Numerical integration of analytic functions

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1 Numerical itegratio of aalytic fuctios Gradimir V. Milovaović, Dobrilo Ð Tošić, ad Miloljub Albijaić Citatio: AIP Cof. Proc. 1479, ); doi: 1.163/ View olie: View Table of Cotets: Published by the America Istitute of Physics. Additioal iformatio o AIP Cof. Proc. Joural Homepage: Joural Iformatio: Top dowloads: Iformatio for Authors:

2 Numerical Itegratio of Aalytic Fuctios Gradimir V. Milovaović, Dobrilo D-. Tošić ad Miloljub Albijaić Mathematical Istitute of the Serbia Academy of Scieces ad Arts, Kez Mihailova 36, P.O. Box 367, 111 Beograd, Serbia Departmet of Mathematics, Faculty of Electrical Egieerig, Uiversity of Belgrade, P.O. Box 35-54, 1112 Belgrade, Serbia Zavod za udžbeike, Obilićev veac 5, 111 Beograd, Serbia Abstract. A weighted geeralized N-poit Birkhoff Youg quadrature of iterpolatory type for umerical itegratio of aalytic fuctios is cosidered. Special cases of such quadratures with respect to the geeralized Gegebauer weight fuctio are derived. Keywords: quadrature formula; weight fuctio; error term; orthogoality; aalytic fuctio; odes; weight coefficiets. PACS: 2.6.Jh, 2.6.Nm INTRODUCTION I 195 Birkhoff ad Youg [2] proposed a quadrature formula of the form z +h z h f z)dz = h 15 ] [ ]} 24 f z )+4 f z + h)+ f z h) f z + ih)+ f z ih) +R BY 5 f ) for umerical itegratio of complex aalytic fuctios i Ω = { z : z z r }, where h r. This five poit quadrature formula is exact for all algebraic polyomials of degree at most five, ad its error term ca be estimated by R BY h 7 5 f ) 189 max f 6) z), z S where S deotes the square with vertices z + i k h, k =,1,2,3 see [16] ad [3, p. 136]). This error estimate is about four teths as large as the correspodig error R ES 5 f ) for the so-called exteded Simpso rule cf. [13, p. 124]) z +h z h f z)dz = h 9 { 114 f z )+34 [ f z + h)+ f z h) ] [ f z + 2h)+ f z 2h) ]} +R ES 5 f ), for which we have R ES h 7 5 f ) 756 f 6) ζ ), < ζ z 2h) < 1. 4h Without loss of geerality, the previous Birkhoff Youg formula ca be reduced to a itegratio over [ 1,1], I f )= f z)dz = f )+ 4 [ ] 1 [ ] f 1)+ f 1) f i)+ f i) +R5 f ). 1) I 1976 Lether [4] poited out that the three poit Gauss-Legedre quadrature which is also exact for all polyomials of degree at most five, is more precise tha 1) ad he recommeded it for umerical itegratio. However, Tošić [15] improved the quadrature 1) i a simple way takig its odes at the poits ±r ad ±ir, with r,1), istead of ±1 ad ±i, ad he derived a oe-parametric family of quadrature rules i the form I f )= r 4 ) f )+ 1 6r r 4 ) [ f r)+ f r) ] + 1 6r r 4 ) [ f ir)+ f ir) ] +R T 5 f ;r). Numerical Aalysis ad Applied Mathematics ICNAAM 212 AIP Cof. Proc. 1479, ); doi: 1.163/ America Istitute of Physics /$3. 146

