Concept

Erreur d'approximation

The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur for a variety of reasons, among them a computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm). In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates the extent to which errors in the input of the algorithm will lead to large errors of the output; numerically stable algorithms to not yield a significant error in output when the input is malformed and vice versa. Given some value v and its approximation vapprox, the absolute error is where the vertical bars denote the absolute value. If the relative error is and the percent error (an expression of the relative error) is An error bound is an upper limit on the relative or absolute size of an approximation error. These definitions can be extended to the case when and are n-dimensional vectors, by replacing the absolute value with an n-norm. As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002 = 0.2%. As a practical example, when measuring a 6 mL beaker, the value read was 5 mL. The correct reading being 6 mL, this means the percent error in that particular situation is, rounded, 16.7%. The relative error is often used to compare approximations of numbers of widely differing size; for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3; in the first case the relative error is 0.003 while in the second it is only 0.000003. There are two features of relative error that should be kept in mind.

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https://fr.wikipedia.org/wiki/Erreur_d'approximation

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https://fr.wikipedia.org/wiki/Erreur_d'approximation

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Concepts associés (3)

Erreur d'arrondi

Une erreur d'arrondi est la différence entre la valeur approchée calculée d'un nombre et sa valeur mathématique exacte. Des erreurs d'arrondi naissent généralement lorsque des nombres exacts sont représentés dans un système incapable de les exprimer exactement. Les erreurs d'arrondi se propagent au cours des calculs avec des valeurs approchées ce qui peut augmenter l'erreur du résultat final. Dans le système décimal des erreurs d'arrondi sont engendrées, lorsqu'avec une troncature, un grand nombre (peut-être une infinité) de décimales ne sont pas prises en considération.

Stabilité numérique

En analyse numérique, une branche des mathématiques, la stabilité numérique est une propriété globale d’un algorithme numérique, une qualité nécessaire pour espérer obtenir des résultats ayant du sens. Une définition rigoureuse de la stabilité dépend du contexte. Elle se réfère à la propagation des erreurs au cours des étapes du calcul, à la capacité de l’algorithme de ne pas trop amplifier d’éventuels écarts, à la précision des résultats obtenus. Le concept de stabilité ne se limite pas aux erreurs d’arrondis et à leurs conséquences.

Chiffre significatif

Le nombre de chiffres significatifs indique la précision d'une mesure physique. Il s'agit des chiffres connus avec certitude ou compris dans un intervalle d'incertitude. La précision (ou l'incertitude) avec laquelle on connaît la valeur d'une grandeur dépend du mesurage (ensemble d'opérations ayant pour but de déterminer la valeur d'une grandeur). Exemple : a cinq chiffres significatifs. Le premier chiffre incertain est le 5.

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Exercices associés (2)

Saut du saumon

The equation for uniformly accelerated rectilinear motion alone an x axis is $\ddot{x} = a ,$ where a is a constant. Show that its general solution is $x(t) = \frac{1}{2}at^2 + v_0t + x_0$ for any v₀ and x₀. Interpret these constants.
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Write the blocks’ total mechanical energy, considering that the thread has constant length.
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Notebooks associés (5)

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Numerical Precision in DFT Computation[1]¶

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In [1]:

first our usual bookkeeping

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                                      [2]Paolo Prandoni, [3]LCAV, EPFL

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                                      [2]Paolo Prandoni, [3]LCAV, EPFL