Premium
On the weighting method for mixed least squares–total least squares problems
Author(s) -
Liu Qiaohua,
Wang Minghui
Publication year - 2017
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2094
Subject(s) - qr decomposition , mathematics , cholesky decomposition , total least squares , least squares function approximation , singular value decomposition , weighting , rayleigh quotient iteration , linear least squares , rayleigh quotient , generalized least squares , non linear least squares , iteratively reweighted least squares , factorization , mathematical optimization , algorithm , iterative method , statistics , estimation theory , power iteration , eigenvalues and eigenvectors , medicine , physics , quantum mechanics , estimator , radiology
Summary It is well known that the standard algorithm for the mixed least squares–total least squares (MTLS) problem uses the QR factorization to reduce the original problem into a standard total least squares problem with smaller size, which can be solved based on the singular value decomposition (SVD). In this paper, the MTLS problem is proven to be closely related to a weighted total least squares problem with its error‐free columns multiplied by a large weighting factor. A criterion for choosing the weighting factor is given; and for the sake of stability in solving the MTLS problem, the Cholesky factorization‐based inverse (Cho‐INV) iteration and Rayleigh quotient iteration are also considered. For large‐scale MTLS problems, numerical tests show that Cho‐INV is superior to the standard QR‐SVD method, especially for the case with big gap between the desired and undesired singular values and the case when the coefficient matrix has much more error‐contaminated columns. Rayleigh quotient iteration behaves more efficient than QR‐SVD for most cases and fails occasionally, and in some cases, it converges much faster than Cho‐INV but still less efficient due to its higher computation cost.