computed singular vectors unless extended precision arithmetic is used. A generalization that computes the SVD of a lower banded matrix is also presented.