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Keywords:
truncated total least squares; multiple right-hand sides; eigenvalues of rank-\$d\$ update; ill-posed problem; regularization; filter factors
Summary:
The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems \$Ax\approx b\$ were analyzed by Fierro, Golub, Hansen, and O'Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of \$A\$ applied to \$b\$. This paper focuses on the situation when multiple observations \$b_1,\ldots ,b_d\$ are available, i.e., the T-TLS method is applied to the problem \$AX\approx B\$, where \$B=[b_1,\ldots ,b_d]\$ is a matrix. It is proved that the filtering representation of the T-TLS solution can be generalized to this case. The corresponding filter factors are explicitly derived.
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