Computational Algorithm for Higher Order Legendre Polynomial and Gaussian Quadrature Method

Transcription

1 Computatioal Algorithm for Higher Order Legre olyomial ad Gaussia Quadrature Method Asif M. Mughal, Xiu Ye ad Kamra Iqbal Dept. of Applied Sciece, Dept. of Mathematics ad Statistics, Dept. of Systems Egieerig Uiversity of Arkasas at Little Rock Abstract: - There are may umerical methods adopted to solve mathematical problems. Early researchers focused o the methods to reduce computatioal costs. I recet years, reductio i computatioal costs makes may umerical methods available which were ot tried for this reaso. The use of higher order Legre polyomials for more tha -7 orders is usually ot commo. The efficiet ad quick umerical methods like Gaussia Quadrature were ot adopted for higher orders. I this paper a very simple computatioal algorithm is adopted for calculatios of higher order Legre polyomials ad its use for Gaussia quadrature umerical itegratio till 44 th order. I. INTRODUCTION There are may efficiet umerical methods which are ot i use much due to computatioal costs ad ature of precise calculatios i the computers. I recet years, reductio i techological costs leads to make may solutios available which were oly tried ad used for very high paid defese ad commercial use. Adrie Marie Legre, a Frech mathematicia (d. 8) whe discovered famous polyomial bear his ame ow was ever aware of that how much it will be used i developig mathematics. This Legre polyomial is beig used by mathematicias ad egieers for variety of mathematical ad umerical solutios. This use of Legre polyomial i ormal texts is usually oly referred till -7 order polyomials, because fidig Legre polyomials for higher orders is ot oly computatioally tough but also time cosumig. The recet research o Legre polyomial such as Ref. [] discusses ad explores the use of Legre series ad best leadig coefficiets of Legre polyomials for differet applicatios. Ref [] uses sixth order Legre polyomial for propagatio of guided waves ad similar method with higher order polyomials leads to better results. I umerical aalysis ad methods, Legre polyomials are used to efficietly calculate umerical itegratios by Gaussia quadrature method. This method is very effective i approximatig itegrals with accuracy ad i small time. Ref [] describes may properties of this method for arbitrary fuctios ad use of Gaussia quadrature for the triagular fiite elemets. The other researches such as Ref [4-6] published valuable use of this method over period of time, but use of this method for higher order computatio requires higher order Legre polyomials ad coefficiets of Gaussia quadrature which are usually ot available easily i refereced text. I this paper, first mathematical backgroud is preseted followed by computatioal algorithm ad at the a example is discussed with umerical itegratios. Roots of Legre polyomials ad coefficiets of Gaussia quadrature are provided i Appix for 44 th order. II.MATHEMATIAL BACKGROUND A. Legre olyomials The th order Legre polyomial is geerally give by the followig equatio. M = ( ) m=! ()! m! ( m)! ( m)(! m)! m m x () Where is the order of polyomial ad M determies the eve ad odd ature of polyomial by fidig which is iteger. ( ) M = or M = () There are differet cases of Legre polyomials ad usually the effect is determied by leadig coefficiets of x. I origial Equatio give i Ref [7], leadig coefficiets of Legre polyomial is ot ad scaled by -, but dividig the whole polyomial with leadig coefficiet with will results i a polyomial give by Eq() with leadig coefficiet. The basic property of Legre polyomials is that these are orthogoal to each other with respect to weight fuctio w(x) = o [-,]. The first two polyomials are always same i all cases but the higher order polyomials are created with recursive algorithm ad ca be ormalized accordigly. The set of Legre polyomial up till ay degree provided a orthogoal basis for differet applicatios. The orm of Legre polyomials is give by = () +

