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

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