An Inverse Singular Value Problem For Upper Triangular Matrices

An inverse eigenvalue(singular value)problem aims to reconstruct a structured matrix from the prescribed eigenvalues(singular values).Inverse eigenvalue problems arise in many applications such as structured design,vibrating systems,control theory,numerical algebra,exploration and remote sensing,geophysics,circuit theory,and finite element model updating etc.Large non-Hermitian linear systems come from various application fields in modern science and engineering.Currently,there exist many iter-ative methods for solving such linear systems,e.g.,various Krylov subspace methods.In particular,the GMRES method proposed by Saad and Schultz in 1986 is one of the most important Krylov subspace methods for solving large non-Hermitian linear systems.In the last decades,the convergence of the GMRES method has been widely studied.So far it has been determined that the convergence of the GMRES method is independent of Ritz values and harmonic Ritz values generated by Arnoldi process.This thesis is concerned with the relationship between the convergence of the GMRES method and the singular values of a series of upper Hessenberg matrices generated by Arnoldi process.To do so,in this thesis,we propose an inverse singular value problem for upper Hessenberg matrices,which is reduced to an inverse singular value problem for upper triangular matrices.In this thesis,we give a constructive proof for the solvability of the inverse singular value problem for upper triangular matrices.Finally,we give the corresponding constructive algorithm and show its effectiveness via numerical examples.The theoretical result of this thesis may shed light on whether the convergence of the GMRES method is dependent on the singular values of a series of upper Hessenberg matrices generated by Arnoldi process.