2 Legre polyomials till th order are give as follows = = x = x = x x 4 6 = x x + 7 = x x + x = x x + x 7 = x x + x x = x x + x x = x x + x x = x x + x x + x The pyramid of Legre polyomial ca be developed at ay required degree. These polyomials are geerated from code give i sec III. The graphs of Legre polyomials itercept the x-axis at times withi [-, ]. The importat thig to ote is that the area covered by these polyomials is differet ad that ca be implied to get umerical itegratio. Fig. shows the plot of Legre polyomials ad it ca be oted that higher degrees has lesser amplitude.. -. & x x Fig. lots of Legre olyomials It is obvious from the figure that icreasig order of Legre polyomials decreases the magitude of oscillatios i the polyomial ad this is the property which was exploited by Karli ad Studde to itroduce Gaussia quadrature algorithm. B. Gaussia Quadrature Method Amog may other umerical itegratio methods, Gaussia quadrature method is used for may itegral approximatios. The mai problem of this method is that it requires computatio of Legre polyomials to get it solutio. The algorithm usually follows by the icreasig the order of polyomial if error boud does t satisfies. Though it is fairly simple algorithm yet it is rarely used i practical applicatios due to its computatioal expesiveess. This method works with the roots of Legre polyomials ad fidig coefficiets for usig these roots. The desired fuctio (itegrad) is represeted i terms of polyomial orthogoal basis. If the itegrad is a polyomial the residual error will be zero ad Gaussia quadrature is applied as i= = c ( ) i x i Where the coefficiets c i are give as j= xi j i (4) = x x j c i dx () x j If the itegrad is ot a polyomial or ot i iterval [-,] the it ca be trasformed i to a itegral over [-,] usig chage of variables, ad i this case error will ot be zero. b a + + = ( b a) t a b ( b a) f dx f dt The error i this case will coverge to zero at higher orders of Legre polyomials ad yield better results. The proof of covergece ad details descriptio is give i Ref. [8] pp-6. I Ref. [9] pp8-9, this method is used for a example to show faster ad direct covergece tha Simpso s rule or Romberg itegratio method. However, this is ot for all fuctios ad lower order Legre polyomial with Gaussia quadrature may yield to a ocoverget behavior as show i example i sectio IV. III. COMUTATIONAL ALGORITHIM Higher order Legre polyomials ca be obtaied by usig ew computatioal techiques ad also ca be applied i differet methods such as Gaussia quadrature for umerical itegratios. I this paper, a MATLAB code is preseted that shows how simply a umerical itegratio usig Gaussia quadrature is possible. The symbolic math toolbox is useful i computig this algorithm. I first coefficiets of Legre polyomial are foud usig Eq () (6)

3 (ad also ormalize it i.e. leadig coefficiet is ). The zeros are stuffed i coefficiets vector to geerate roots of polyomial (required by the MATLAB code). The roots are the used to geerate coefficiets of Gaussia quadrature method as give by Eq (6). Simultaeously Gaussia quadrature itegratio of a required fuctio is also geerated. This method work recursively with eve ad odd polyomials ad get results till the required degree. It is also importat ote that the computatioal efficiecy of machie o which MATLAB code works also take ito cosideratios. This code was ru o MATLAB v7. o Su Solaris System to get roots ad coefficiets give i Appix till 44 th order of the polyomial. A. MATLAB CODE clear all clc GQ=; sol=; F=ilie('f(x),'x'); =; MN=44; c=zeros(mn,mn); pl=zeros(mn,mn); for =:MN if mod(,)== a=; M=/; for m=:m a(m+)=power(-,m)*factorial(*- *m)/(factorial(m)*factorial(-m)*factorial(-*m)); b=a./a(); j=; i=; rt=zeros(,*legth(b)-); while j<=legth(rt) rt(j)=b(i); j=j+; i=i+; p=roots(rt); for u=:legth(p) pl(u,)=p(u); syms x; i=; j=; G=; H=; for i=: for j=: if j==i l=;%disp(i); G=G*(x-p(j))/(p(i)-p(j)); c(i,)=double(it(g,x,-,)); H=G; G=; sum=; for i=: sum=sum+c(i,)*f(p(i)); GQ()=sum; a=; M=(-)/; for m=:m a(m+)=power(-,m)*factorial(*- *m)/(factorial(m)*factorial(-m)*factorial(-*m)); b=a./a(); j=; i=; rt=zeros(,*legth(b)); while j<=legth(rt) rt(j)=b(i); j=j+; i=i+; p=roots(rt); for u=:legth(p) pl(u,)=p(u); syms x; i=; j=; c=; G=; H=; for i=: for j=: if j==i l=;%disp(i); G=G*(x-p(j))/(p(i)-p(j)); c(i,j)=double(it(g,x,-,)); H=G; G=; sum=; for i=: sum=sum+c(i,j)*f(p(i)); GQ()=sum; sol=gq() %END ROGRAM

