A note on Tikhonov regularization of large linear problems (Q1415410)

This short note completes the analysis of bounds of Tikhonov residual \(\varphi(\alpha)=\| b-Ax_\alpha\| _2^2\) given in the paper of \textit{D. Calvetti} and \textit{L. Reichel} [ibid. 43, No. 2, 263--283 (2003; Zbl 1038.65048)], where \(x_\alpha\) is a solution of Tikhonov regularization of the large linear problem \(Ax=b\). The lower and upper bounds for \(\varphi(\alpha)\) are given by the \(l\)-point Gauss and Gaus-Radau quadratures, respectively. The author proves that these bounds depend monotonically on \(l\), i.e. the upper bound is decreasing monotonically, while the lower bound is increasing monotonically with respect to \(l\). The last statement was proved by Calvetti and Reichel [loc. cit.] on the another way.