Alied Mathematics and Comutation 18 (011) 944 948 Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: www.elsevier.com/locate/amc A generalized Birkhoff Young Chebyshev quadrature formula for analytic functions q Gradimir V. Milovanović a,, Aleksandar S. Cvetković b, Marija P. Stanić c a Faculty of Comuter Sciences, Megatrend University, Bulevar Umetnosti 9, 1100 Novi Beograd, Serbia b Deartment of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 1110 Beograd, Serbia c Deartment of Mathematics and Informatics, Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia article info abstract Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Quadrature formula Error term Chebyshev olynomial Orthogonality Analytic function A generalized N-oint Birkhoff Young quadrature of interolatory tye, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some secial cases of this quadrature formula are derived. Ó 011 Elsevier Inc. All rights reserved. 1. Introduction For numerical integration over a line segment in the comlex lane, Birkhoff and Young [1] roosed a quadrature formula of the form Z z0 þh z 0 h f ðzþdz h f 1 4f ðz 0Þþ4½fðz 0 þ hþþfðz 0 hþš½f ðz 0 þ ihþþfðz 0 ihþšg where f(z) is a comlex analytic function in X ={z : jz z 0 j 6 r} and jhj 6 r. This five oint quadrature formula is exact for all algebraic olynomials of degree at most five, and for its error R BY ðf Þ can be roved the following estimate [10] (see also [,. 136]) jr BY jhj ðf Þj 6 1890 max f ð6þ ðzþj zs where S denotes the square with vertices z 0 +i k h, k = 0,1,,3. Without loss of generality we can consider the integration over [,1] for analytic functions in a unit disk X ={z : jzj 6 1}, so that the revious Birkhoff Young formula becomes f ðzþdz ¼ 8 f ð0þþ 4 f ð1þþfðþ 1 ½ Š 1 f ðiþþfðiþ 1 ½ ŠþR ðf Þ: ð1:1þ q The authors were suorted in art by the Serbian Ministry of Science and Technological Develoments (Project: Aroximation of integral and differential oerators and alications, Grant Number #1401). Corresonding author. E-mail addresses: gvm@megatrend.edu.rs (G.V. Milovanović), acvetkovic@mas.bg.ac.rs (A.S. Cvetković), stanicm@kg.ac.rs (M.P. Stanić). 0096-3003/$ - see front matter Ó 011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.011.0.00
G.V. Milovanović et al. / Alied Mathematics and Comutation 18 (011) 944 948 94 This five oint formula can also be used to integrate real harmonic functions (see [1]). We mention here that Lyness and Delves [] and Lyness and Moler [6], and later Lyness [4], develoed formulae for numerical integration and numerical differentiation of comlex functions. In 196 Lether [3] ointed out that the three oint Gauss Legendre quadrature which is also exact for all olynomials of degree at most five is more recise than (1.1) and he recommended it for numerical integration. However, Tošić [9] imroved the quadrature (1.1) in the form where f ðzþdz ¼ Af ð0þþbfðrþþf ½ ðrþ ŠþCfðirÞþf ½ ðirþšþr T ðf rþ A ¼ 1 1 B ¼ 1 r 4 6r þ 1 10r C ¼ 1 4 6r þ 1 ð0 < r < 1Þ 10r 4 and the error-term is given by the exresion R T ðf rþ ¼ 3 6! r4 þ f ð6þ ð0þþ! 8! r4 þ f ð8þ ð0þþ ð1:þ 9! Evidently, for r = 1 this formula reduces to (1.1) and for r ¼ ffiffiffiffiffiffiffiffi 3= to the Gauss Legendre formula (then C = 0). Moreover, for r ¼ ffiffiffiffiffiffiffiffi 4 3= the first term on the right-hand side in (1.) vanishes and the formula reduces to the modified Birkhoff Young quadrature of the maximum accuracy (named MF in [9]), with the coefficients A ¼ 16 1 B ¼ 1 r ffiffiffi! 6 þ C ¼ 1 r ffiffiffi! 