18 Numerical Integration of Functions

Transcription

1 Slightly moifie //9, /8/6 Firstly written at Marh 5 8 Numerial Integration of Funtions Introution Romberg Integration Gauss Quarature Aaptive Quarature Case Stuy: Root-Mean-Square Current DM869/Computational Numerial Analysis/7_na.o 7

2 Slightly moifie //9, /8/6 Firstly written at Marh 5 Introution two forms of numerial integration table of values funtion Riharson extrapolation - ombining two numerial integral estimates to obtain a thir, more aurate value - an effiient omputational algorithm for Riharson's extrapolation Romberg integration Gauss quarature aaptive quarature Romberg Integration one tehnique esigne to attain effiient numerial integrals of funtions base on suessive appliation of the trapezoial rule Riharson's extrapolation using two estimates of an integral to ompute a thir, more aurate approximation Let us onsier: I I ( h E( h where I = the exat value of the integral I(h = the approximation from an n-segment appliation of the trapezoial rule h = (b a/n, step size E(h = the trunation error Making two separate estimates an having exat values for the error: I( h E( h I( h E( h Reall that b a E h f an assume that f is onstant regarless of step size. Then we an have: E( h h or E( h h whih yiels: E ( h h I( h E( h I( h E( h h I( h I( h or E( h ( h / h E( h 7 DM869/Computational Numerial Analysis/7_na.o h h

3 Slightly moifie //9, /8/6 Firstly written at Marh 5 This estimate of the truation error an be substitute into I I( h E( h to yiel an improve estimate of the integral: I I( h [ I ( h I( h ] error ~ O(h ( h / h for the speial ase where h = h /: I I( h I( h Example Use Riharson extrapolation to evaluate the integral of f ( x. 5x x 675x 9x x from a = to b =.8. Note that the exat value is Sol. Single an omposite trapezoial rule: Segments h Integral t %.9% 9.5% for & segments: I (.688 ( E t = =.767 ( t = 6.6% for & segments: I (.88 ( E t = =.767 ( t =.% Riharson extrapolation again for two improve integrals 6 I I m I l O(h an again Riharson extrapolation I I m I l O(h where I m = the more aurate estimate I l = the less aurate estimate DM869/Computational Numerial Analysis/7_na.o 75

4 Slightly moifie //9, /8/6 Firstly written at Marh 5 Example Repeat the above example to ompute an integral with O(h 6. Sol. 6 I (.667 ( the exat value of the integral The Romberg integration algorithm general form well suite for omputer implementation I j, k k I j, k k I j, k where I j+,k- = the more aurate integral I j,k- = the less aurate integral I j,k = the improve integral k signifying the level of the integration e.g., k = the original trapezoial rule estimates k = the O(h estimates k = the O(h 6 estimates I, I, ex for k = an j = : I, I( h I( h Graphial epition of the sequene of integral estimates generate using Romberg integration. (a First iteration. (b Seon iteration. ( Thir iteration. 76 DM869/Computational Numerial Analysis/7_na.o

5 Slightly moifie //9, /8/6 Firstly written at Marh 5 termination (or stopping riterion - an estimate of the perent relative error ε a I, k I I, k, k % Romberg integration is more effiient than the trapezoial rule an Simpson's rules - Simpson's rule requires a 8-segment appliation for the above example - but, Romberg integration only requires 5 funtion evaluation <An M-file to implement Romberg integration> DM869/Computational Numerial Analysis/7_na.o 77

6 Slightly moifie //9, /8/6 Firstly written at Marh 5 Gauss Quarature (a Graphial epition of the trapezoial rule as the area uner the straight line joining fixe en points. (b An improve integral estimate obtaine by taking the area uner the straight line passing through two intermeiate points. By positioning these points wisely, the positive an negative errors are better balane, an an improve integral estimate results. the trapezoial rule preetermine or fixe points for the integration interval sometimes resulting in a large error if taking the enpoints freely an wisely an improve estimate by balaning the positive an negative errors Metho of unetermine oeffiients Reall the trapezoial rule: Two integrals that shoul be evaluate exatly by the trapezoial rule: (a a onstant an (b a straight line. f ( a f ( b I ( b a or I f a f ( ( b The trapezoial rule yiels exat results when the funtion being integrate is a onstant or a straight line. Thus, an ( b a / ( ba / x b a b a b a ( b a / b a b a xx ( ba / 78 DM869/Computational Numerial Analysis/7_na.o

