The matrix-restricted total least-squares problem

We present and study the matrix-restricted total least squares   (MRTLS) devised to solve linear systems of the form Ax≈bAx≈b where AA and bb are both subjected to noise and AA has errors of the form DECDEC. DD and CC are known matrices and EE is unknown. We show that the MRTLS problem amounts to solving a problem of minimizing a sum of fractional quadratic terms and a quadratic function and compare it to the related restricted TLS problem of Van Huffel and Zha [The restricted total least squares problem: formulation, algorithm, and properties, SIAM J. Matrix Anal. Appl. 12(2) (1991) 292–309.]. Finally, we present an algorithm for solving the MRTLS, which is based on a reduction to a single-variable minimization problem. This reduction is shown to have the ability of eliminating local optima points.