10 The total least squares problem

Abstract

The total least squares (TLS) method is one of the several linear parameter estimation techniques that have been devised to compensate for data errors. One of the main reasons for its popularity is the availability of efficient and numerical robust algorithms in which the singular value decomposition plays a prominent role. Another reason is the fact that TLS is an application oriented procedure. It is ideally suited for situations in which all data are corrupted by noise, which is almost always the case in engineering applications. In this sense, it is a powerful extension of the classical least squares method, which corresponds only to a partial modification of the data. The chapter discusses the main principle of the TLS problem, the basic algorithm for solving the TLS problem by making substantial use of the singular value decomposition (SVD), the extensions of the basic TLS algorithm, some methods that enhances the efficiency of the algorithms, some iterative methods that are suitable for large sparse problems, and TLS applications. The solution to the TLS problem can be determined from the SVD of the matrix [A; b]. A simple algorithm outlines the computations of the solution of the basic TLS problem. By ‘basic” is meant that only one right-hand side vector b is considered (one-dimensional) and that the TLS problem is solvable (generic) and has a unique solution.