Definition Of Numerical Error
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of computations involving floating-point or integer values. The second usually called truncation error is the difference between the exact mathematical definition numerical order solution and the approximate solution obtained when simplifications are made to definition numerical data the mathematical equations to make them more amenable to calculation. The term truncation comes from the fact
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that either these simplifications usually involve the truncation of an infinite series expansion so as to make the computation possible and practical, or because the least significant bits
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of an arithmetic operation are thrown away. Floating-point numerical error is often measured in ULP (unit in the last place). See also[edit] numerical analysis round-off error Kahan summation algorithm impact of error References[edit] Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, ISBN 0-89871-355-2 "Computational Error And Complexity In Science And Engineering", V. Lakshmikantham, S.K. Sen, ISBN definition of numerical variable 0444518606 This software-engineering-related article is a stub. You can help Wikipedia by expanding it. v t e This applied mathematics-related article is a stub. You can help Wikipedia by expanding it. v t e Retrieved from "https://en.wikipedia.org/w/index.php?title=Numerical_error&oldid=723542893" Categories: Computer arithmeticNumerical analysisSoftware engineering stubsApplied mathematics stubsHidden categories: All stub articles Navigation menu Personal tools Not logged inTalkContributionsCreate accountLog 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 Add links This page was last modified on 3 June 2016, at 17:23. 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
Truncation error arises in Euler's method because the curve is not generally a straight-line between the neighbouring grid-points and , as assumed above. The error associated with this approximation can easily be assessed by Taylor expanding about :
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(11) A comparison of Eqs.(10) and (11) yields (12) In other words, definition of numerical coefficient every time we take a step using Euler's method we incur a truncation error of , where is the step-length. Suppose definition of numerical age that we use Euler's method to integrate our o.d.e. over an -interval of order unity. This requires steps. If each step incurs an error of , and the errors are simply cumulative (a fairly conservative https://en.wikipedia.org/wiki/Numerical_error assumption), then the net truncation error is . In other words, the error associated with integrating an o.d.e. over a finite interval using Euler's method is directly proportional to the step-length. Thus, if we want to keep the relative error in the integration below about then we would need to take about one million steps per unit interval in . Incidentally, Euler's method is termed a first-order integration method because http://farside.ph.utexas.edu/teaching/329/lectures/node33.html the truncation error associated with integrating over a finite interval scales like . More generally, an integration method is conventionally called th order if its truncation error per step is . Note that truncation error would be incurred even if computers performed floating-point arithmetic operations to infinite accuracy. Unfortunately, computers do not perform such operations to infinite accuracy. In fact, a computer is only capable of storing a floating-point number to a fixed number of decimal places. For every type of computer, there is a characteristic number, , which is defined as the smallest number which when added to a number of order unity gives rise to a new number: i.e., a number which when taken away from the original number yields a non-zero result. Every floating-point operation incurs a round-off error of which arises from the finite accuracy to which floating-point numbers are stored by the computer. Suppose that we use Euler's method to integrate our o.d.e. over an -interval of order unity. This entails integration steps, and, therefore, floating-point operations. If each floating-point operation incurs an error of , and the errors are simply cumulative, then the net round-off error is . The total error, , associated with integrating our o.d.e. over an -interval of order unity
use fixed number of bits and hence fixed number of binary digits to represent numbers. In a numerical computation round-off errors are introduced at every stage of computation. Hence though an individual round-off http://nptel.ac.in/courses/122104019/numerical-analysis/Rathish-kumar/numerr/new4.htm error due to a given number at a given numerical step may be small but the cumulative effect can be significant. When the number of bits required for representing a number are less then the number is usually rounded to fit the available number of bits. This is done either by chopping or by symmetric rounding. Chopping: Rounding a number by chopping definition of amounts to dropping the extra digits. Here the given number is truncated. Suppose that we are using a computer with a fixed word length of four digits. Then the truncated representation of the number will be . The digits will be dropped. Now to evaluate the error due to chopping let us consider the normalized representation of the given number i.e. chopping error definition of numerical in representing . So in general if a number is the true value of a given number and is the normalized form of the rounded (chopped) number and is the normalized form of the chopping error then Since , the chopping error Symmetric Round-off Error : In the symmetric round-off method the last retained significant digit is rounded up by 1 if the first discarded digit is greater or equal to 5.In other words, if in is such that then the last digit in is raised by 1 before chopping . For example let be two given numbers to be rounded to five digit numbers. The normalized form x and y are and . On rounding these numbers to five digits we get and respectively. Now w.r.t here In either case error . Truncation Errors: Often an approximation is used in place of an exact mathematical procedure. For instance consider the Taylor series expansion of say i.e. Practically we cannot use all of the infinite number of terms in the series for computing the sine of angle x. We usually terminate the process after a certain num