Numerical Methods for Inverse Singular Value Problems

Two numerical methods one continuous and the other discrete are proposed for solving inverse singular value problems The rst method consists of solving an ordinary di erential equation obtained from an explicit calculation of the projected gradient of a certain objective function The second method generalizes an iterative process proposed originally by Friedland et al for solving inverse eigenvalue prob lems With the geometry understood from the rst method it is shown that the second method also the method proposed by Friedland et al for inverse eigenvalue problems is a variation of the Newton method While the continuous method is expected to converge globally at a slower rate in nding a stationary point of the objective function the discrete method is proved to converge locally at a quadratic rate if there is a solution Some numerical examples are presented Introduction For decades there has been considerable discussion about inverse eigenvalue prob lems Some theoretical results and computational methods can be found for example in the articles and the references contained therein Recently Friedland et al have surveyed four quadratically convergent numerical methods for the following inverse eigenvalue problem IEP Given real symmetric matrices A A An R n n and real numbers n nd values of c c cn T R such that the eigenvalues of the matrix A c A c A cnAn are precisely n In particular the so called Method III proposed in has been suggested to be a new method Also included in is a good collection of interesting applications where the IEP may arise In this paper we want to consider the inverse singular value problem a question very analogous to the IEP The problem is stated as follows ISVP Given real general matrices B B Bn R m n m n and non negative real numbers n nd values of c c cn T R such that the singular values of the matrix B c B c B cnBn are precisely n At the present time we do not know of any physical application of the ISVP But we think the problem is of interest in its own right Using the fact that the eigenvalues of the symmetric matrix A A are plus and minus of the singular values of the matrix A an ISVP can always be solved by conversion to an IEP On the other hand it can easily be argued by counterexamples that the IEP will not always have a solution Existence questions for the IEP therefore should be considered under more restricted condition The inverse Toeplitz eigenvalue problem ITEP for example is a special case of the IEP where A and Ak A k ij with A k ij if ji jj k otherwise Even though the ITEP is so specially structured the question of whether symmetric Toeplitz matrices can have arbitrary real eigenvalues is still an open problem for n Likewise the existence question for the ISVP might also be an interesting research topic As yet we have not been aware of any result in the literature The present paper is devoted to the numerical computation only The following notations will be used throughout the discussion O n stands for the manifold of all orthogonal matrices in R n ij R m n stands for the diagonal matrix in which ij i if i j n otherwise The setMs de ned by Ms fU V T jU O m V O n g obviously is equal to the set of all matrices in R n whose singular values are precisely n We use B to denote the a ne subspace B fB c jc Rg Clearly solving the ISVP is equivalent to nding an intersection of the two sets Ms and B In this paper we propose two di erent ways to nd such an intersection if it exists Our rst approach is motivated by a recent study of the projected gradient method The ISVP is cast as an equality constrained optimization problem in which the distance measured in the Frobenius norm betweenMs and B is minimized We show that the gradient of the distance function can be projected explicitly onto the feasible set without using Lagrange multipliers Consequently we are able to derive an ordinary di erential equation which characterizes a steepest descent ow for the distance function The steepest descent ow is easy to follow by using any available ODE software Our rst method for the ISVP is embedded in this continuous real ization process The formulation of the di erential system is presented in Section A similar approach for the IEP has already been discussed in Our second approach is simply a generalization of the so called Method III in Method III has been thought to be a new method In the course of trying to understand why Method III works we begin to realize based on the knowledge we learn from that Method III can be interpreted geometrically as a classical Newton method We emphasize the word classical because the geometry involved in Method III is closely related to that of the Newton method for one dimensional nonlinear equations This interpretation for the IEP will be explained in Section Once the geometry is understood Method III can easily be generalized to an iterative process for the ISVP Furthermore the method can be shown to converge quadratically The discussion of our second approach is presented in Section One other important result of Friedland et al is the modi cation of Method III so as to retain quadratic convergence when multiple eigenvalues are present We certainly can do similar modi cation in our method when multiple singular values are present This modi cation is described in Section The behavior of our modi ed method is expected to be similar to that in Indeed a proof of quadratic convergence can be established in the same spirit as in We shall not provide the proof in this paper The numerical examples reported in Section however should illustrate our results Both of the continuous approach and the iterative approach generate sequences of matrices in the manifoldMs But schematically the continuous approach evolves explicitly in the manifoldMs whereas the iterative approach is an implicit lifting of evolution in the a ne subspace B It is also worth noting that the continuous method converges globally but slowly whereas the iterative method converges quadratically but locally These features can therefore be taken advantage of in such a way that the continuous method is used rst to a low order of accuracy to help get into the domain of convergence of the discrete method which then will be turned on to improve the accuracy A Continuous Approach for ISVP In this section we shall solve the ISVP by minimizing the distance between Ms and B The distance is measured in term of the norm induced by the Frobenius inner product