Structure Total Least Squares Method For Multiple Right-hand Sides

Total least squares (TLS) technique is an efficient method for solving an overdetermined linear system Ax≈b, with possible errors in both A and b . This method has been applied to signal processing, system identification and parameter estimation problem, etc. In many of these applications, however, the matrix A or the augmented matrix [ A, b ] has a special structure, such as Toeplitz or sparse structure. Furthermore, errors have the same structure as A or [ A, b ]. The structured total least squares method is a generalization of the total least squares technique.In this paper, the structured total least squares technique is generalized to solve an overdetermined linear system with multiple right-hand sides AX≈B, with possible errors in both A and B. A new algorithm (called Structured Block Total Least Squares Method (SBTLS)) for computing the solution to the overdetermined linear system AX≈B is presented. The extended approach preserves any affine structure of A or [ A, B ], and minimizes a measure of the error in the p-norm, where p = 1,2,∞. The convergence of the algorithm and the optimality of computational results are analyzed for the 2-norm. It is shown that the SBTLS algorithm is equivalent to the Gauss-Newton method. We show how to modify the SBTLS algorithm so that it can treat the cases of B having a special structure or [ A, B ] having the Toeplitz structure. The results on Toeplitz structure apply to Hankel structure in a straightforward manner. Two numerical examples are reported to show the convergence and effectiveness of the developed algorithm.