Difference between revisions of "Discrepancy of an approximation"
From Encyclopedia of Mathematics
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| − | One of the characteristics of the quality of an approximate solution | + | {{TEX|done}} |
| + | One of the characteristics of the quality of an approximate solution $\overline u$ of an operator equation $P(u)=0$ (e.g. a linear algebraic system, a differential equation). The discrepancy is defined as the quantity $P(\overline u)$ or a norm of this quantity, e.g., $\|P(\overline u)\|_2$. If the estimate | ||
| − | + | $$\|u_1-u_2\|_1\leq C\|P(u_1)-P(u_2)\|_2$$ | |
holds, then the error of the solution may be estimated in terms of the discrepancy: | holds, then the error of the solution may be estimated in terms of the discrepancy: | ||
| − | + | $$\|\overline u-u\|_1\leq C\|P(\overline u)\|_2.$$ | |
If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution. | If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution. | ||
Latest revision as of 08:54, 25 November 2018
One of the characteristics of the quality of an approximate solution $\overline u$ of an operator equation $P(u)=0$ (e.g. a linear algebraic system, a differential equation). The discrepancy is defined as the quantity $P(\overline u)$ or a norm of this quantity, e.g., $\|P(\overline u)\|_2$. If the estimate
$$\|u_1-u_2\|_1\leq C\|P(u_1)-P(u_2)\|_2$$
holds, then the error of the solution may be estimated in terms of the discrepancy:
$$\|\overline u-u\|_1\leq C\|P(\overline u)\|_2.$$
If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution.
References
| [1] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian) |
| [2] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |
How to Cite This Entry:
Discrepancy of an approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrepancy_of_an_approximation&oldid=14211
Discrepancy of an approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrepancy_of_an_approximation&oldid=14211
This article was adapted from an original article by N.S. Bakhvalov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article