We have developed algorithms to count singular values of a bidiagonalmatrix which are greater than a speci ed value This requires the transformation of the singular value problem to an equivalent symmetric eigenvalue problem The counting of sin gular values is paramount in the design of bisection and multisection type algorithms for computing singular values on serial and parallel machines The ...

In this paper we derive a new algorithm for constructing a unitary decomposition of a sequence of matrices in product or quotient form. The unitary decomposition requires only unitary left and right transformations on the individual matrices and amounts to computing the generalized singular value decomposition of the sequence. The proposed algorithm is related to the classical Golub–Kahan proce...

The symmetric Lanczos method is commonly applied to reduce large-scale symmetric linear discrete ill-posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill-posed problems in terms of ...

The Golub-Kahan-Lanczos algorithm with reorthogonalization (GKLR algorithm) is an algorithm for computing a subset of singular triplets for large-scale sparse matrices. The reorthogonalization tends to become a bottleneck of elapsed time, as the iteration number of the GKLR algorithm increases. In this paper, OpenMP-based parallel implementation of the classical Gram-Schmidt algorithm with reor...

In this paper we derive a new algorithm for constructing a uni-tary decomposition of a sequence of matrices in product or quotient form. The unitary decomposition requires only unitary left and right transformations on the individual matrices and amounts to computing the generalized singular value decomposition of the sequence. The proposed algorithm is related to the classical Golub-Kahan proc...

where U ∈ R is left orthogonal, V ∈ R is orthogonal, and B ∈ R is bidiagonal. When the Lanczos recurrence is implemented in finite precision arithmetic, the columns of U and V tend to lose orthogonality, making a reorthogonalization strategy necessary to preserve convergence of the singular values. A new strategy is proposed for recovering the left singular vectors. When using that strategy, it...

In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in application meshless method solving PDEs three-dimensional space using multiquadric radial basis functions. It is well known that truncated singular value decomposition (TSVD) most common effective solver ill-conditioned systems, but unfortunately operation count system with ...