Backward Error Analysis Wiki
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issue is particularly prominent in applied areas such as numerical analysis and statistics. Contents 1 Error analysis in numerical modeling 1.1 Forward error analysis 1.2 Backward error analysis 2 Applications 2.1 Global positioning system 2.2 Molecular dynamics
Backward Error Correction
simulation 2.3 Scientific data verification 3 See also 4 References 5 External links Error analysis error propagation wiki in numerical modeling[edit] In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the
Percent Error Wiki
model as the parameters to the model vary about a mean. For instance, in a system modeled as a function of two variables z = f ( x , y ) {\displaystyle \scriptstyle z\,=\,f(x,y)} . Error analysis error analysis in mathematics deals with the propagation of the numerical errors in x {\displaystyle \scriptstyle x} and y {\displaystyle \scriptstyle y} (around mean values x ¯ {\displaystyle \scriptstyle {\bar {x}}} and y ¯ {\displaystyle \scriptstyle {\bar {y}}} ) to error in z {\displaystyle \scriptstyle z} (around a mean z ¯ {\displaystyle \scriptstyle {\bar {z}}} ).[1] In numerical analysis, error analysis comprises both forward error analysis and backward error analysis. Forward error analysis[edit] Forward error analysis involves the forward error analysis analysis of a function z ′ = f ′ ( a 0 , a 1 , … , a n ) {\displaystyle \scriptstyle z'=f'(a_{0},\,a_{1},\,\dots ,\,a_{n})} which is an approximation (usually a finite polynomial) to a function z = f ( a 0 , a 1 , … , a n ) {\displaystyle \scriptstyle z\,=\,f(a_{0},a_{1},\dots ,a_{n})} to determine the bounds on the error in the approximation; i.e., to find ϵ {\displaystyle \scriptstyle \epsilon } such that 0 ≤ | z − z ′ | ≤ ϵ {\displaystyle \scriptstyle 0\,\leq \,|z-z'|\,\leq \,\epsilon } . Backward error analysis[edit] Backward error analysis involves the analysis of the approximation function z ′ = f ′ ( a 0 , a 1 , … , a n ) {\displaystyle \scriptstyle z'\,=\,f'(a_{0},\,a_{1},\,\dots ,\,a_{n})} , to determine the bounds on the parameters a i = a i ¯ ± ϵ i {\displaystyle \scriptstyle a_{i}\,=\,{\bar {a_{i}}}\,\pm \,\epsilon _{i}} such that the result z ′ = z {\displaystyle \scriptstyle z'\,=\,z} .[2] Backward error analysis, the theory of which was developed and popularized by James H. Wilkinson, can be used to establish that an algorithm implementing a numerical function is numerically stable.[3] The basic approach is to show that although the calculated result, due to roundoff errors, will not be exactly correct, it is the exact solution to a nearby problem with slightly
the
Types Of Mathematical Error
model vary about a mean. Error analysis (linguistics) error analysis calculus studies the types and causes of language errors. Error analysis for
Error Analysis In Mathematics Pdf
the Global Positioning System This disambiguation page lists articles associated with the title Error analysis. If an internal link https://en.wikipedia.org/wiki/Error_analysis_(mathematics) led you here, you may wish to change the link to point directly to the intended article. Retrieved from "https://en.wikipedia.org/w/index.php?title=Error_analysis&oldid=724970265" Categories: Disambiguation pagesHidden categories: All article disambiguation pagesAll disambiguation pages Navigation menu Personal tools Not logged inTalkContributionsCreate accountLog https://en.wikipedia.org/wiki/Error_analysis in Namespaces Article Talk Variants Views Read Edit View history More Search Navigation Main pageContentsFeatured contentCurrent eventsRandom articleDonate to WikipediaWikipedia store Interaction HelpAbout WikipediaCommunity portalRecent changesContact page Tools What links hereRelated changesUpload fileSpecial pagesPermanent linkPage informationWikidata itemCite this page Print/export Create a bookDownload as PDFPrintable version Languages Español Edit links This page was last modified on 12 June 2016, at 19:18. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view
the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has order one and is A-stable. Contents 1 Description 2 Derivation https://en.wikipedia.org/wiki/Backward_Euler_method 3 Analysis 4 Extensions and modifications 5 See also 6 Notes 7 References Description[edit] Consider https://en.wikipedia.org/wiki/Condition_number the ordinary differential equation d y d t = f ( t , y ) {\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}=f(t,y)} with initial value y ( t 0 ) = y 0 . {\displaystyle y(t_{0})=y_{0}.} Here the function f {\displaystyle f} and the initial data t 0 {\displaystyle t_{0}} and y 0 {\displaystyle y_{0}} are known; error analysis the function y {\displaystyle y} depends on the real variable t {\displaystyle t} and is unknown. A numerical method produces a sequence y 0 , y 1 , y 2 , … {\displaystyle y_{0},y_{1},y_{2},\ldots } such that y k {\displaystyle y_{k}} approximates y ( t 0 + k h ) {\displaystyle y(t_{0}+kh)} , where h {\displaystyle h} is called the step size. The backward Euler method computes the approximations using y k + error analysis in 1 = y k + h f ( t k + 1 , y k + 1 ) . {\displaystyle y_{k+1}=y_{k}+hf(t_{k+1},y_{k+1}).} [1] This differs from the (forward) Euler method in that the latter uses f ( t k , y k ) {\displaystyle f(t_{k},y_{k})} in place of f ( t k + 1 , y k + 1 ) {\displaystyle f(t_{k+1},y_{k+1})} . The backward Euler method is an implicit method: the new approximation y k + 1 {\displaystyle y_{k+1}} appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown y k + 1 {\displaystyle y_{k+1}} . Sometimes, this can be done by fixed-point iteration: y k + 1 [ 0 ] = y k , y k + 1 [ i + 1 ] = y k + h f ( t k + 1 , y k + 1 [ i ] ) . {\displaystyle y_{k+1}^{[0]}=y_{k},\quad y_{k+1}^{[i+1]}=y_{k}+hf(t_{k+1},y_{k+1}^{[i]}).} If this sequence converges (within a given tolerance), then the method takes its limit as the new approximation y k + 1 {\displaystyle y_{k+1}} .[2] Alternatively, one can use (some modification of) the Newton–Raphson method to solve the algebraic equation. Derivation[edit] Integrating the differential equation d y d t = f ( t , y ) {\d
change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem – given f ( x ) = y , {\displaystyle f(x)=y,} one is solving for x, and thus the condition number of the (local) inverse must be used. In linear regression the condition number can be used as a diagnostic for multicollinearity.[1][2] The condition number is an application of the derivative, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several variables. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability. In general, a backward stable algorithm can be expected to accurately solve well-conditioned problems. Numerical analysis textbooks give formulas for the condition numbers of problems and identify the backward stable algorithms. As a rule of thumb, if the condition number κ ( A ) = 10 k {\displaystyle \kappa (A)=10^{k}} , then you may lose up to k {\displaystyle k} digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods.[3] However, the condition number does not give the exact value of the maximum inaccuracy that may occur in the algorithm. It generally just bounds it with an estimate (whose computed value depends on the choice of the norm to measure the inaccuracy). Contents 1 Matrices 2 Non-linear 2.1 One variable 2.2 Several variables 3 See also 4 References 5 External links Matrices[edit] F