This paper is the result of contrived efforts to break the barrier between numerical accuracy and run time efficiency in computing the fundamental decomposition of numerical linear algebra – the singular value decomposition (SVD) of a general dense matrix. It is an unfortunate fact that the numerically most accurate one–sided Jacobi SVD algorithm is several times slower than generally less accu...

The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but they are still challenged even for medium matrix sizes due to the difficulty of the problem. We propose a nov...

PURPOSE Developing a computationally efficient automated method for the optimal choice of regularization parameter in diffuse optical tomography. METHODS The least-squares QR (LSQR)-type method that uses Lanczos bidiagonalization is known to be computationally efficient in performing the reconstruction procedure in diffuse optical tomography. The same is effectively deployed via an optimizati...

The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but algorithmic research is ongoing and without production level software. We develop a high quality SVD software...

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization parameter for the projected problem by unbiased predictive risk estimation, generalized...

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization parameter for the projected problem by unbiased predictive risk estimation, generalized...

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. BL...

In this paper are presented current achievements and the state-of-the-art algorithms and implementations for dense linear algebra on traditional architectures such as single-core machines or distributed memory parallel machines. Also, this paper summarizes the current implementations and publicly available libraries for basic linear algebra for multi-core and many-core architectures (e.g. graph...

Recent progress in the serial one-sided Jacobi method is the consequence of two main ideas. The first one is that of preconditioning of an original matrix by one (two) QR (and LQ) decomposition(s) with column pivoting. Drmač and Veselić [1] have shown (experimentally and, to some degree, also theoretically) that such a preconditioning leads to a significant concentration of an off-diagonal matr...

The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but algorithmic research is ongoing and without production level software. We show that a more efficient, robust,...