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A study of Auchmuty’s error estimate. (English) Zbl 0985.65039

G. Auchmuty [Numer. Math. 61, No. 1, 1-6 (1992; Zbl 0747.65027)] derived the error estimate

x-x * p =cr(x) 2 2 A T r(x) q -1

for some approximation x n to the exact solution x * n of the linear system Ay=b with the regular system matrix A n×n and the right-hand side b n , where 1p, 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

C p (A)=supA T z q A -1 z p z 2 -1 ·

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.

MSC:
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
65H10Systems of nonlinear equations (numerical methods)