In this paper, we study the algebraic properties for the solutions of equality constrained least squares problem (LSE) minfesⅡKf-gⅡ2 with S={F:||fSKfg∈−LF-H||fSKfg∈−2=MINYEXN||fSKfg∈−Ly-h||fSKfg∈−2}and corresponding weighted least squares problem (WLS), in which L and may not be of full rank. General formulas of the solutions are given, the algebraic relations between the LSE and the WLS solutions are obtained.

This paper presents an analysis of the solutions of the total least squares problem (TLS) AX\approx B in cases where the matrix (A,B) may have multiple smallest singular values. General formulas for the minimum norm TLS solutions are given; the difference between the TLS and the LS solutions is obtained; the error bounds for the perturbed TLS solutions with or without minimal length are deduced. The analysis is useful especially for rank deficient problems and generalizes the results of Golub and Van Loan, Van Huffel and Vandewalle and Zoltowski. Numerical results for a practical application are also given to verify the error bounds.

This paper presents perturbation theory for the equality constrained least squares problem (LSE) minⅡfESⅡKF-gⅡ−gⅡ2 with S={f:ⅡLf-hⅡ2=minⅡLy-hⅡ2}, in which L and may not be of full rank. The analysis generalizes the results of Elden. Perturbation theorems for the LSE problem are proved. Numerical experiments are also given to verify the error bounds.

This paper studies the algebraic properties and perturbation theory of the generalized total least squares problem (GTLS) AX≈B, in which is free of error, and the error contained in is of the form EC with C a given nonsingular matrix. The problem was proposed by Van Huffel and Vandewalle. The solvability conditions, formulas for the GTLS solutions, their residuals, and the minimum norm correction matrices are obtained, and a perturbation theory for the GTLS problem is given.

In this paper, a novel numerical algorithm is described for computing the scattering frequencies of a bounded, cylindrically symmetric potential with compact support in R^3. The potential can be time independent, or it can depend periodically on time. The algorithm is based on a characterization of the scattering frequencies as the exponents in the long-time asymptotic expansion of the solution of the initial value problem for the time-dependent scalar wave equation perturbed by the potential. It uses a discretization of the initial value problem for the time-dependent wave equation perturbed by the potential combined with a procedure for extracting the exponents from the computed solution. A number of numerical computations are also presented to document the performance and reliability of the algorithm.