Editing Approximation error (section)

== Generalizations ==
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The fundamental definitions of absolute and relative error, as presented primarily for scalar (one-dimensional) values, can be naturally and rigorously extended to more complex scenarios where the quantity of interest <math>v</math> and its corresponding approximation <math>v_{\text{approx}}</math> are [[Euclidean vector|''n''-dimensional vectors]], matrices, or, more generally, elements of a [[normed vector space]]. This important generalization is typically achieved by systematically replacing the [[absolute value]] function (which effectively measures magnitude or "size" for scalar numbers) with an appropriate [[norm (mathematics)|vector ''n''-norm]] or matrix norm. Common examples of such norms include the L<sub>1</sub> norm (sum of absolute component values), the L<sub>2</sub> norm (Euclidean norm, or square root of the sum of squared components), and the L<sub>∞</sub> norm (maximum absolute component value). These norms provide a way to quantify the "distance" or "difference" between the true vector (or matrix) and its approximation in a multi-dimensional space, thereby allowing for analogous definitions of absolute and relative error in these higher-dimensional contexts.<ref name="GOLUB_MAT_COMP2.2.3">{{cite book |last=Golub |first=Gene |title=Matrix Computations |edition=Third |author2=Charles F. Van Loan |publisher=The Johns Hopkins University Press |year=1996 |isbn=0-8018-5413-X |location=Baltimore |pages=53 |author-link=Gene H. Golub}}
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