3 6 3 and with the error-term R MF ðf Þ¼RT f ffiffiffiffiffiffiffiffi 4 1 3= ¼ 93800 f ð8þ 1 ð0þþ 611600 f ð10þ ð0þþ This formula was extended by Milovanović and Ðor dević [8] to the following quadrature formula of interolatory tye: f ðzþdz ¼ Af ð0þþc 11 ½f ðr 1 Þþfðr 1 Þ where 0 < r 1 < r < 1. They roved that for sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 ¼ r 1 ¼ 63 4 ffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 114 and r ¼ r 143 ¼ 63 þ 4 ffiffiffiffiffiffiffiffi 4 114 143 this formula has the algebraic recision = 13, with the error-term R 9 ðf r 1 r Þ¼ 1 8166106600 f ð14þ ð0þþ3:6 10 4 f ð14þ ð0þ: ŠþC 1 ½f ðir 1 Þþfðir 1 ÞŠþC 1 ½f ðr Þþfðr ÞŠþC ½f ðir Þþfðir ÞŠþR 9 ðf r 1 r Þ In this aer we consider a generalized quadrature with resect to the Chebyshev weight.. Generalized Birkhoff Young Chebyshev quadrtaure formula For analytic functions in the unit disk X ={z : jzj 6 1}, we consider numerical integration Iðf Þ :¼ f ðzþ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ Q N ðf ÞþR N ðf Þ ð:1þ where Q N is the N-oint quadrature formula of interolatory tye with nodes at the zeros of a monic olynomial of degree N, x N ðzþ ¼z m nm ðz 4 Þ¼z m Yn ðz 4 r k Þ 0 < r 1 < < r n < 1 ð:þ k¼1 where n =[N/4] and m = N 4[N/4], i.e., N ¼ 4n þ m n N m f0 1 3g, and R N (f) is the corresonding remainder term. According to (.) the quadrature formula in (.1) has the form Q N ðf Þ¼ Xm C j f ðjþ ð0þþ Xn j¼0 k¼1 fa k ½f ðx k Þþfðx k ÞŠþB k ½f ðix k Þþfðix k ÞŠg where x k ¼ 4 ffiffiffiffi r k k ¼ 1... n. For m = 0, the first sum in QN (f) is emty. Also, in order to have Q N (f)=i(f) = 0 for f(z)=z, it must be C 1 = 0, so that Q 4n+1 (f) Q 4n+ (f).
946 G.V. Milovanović et al. / Alied Mathematics and Comutation 18 (011) 944 948 Theorem.1. For any N N there exists a unique interolatory quadrature Q N (f) with a maximal degree of recision d =6n + s, where n =[N/4], m = N 4[N/4] {0,1,,3}, and s ¼ ð:3þ m 1 m ¼ 0 m m ¼ 1 3: The nodes of such a quadrature are characterized by the following orthogonality relation: Z 1 T T k z sþ1 nm ðz 4 k ðzþz sþ1 nm ðz 4 Þ Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 0 k ¼ 0 1... n 1 ð:4þ where T k is the Chebyshev olynomials of the first kind of degree k. Proof. Let P d denote the set of algebraic olynomials of degree at most d. For a given N N, suose that f P d, where d P N =4n + m (n =[N/4], m = N 4[N/4]). Then, it can be exressed in the form f ðzþ ¼uðzÞx N ðzþþvðzþ ¼uðzÞz m nm ðz 4 ÞþvðzÞ u P dn v P N from which, alying (.1), we get Iðf Þ¼ uðzþz m nm ðz 4 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz þ IðvÞ: Note that I(v)=Q N (v) (interolatory quadrature!) and v(z)=f(z) at the zeros of x N, and therefore Q N (v)=q N (f), so that for each f P d we have Iðf Þ¼ uðzþz m nm ðz 4 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz þ Q N ðf Þ: It is clear that the quadrature formula Q N (f) has a maximal degree of recision if and only if uðzþz m nm ðz 4 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 0 for a maximal degree of the olynomial u P dn. According to the values of m, the revious orthogonality condition can be considered as hðz Þz m nm ðz 4 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 0 and zhðz Þz m nm ðz 4 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 0 for m = 0, and m = 1, 3, resectively, where h P n. It can be reresented in a comact form hðz Þz sþ1 nm ðz 4 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 0 h P n : ð:þ Thus, the maximal degree of the olynomial u P dn is n 1 m ¼ 0 d max N ¼ n m ¼ 1 3 i.e., d max =6n + s, where s is defined by (.3). Notice that s +1 {0,,4}. The orthogonality conditions (.) can be exressed in terms of Chebyshev olynomials of the first kind, i.e., in the form (.4), where the inner roduct is defined in a usual way as ðf gþ ¼ R 1 f ðzþgðzþð1 z Þ = dz. h According to (.), the olynomial z s+1 n,m (z 4 ) can be exressed in the form z sþ1 nm ðz 4 Þ¼ Xn ðþ j r j z 4ðnjÞþsþ1 j¼0 where r j are the so-called elementary symmetric functions, defined by r j ¼ X r k1 r kj j ¼ 1... n ðk 1...k j Þ ð:6þ and the summation is erformed over all combinations (k 1,...,k j ) of the basic set {1,...,n}. Thus, r 1 ¼ r 1 þ r þþr n r ¼ r 1 r þþr n r n... r n ¼ r 1 r r n and for the convenience we ut r 0 =1.
