Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

Reliable estimates for the condition number of a large, sparse, real matrix A are important in many applications. To get an approximation for the condition number κ(A), an approximation for the smallest singular value is needed. Standard Krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value. Therefore, we study extended Krylov subspaces which tu...

L-curve Curvature Bounds via Lanczos Bidiagonalization

The L-curve is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. The location of the vertex of the L-curve typically yields a suitable value of the regularization parameter. However, the computation of the L-curve and of its curvature is quite costly ...

Extended Lanczos Bidiagonalization for Dimension Reduction in Information Retrieval

We describe an extended bidiagonalization scheme designed to compute low-rank approximations of very large data matrices. Its goal is identical to that of the truncated singular value decomposition, but it is significantly cheaper. It consists in an extension of the standard Lanczos bidiagonalization that improves its approximation capabilities, while keeping the computational cost reasonable. ...

Lanczos Bidiagonalization With Partial Reorthogonalization

A Lanczos bidiagonalization algorithm for Hankel matrices

This paper presents a fast algorithm for bidiagonalizing a Hankel matrix. An m×n Hankel matrix is reduced to a real bidiagonal matrix in O((m+ n)n log(m+ n)) floating-point operations (flops) using the Lanczos method with modified partial orthogonalization and reset schemes to improve its stability. Performance improvement is achieved by exploiting the Hankel structure, as fast Hankel matrix–ve...