Abstract
For large-scale problems, how to establish an algorithm with high accuracy and stability is particularly important. In this paper, the Householder bidiagonalization total least squares (HBITLS) algorithm and nonlinear iterative partial least squares for total least squares (NIPALS-TLS) algorithm were established, by which the same approximate TLS solutions was obtained. In addition, the propagation of the roundoff error for the process of the HBITLS algorithm was analyzed, and the mixed forward-backward stability of these two algorithms was proved. Furthermore, an upper bound of roundoff error was derived, which presents a more detailed and clearer approximation of the computed solution.
Highlights
We find that the Householder bidiagonalization total least squares (HBITLS) and NIPALS-total least squares (TLS) algorithms compute the same approximate solutions for the TLS problems
In many practical problems, the stop criterion can be safely selected on the basis of the rounding error analysis of the original problem, thereby diminishing the need for an extremely precise approximation of the algebraic problem solution [4]
As far as we know, the roundoff error analysis of the approximation TLS solutions obtained by using the Householder bidiagonalization procedure was not systematically performed in the literature
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The orthonormal properties of the Krylov basis strongly support the use of these Householder matrix-based algorithms This is true when we need to be sure that the perturbed problem we are solving has to conserve some spectral similarity properties. We find that the HBITLS and NIPALS-TLS algorithms compute the same approximate solutions for the TLS problems. When it comes to practical problems, the arithmetic will be inaccurate and there will be errors in each step of the calculation. As far as we know, the roundoff error analysis of the approximation TLS solutions obtained by using the Householder bidiagonalization procedure was not systematically performed in the literature.
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