Abstract Randomized methods can be competitive for the solution of problems with a large matrix low rank. They also have been applied successfully to large-scale linear discrete ill-posed by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails computation an approximation partial singular value decomposition A that is numerical The present paper compares random...

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair . We make numerical analysis the underlying JBD process establish relationships between it two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on results analysis, we investigate convergence approximate show that, u...

Many popular solution methods for large discrete ill-posed problems are based on Tikhonov regularization and compute a partial Lanczos bidiagonalization of the matrix. The computational effort required by these methods is not reduced significantly when the matrix of the discrete ill-posed problem, rather than being a general nonsymmetric matrix, is symmetric and possibly indefinite. This paper ...

Computing a small number of singular values is required in many practical applications and it is therefore desirable to have efficient and robust methods that can generate such truncated singular value decompositions. A new method based on the Lanczos bidiagonalization and the Krylov-Schur method is presented. It is shown how deflation strategies can be easily implemented in this method and pos...

We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation of the svd of B using an existing subroutine library such as lapack provides the singular values of A. The singular vectors of A are obtained trivially from ...

Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as...

In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of a large matrix A. Our method combines the advantages of a block routine, implicit shifting, and the application of Leja points as shifts in the accelerating polynomial. The method neither requir...

We describe efficient implementations of the Subtractive Optimally Localized Averages (SOLA) mollifier method for solving linear inverse problems in, e.g., inverse helioseismology. We show that the SOLA method can be regarded as a constrained least squares problem, which can be solved by means of standard “building blocks” from numerical linear algebra. We compare the standard implementation of...