G. Auchmuty [Numer. Math. 61, No. 1, 1-6 (1992; Zbl 0747.65027)] derived the error estimate
for some approximation to the exact solution of the linear system with the regular system matrix and the right-hand side , where , , and denotes the residual. The unknown constant is contained in the interval , where
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.