7 Slightly moifie //9, /8/6 Firstly written at Marh 5 Solving for the two equations with two unknowns: b a Finally, we have b a b a I f ( a f ( b equivalent to the trapezoial rule Derivation of the two-point Gauss-Legenre formula the objet of Gauss quarature - to etermine the oeffiients of an equation of the form I f ( x f ( x where the 's = the unknown oeffiients - note that not fixe x an x four unknowns requiring four onitions y = onstant, y = x, y = x, an y = x x x x xx x x x x x x x x x x = x sine x x x x.5775 Graphial epition of the unknown variables x an x for integration by Gauss quarature. x.5775 Therefore, the two-point Gauss-Legenre formula is I f ( f ( DM869/Computational Numerial Analysis/7_na.o 79

8 Slightly moifie //9, /8/6 Firstly written at Marh 5 general formulation in the integral limits assuming that x a x a - the lower limit: x = a then x = a a ( a - the upper limit: x = b then x = b a ( a Solving: a a b a b a Then we have: ( b a ( b a x x b a x x Example Use the two-point Gauss-Legenre formula to evaluate the integral of f ( x. 5x x 675x 9x x between the limits x = to.8. Note that the exat value is Sol. Changing the limits to an +: x x.. an x. x.8 (. 5x x - 675x [. 5(..x (..x 9(..x 9x (..x 5 x x 675(..x 5 ].x.567 at x / an.587 at x / =.8578 a perent relative error of.% DM869/Computational Numerial Analysis/7_na.o

9 Slightly moifie //9, /8/6 Firstly written at Marh 5 Higher-point formulas I f ( x f ( x n f ( xn where n = the number of points <Weighting fators an funtion arguments use in Gauss-Legenre formulas.> Points Weighting fators Funtion arguments Trunation error 5 6 =. =. = = = =.7858 =.655 =.655 =.7858 =.6969 = = = =.6969 =.75 =.6766 =.6799 =.6799 = =.75 x = x = x = x =. x = x =.866 x =.998 x =.998 x =.866 x = x =.5869 x =. x =.5869 x = x =.9695 x = x = x = x = x 5 =.9695 f ( ( f (6 ( f (8 ( f ( ( f ( ( Example Use the three-point Gauss-Legenre formula to evaluate the integral of f ( x. 5x x 675x 9x x between the limits x = to.8. Note that the exat value is Sol. I f ( f ( f ( I not appropriate when the funtion is unknown not suite for engineering problem with tabulate ata DM869/Computational Numerial Analysis/7_na.o 8

10 Slightly moifie //9, /8/6 Firstly written at Marh 5 Aaptive Quarature omposite Simpson's / rule is applie for equally spae points not take into aount relatively abruptly hange regions where more refine spaing might be require aaptive quarature methos - automatially ajusting the step size - small steps in regions of sharp variations - larger steps in regions of graual variations - applying the / rule at two levels of refinement - estimating the trunation errors for the two levels - refining the step size an repeating until the error falls to aeptable levels, if the error is too large MATLAB funtions: qua an qual qua - aaptive Simpson quarature - more effiient for low auaries or nonsmooth funtions qual - Lobatto quarature - more effiient for high auraies an smooth funtions q = qua(fun, a, b, tol, trae, p, p,... where fun the funtion to be integrate a an b the integration bouns tol the esire absolute error tolerane (efault = -6 trae the optional variable p, p,... parameters passe to fun 8 DM869/Computational Numerial Analysis/7_na.o

11 Slightly moifie //9, /8/6 Firstly written at Marh 5 Example Use qua to integrate the following funtion: f ( x s ( x q. ( x r. between the limits x = to. Note that for q =., r =.9, an s = 6, this is the built-in humps funtion that MATLAB uses to emonstrate some of its numerial apabilities. The humps funtion exhibits both flat an steep regions over a relatively short x range. Hene, it is useful for emonstrating an testing routines like qua an qual. Note that the humps funtion an be integrate analytially between the given limits to yiel an exat integral of Sol. Using built-in version of humps along with the efault tolerane: >> format long >> qua(@humps,, ans = Using a looser tolerane: funtion y = myhumps(x,q,r,s y =./((x-q.^ +. +./((x-r.^ +. s; >> qua(@myhumps,,,e-,[],.,.9,6 ans = DM869/Computational Numerial Analysis/7_na.o 8

12 8 DM869/Computational Numerial Analysis/7_na.o HK Kim Slightly moifie //9, /8/6 Firstly written at Marh 5