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Zbl 0985.65039
Galántai, A.
A study of Auchmuty's error estimate. (English)
[J] Comput. Math. Appl. 42, No.8-9, 1093-1102 (2001). ISSN 0898-1221

{\it G. Auchmuty} [Numer. Math. 61, No. 1, 1-6 (1992; Zbl 0747.65027)] derived the error estimate $$\|x- x^*\|_p= c\|r(x)\|^2_2 \|A^T r(x)\|^{-1}_q$$ for some approximation $x\in \bbfR^n$ to the exact solution $x^*\in \bbfR^n$ of the linear system $Ay= b$ with the regular system matrix $A\in \bbfR^{n\times n}$ and the right-hand side $b\in \bbfR^n$, where $1\le p\le\infty$, $p^{-1}+ q^{-1}= 1$, and $r(x)= Ax- b$ denotes the residual. The unknown constant $c$ is contained in the interval $[1, C_p(A)]$, where $$C_p(A)= \sup\|A^T z\|_q \|A^{-1}z\|_p \|z\|^{-1}_2.$$ 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.
[Ulrich Langer (Linz)]

MSC 2000:: *65F35 Matrix norms, etc. (numerical linear algebra)
65H10 Systems of nonlinear equations (numerical methods)

Keywords: a posteriori error estimates; probabilistic analysis; numerical examples; linear system; nonlinear systems

Citations: Zbl 0747.65027

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