On the Lanczos and Golub-Kahan reduction methods applied to discrete ill-posed problems

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 eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. Similarly, we show that the solution subspace determined by the LSQR method when applied to the solution of linear discrete ill-posed problems with a nonsymmetric matrix often can be used instead of the solution subspace determined by the singular value decomposition without significant, if any, reduction of the quality of the computed solution.