A study of Auchmuty's error estimate (Q5948885)

\textit{G. Auchmuty} [Numer. Math. 61, No. 1, 1-6 (1992; Zbl 0747.65027)] derived the error estimate NEWLINE\[NEWLINE\|x- x^*\|_p= c\|r(x)\|^2_2 \|A^T r(x)\|^{-1}_qNEWLINE\]NEWLINE for some approximation \(x\in \mathbb{R}^n\) to the exact solution \(x^*\in \mathbb{R}^n\) of the linear system \(Ay= b\) with the regular system matrix \(A\in \mathbb{R}^{n\times n}\) and the right-hand side \(b\in \mathbb{R}^n\), where \(1\leq p\leq\infty\), \(p^{-1}+ q^{-1}= 1\), and \(r(x)= Ax- b\) denotes the residual. The unknown constant \(c\) is contained in the interval \([1, C_p(A)]\), where NEWLINE\[NEWLINEC_p(A)= \sup\|A^T z\|_q \|A^{-1}z\|_p \|z\|^{-1}_2.NEWLINE\]NEWLINE The author gives a new derivation of Auchmuty's estimate, provides a geometrical interpretation, makes some kind of probabilistical analysis, generalize it to nonlinear systems, and concludes with numerical testing.