4 This code geerates matrices for roots of Legre polyomials ad coefficiets c i for Gaussias quadrature method for to MN, (th) order. Solutio of Gaussia quadrature GQ is ot required i this code, it ca be doe separately ad easily, but it is provided i this to see the tr of covergece with respect to order of polyomials. The roots of ay m order polyomial ca be store i a vector form i oe variable e.g. p ad a simple MATLAB commad rats(poly(p)) will give a mth order polyomial i fractios. IV. EXAMLE The code give i previous sectio ca be computed for ay fuctio of x provided that it must be geerated accordig to Eq (6), ad itegratio limits are appropriately trasformed with i [-,] boud. There are may fuctios for which Gaussia quadrature coverges i a low order ad proves the better results tha Simpso s rule or Romberg itegratio. But there are may for which if roots of Legre polyomials ad coefficiets of Gaussia quadrature are available, it ca compute the results with less computatioal delays. Oe such example is the fuctio give i Eq (7), this seems to be a very easy problem ad the exact solutio of this ca be obtaied very easily. dx x + I (7) = The exact solutio of this itegral is give as = I = ta (8) I this case we are cosiderig a absolute error boud less tha -. The itegral of Eq (7) ca be trasformed accordig to Eq(6) ad it will be give as I = dt t + Trasformig the itegrad f(x) i Eq(7) ad Eq(9) actually does t chage the fuctioal properties as show i Fig. f(x) Origial - - x f(t) (9) After Chage of Variable t Fig. Itegrad before ad after chage of variable It ca be see the fuctio from - to remais the same withi - to after chage of variable ad i origial case the maximum value is ad after chage of variable the maximum value is. Such fuctios to umerical itegrate with low order ad usig Simpso s Rule ad Romberg Itegratio the values are give i Table. These methods are ot discussed here ad their discussio is available i Ref [8], pp96-. Also to ote that i case of Romberg itegratio method, it is ot simple as it computes x matrices. The covergece of this problem with respect to order of Gaussia quadrature is show i Table. It is also oted that i case of available roots ad coefficiets, the calculatio method is simpler tha Simpso s rule ad Romberg Itegratio method. I the above code, i a ilie fuctio, left had side of Eq (7) is used istead of f(x) before ruig the code for this example. At = 44, the absolute error is with respect to exact solutio, better tha the similar order of Simpso s rule or Romberg itegratio method. TABLE Itegratio with Simpso s Rule ad Romberg Method Simpso s Rule = Simpso s Rule = Romberg Method = Romberg Method =9.949 TABLE Gaussia Quadrature Method GQ GQ

5 V. CONCLUSION The roots or Legre polyomial ca be calculated easily with the use of simple MATLAB code ad higher order polyomials are available to use i differet applicatios. The higher order roots are successfully used to apply i Gaussia quadrature itegratio method for a problem which seems to be usolvable by this method i lower orders. Ay order polyomial ca be geerated for other applicatios usig this code. AENDIX - TABLE Calculated roots of Legre olyomial ad Gaussia quadrature coefficiets for 44 th order. c 44 r REFERENCES [] Gavi Brow, Stamatis Koumados, ad Kuyag Wag. O the positivity of some basic legre polyomial sums Joural of Lodo Mathematical Society, 9, pp99-94, Cambridge Uiversity ress, 999. [].Jea E. Lefebvre, Victor Zhag, Joseph Gazalet, Tadeusz Gryba, ad Veroique Sadaue, Acoustic Wave ropagatio i Cotiuous Fuctioally Graded lates: A Extesio of the Legre olyomial Approach, IEEE Trasactios O Ultrasoics, Ferroelectrics, Ad Frequecy Cotrol, Vol. 48, No., September [] D. A. Duavat. High degree efficiet symmetrical Gaussia quadrature rules for the triagle. Iteratioal Joural of Numerical Methods i Egieerig, Vol., No. 6, pp9-48, 98. [4] J. Ma, V. Rokhli, S. Wadzura Geeralized Gaussia Quadrature Rules for Systems of Arbitrary Fuctios. SIAM Joural o Numerical Aalysis, Vol. No., pp [] Dirk. Laurie, Calculatio Of Gauss-Krorod Quadrature Rules, Mathematics Of Computatio, Vol. 66, No. 9, pp -4, July 997. [6] Carlos F. Borges, O a class of Gauss-like quadrature rules, Numerische Mathematik, Volume 67, No., pp 7-88, April 994 [7] Erwi Kreyszig, Advaced Egieeig Mathematics, 7 th Ed, J. Weily, 99. [8] J.D. Faires, R. Burde, Numerical Aalysis, 7 th Ed., Brooks/Cole ublishig,. [9] J.D. Faires, R. Burde, Numerical Methods, d Ed., Brooks/Cole ublishig, 998.