Numerical Differentiation Error Propagation
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Error Propagation Formula Physics
question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign error propagation calculus up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Error in a numerical derivative up vote 1 down vote favorite 1 I have a graph of error propagation calculator data, say temperature ($T$) vs time($t$), I know the error bounds in each $\Delta T$. The range of t is from 0 $\to$ 1600 s, with small steps say 0.001 s. If I numerically take the derivative at some point on the graph, what is the error associated with that value. The data is not suitable to be fit with a function or curve. derivatives data-analysis error-propagation share|cite|improve this question edited Nov 11 '12 at 11:11 asked Nov 11 '12 at 10:59
Using Differentials To Estimate Error
John Echo 72111 For which times $t$ do you have corresponding values of $T$? And what do you mean when you say you know the error -- you mean you have some bound on the error? I doubt you know the error exactly. –littleO Nov 11 '12 at 11:06 yes I have error bounds, error bars on the graph. –John Echo Nov 11 '12 at 11:10 Ah. There are different methods for computing derivatives numerically. Do you know which method you want to use? –littleO Nov 11 '12 at 11:14 I'm using OriginLab8 and I think it takes the slope corresponding to 2-3 points. –John Echo Nov 11 '12 at 11:30 add a comment| 1 Answer 1 active oldest votes up vote 2 down vote accepted Here's a formula from Numerical Analysis by Burden and Faires (chapter 4.1). \begin{align*} f'(x_0) &= \frac{f(x_0 + h) - f(x_0 - h)}{2h} - \frac{h^2}{6} f^{(3)}(\xi_0). \end{align*} Notice that if the third derivative of $f$ is huge, the error might be huge. There are other formulas for numerically computing derivatives, and they have similar expressions for the error. Here's one more example: \begin{equation} f'(x_0) = \frac{1}{12h}\left[ f(x_0 - 2h) - 8f(x_0 - h) + 8f(x_0 + h) - f(x_0 + 2h) \right] + \frac{h^4}{30} f^{(5)}(\xi). \end{equation} If the fifth derivative of $f$ is huge, the error might be huge. share|cite|improve this answer answered Nov 11 '12 at 11:33 littleO 16.9k3
propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument percent error calculus precision) which propagate to the combination of variables in the function. The uncertainty u can how to find log error in physics be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be defined by
Error Propagation Calculus Examples
the relative error (Δx)/x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, the positive square root of variance, σ2. The value of http://math.stackexchange.com/questions/234787/error-in-a-numerical-derivative a quantity and its error are then expressed as an interval x ± u. If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are ± one standard deviation from the value, https://en.wikipedia.org/wiki/Propagation_of_uncertainty that is, there is approximately a 68% probability that the true value lies in the region x ± σ. If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.[1] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Caveats and warnings 2.3.1 Reciprocal 2.3.2 Shifted reciprocal 3 Example formulas 4 Example calculations 4.1 Inverse tangent function 4.2 Resistance measurement 5 See also 6 References 7 Further reading 8 External links Linear combinations[edit] Let { f k ( x 1 , x 2 , … , x n ) } {\displaystyle \ ρ 5(x_ ρ 4,x_ ρ 3,\dots ,x_ ρ 2)\}} be a set of m functions which are linear combinations of n {\displaystyle n} variables x 1 , x 2 , … , x n {\displaystyle x_ σ 7,x_ σ 6,\dots ,x_ σ 5} with combination coefficients A k 1 , A k 2 , … , A k n , ( k = 1 … m ) {\displaystyle A_ σ 1,A_ σ 0,\dots ,A_ ρ 9,(k=1\dots m)} . f k = ∑ i n A k i x i
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