G.V. Milovanović et al. / Alied Mathematics and Comutation 18 (011) 944 948 94 In the sequel we need the inner roduct (T k,z m ), 0 6 k 6 m. Using the formula (cf. [,. 10]) x k ¼ 1 X½k=Š k i¼0 k i Tki ðxþ 1 þ d ki where d k,i is Kronecker s delta, we get ðt k z m Þ¼ m 0 6 k 6 m: m m k Then, using (.6), the orthogonality conditions (.4) give the following system of linear equations: X n j¼1 ðþ j T k z 4ðnjÞþsþ1 rj ¼ðT k z 4nþsþ1 Þ k ¼ 0 1... n 1 i.e., Ar = b, where A ¼½a kj Š n kj¼1 r ¼ ½ r 1 r r n Š T b ¼ ½b 1 b b n Š T, and! 4ðn jþþsþ1 a kj ¼ðÞ j 4j k j ¼ 1... n ðn jþþðsþ1þ=kþ1 b k ¼! 4n þ s þ 1 k ¼ 1... n: n þðsþ1þ=kþ1 3. Secial cases In order to calculate arameters of quadrature formula (.1), first we calculate values of r k, k =1,...,n. Knowing values of r k, k =1,...,n, we calculate r k, k =1,...,n. InTable 3.1 we give values of r k, k =1,...,n, for n = 1,,3,4. The arameters of the quadrature formula (.1) as well as the corresonding maximal degree of exactness d = 6n + s are resented in Table 3. for n =1,m = 0,1,,3, and in Table 3.3 for n =,m = 0,1,,3. Table 3.1 The values of r k, k =1,...,n, for n = 1,,3,4. n m r 1 r r 3 r 4 1 0 3/8 1, /8 3 3/48 0 /8 /18 1, 1/0 1/18 3 33/8 33/18 3 0 9/4 99/6 33/4096 1, 143/96 143/6 143/096 3 13/8 18/194 143/1 4 0 39/ 11/18 6/4096 39/368 1, 8/44 1/19 1/104 1/368 3 33/16 969/04 33/104 161/98304 Table 3. Parameters and the maximal degree of exactness of the generalized Birkhoff Young Chebyshev quadrature formula for n = 1,m = 0,1,,3. m x 1 A 1 B 1 C 0 C d qffiffi 0 4 3 1 8 þ 1 ffiffi 1 6 1 ffiffi 6 qffiffi 1, ffiffiffiffi 4 3þ 10 3 8 0 ffiffiffiffi 10 0 qffiffiffiffi 3 ffiffiffiffiffiffi 1 4 3 3ð1þ 10Þ 3ð ffiffiffiffiffiffi 1 10Þ 1 3 490 490 3 8 9
948 G.V. Milovanović et al. / Alied Mathematics and Comutation 18 (011) 944 948 Table 3.3 Parameters and the maximal degree of exactness of the generalized Birkhoff Young Chebyshev quadrature formula for n =,m = 0,1,,3. m 0 1, 3 x 1 4 ffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 4 4 14 1 4 66 11 x 4 ffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 3 1 4 4þ 14 1 4 66þ 11 A 1 ffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ 3 þ ð49þ 3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1309þ 14þ 10ð1066þ3 14Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6þ4 11Þþ 618ð146984þ3611 11Þ B 1 ffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ 3 ð49þ 3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1309þ 14 10ð1066þ3 14Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6þ4 11Þ 618ð146984þ3611 11Þ A ffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 þ ð49 3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1309 14þ 10ð10663 14Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 64 11Þþ 618ð1469843611 11Þ B ffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 ð49 3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1309 14 10ð10663 14Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 64 11Þ 618ð1469843611 11Þ C 0 4 80 1 31 C d 11 13 1 References [1] G. Birkhoff, D.M. Young, Numerical quadrature of analytic and harmonic functions, J. Math. Phys. 9 (190) 1 1. [] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 19. [3] F. Lether, On Birkhoff Young quadrature of analytic functions, J. Comut. Al. Math. (196) 81 84. [4] J.N. Lyness, Quadrature methods based on comlex function values, Math. Comut. 3 (1969) 601 619. [] J.N. Lyness, L.M. Delves, On numerical contour integration round a closed contour, Math. Comut. 1 (196) 61. [6] J.N. Lyness, C.B. Moler, Numerical differentiation of analytic functions, SIAM J. Numer. Anal. 4 (196) 0 10. [] G.V. Milovanović, Numerical Analysis, Part II, Naučna Knjiga, Belgrade, 198 (Third Edition 1991) (Serbian). [8] G.V. Milovanović, R.Ž. Ðor dević, On a generalization of modified Birkhoff Young quadrature formula, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fis. No. 3 No. 6 (198) 130 134. [9] D.Ð. Tošić, A modification of the Birkhoff Young quadrature formula for analytic functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fis. No. 60 No. 633 (198) 3. [10] D.M. Young, An error bound for the numerical quadrature of analytic functions, J. Math. Phys. 31 (19) 4 44.