3 Evidetly, for r = 1 it reduces to 1) ad for r = 3/5 to the three poit Gauss-Legedre formula. Sice the error-term R T 5 f ;r) ca be expressed as R T 5 f ;r)= 2 3 6! r4 + 2 ) f 6) )+ 2 7! 5 8! r4 + 2 ) f 8) )+, 2) 9! it is clear that for r = 4 3/7 the first term o the right-had side i 2) vaishes ad the correspodig formula reduces to the modified Birkhoff-Youg quadrature rule of the maximum degree of precisio seve amed MF i [15]), I f )= 16 ) 15 f ) [ f 4 3/7)+ f 4 ] 3/7) + 1 ) 7 7 [ f i 4 3/7)+ f i 4 ] 3/7) +R MF 5 f ), 3 with the error-term R MF 5 f )=R T 5 f ; 4 3/7) f 8) ). This formula was exteded by Milovaović ad D- ord - ević [14] to the followig quadrature rule of iterpolatory type I f )=Af)+B [ f r 1 )+ f r 1 ) ] +C [ f ir 1 )+ f ir 1 ) ] +D [ f r 2 )+ f r 2 ) ] + E [ f ir 2 )+ f ir 2 ) ] +R 9 f ;r 1,r 2 ), where < r 1 < r 2 < 1. For r 1 = r1 = )/143 ad r 2 = r2 63 = )/143, this formula has the algebraic precisio d = 13, with the error-term R 9 f ;r1,r 2 ) f 14) ). Quadrature formulae of Birkhoff Youg type for aalytic fuctios have bee ivestigated i several papers [1], [9], [1], [11], [12]. These formulas ca also be used to itegrate real harmoic fuctios see [2]). I additio, we metio also that Lyess ad Delves [6] ad Lyess ad Moler [7], ad later Lyess [5], developed formulae for umerical itegratio ad umerical differetiatio of complex fuctios. I this paper we cosider a geeralized quadrature formula of Birkhoff Youg type for itegratig aalytic fuctios with respect to a give weight fuctio. GENERALIZED BIRKHOFF YOUNG QUADRTAURE FORMULA I this sectio we cosider the followig N-poit geeralized quadrature formula of iterpolatory type I f ) := f z)wz)dz = Q N f )+R N f ), 3) 1 for umerical itegratio of aalytic fuctios, which are aalytic i the uit disk Ω = { z : z 1 }. The weight fuctio w : 1,1) R + is a arbitrary eve positive fuctio, for which all momets μ k = 1 zk wz)dz, k =,1,..., exist. Notice that μ 2k+1 = ad μ 2k > for each k N. The quadrature formula 3) has the odes at the zeros of a moic polyomial of degree N, i.e., ν 1 Q N f )= j= C j f j) )+ ω N z)=z ν p,ν z 4 )=z ν z 4 r k ), < r 1 < < r < 1, 4) A k f xk )+ f x k ) ] [ + B k f ixk )+ f ix k ) ]}, x k = 4 r k, k = 1,...,, where =[N/4] ad ν = N 4[N/4] {,1,2,3}), i.e., N = 4 + ν. The correspodig remaider term is deoted by R N f ). Practically, we cosider four subclasses of these quadratures, Q 4+ν f ), ν =,1,2,3. Notice that i Q 4 f ) the first sum is empty. Also, i order to have Q N f )=I f )= for f z) =z, it must be C 1 =, so that Q 4+1 f ) Q 4+2 f ). Recetly Milovaović [1] has proved the existece ad uiqueess of these quadratures Q N f ), with a maximal degree of precisio d = 6 + s, where =[N/4], ν = N 4[N/4] {,1,2,3}, ad s = { ν 1, ν =,2, ν, ν = 1,3, 5) 147

4 as well as a characterizatio of such geeralized quadratures i terms of multiple orthogoal polyomials, usig the the orthogoality coditios [1, Theorem 4.3] t k p,ν t 2 )t s/2 w t)dt =, k =,1..., 1. 6) I this paper we give a method for umerical costructio of quadratures Q 4+ν f ), ν =,1,2,3 usig the momets of the weight fuctio. Accordig to 4), the polyomial p,ν t 2 ) ca be expressed i the form p,ν t 2 )= j= 1) j σ j t 2 j), 7) where σ j are the so-called elemetary symmetric fuctios, defied by σ j = k1,...,k j ) r k1 r k j, j = 1,...,, ad the summatio is performed over all combiatios k 1,...,k j ) of the basic set {1,...,}. Thus, σ 1 = r 1 + r r, σ 2 = r 1 r r 1 r,..., σ = r 1 r 2 r, ad for the coveiece we put σ = 1. Startig from the orthogoality coditios 6), we obtai the followig system of liear equatios i.e., 1) j σ j j= j=1 t k+2 j)+s/2 w t)dt =, k =,1..., 1, 1) j 1 m k, j σ j = m k,, k =,1..., 1, 8) where m k, j = tk+2 j)+s/2 w t)dt = 2 z2k+4 j)+s+1 wz)dz. Sice s+1 is always a eve umber see 5)), the coefficiets i the previous system of equatios ca be expressed oly i terms of the momets of the weight fuctio w. Namely, m k, j = μ 2k+4 j)+s+1, k =,1,..., 1; j =,1,...,. TABLE 1. The values of σ k, k = 1,...,, for = 1,2,3,4 for the Legedre weight wz)=1 ad wz)= z weight Legedre γ =, α = ) wz)= z γ = 1, α = ) ν σ 1 σ 2 σ 3 σ 4 σ 1 σ 2 σ 3 σ 4 1 1/5 1/3 1, 2 3/7 1/2 3 5/9 3/5 2 7/99 1/33 4/5 1/15 1, 2 126/143 15/143 2/21 1/7 3 66/65 7/39 15/14 3/ /85 63/221 1/221 5/4 5/14 1/84 1, 2 429/ /1615 7/323 7/5 1/2 1/ / / /323 84/55 7/11 2/ / /523 66/7429 5/ /33 28/33 4/33 1/495 1, 2 24/ / /1925 9/ /13 756/715 28/143 1/ / / / /135 18/91 18/143 4/143 15/11 I the case of the geeralized Gegebauer weight fuctio wz)= z γ 1 z 2 ) α, γ,α > 1 see [8, pp ]), we obtai m k, j = t k+2 j)+β 1 t) α Γα + 1)Γk + 2 j)+β + 1) dt =, Γk + 2 j)+α + β + 2) where β =s + γ)/2. The coefficiets σ k, k = 1,...,, of the ode polyomial for some specific parameters γ ad α ad 4 are preseted i Tables 1 ad 2. The determiatio of the weight coefficiets A k, B k, C j i the quadrature sum Q N f ) is a liear problem ad it ca be solved by iterpolatio cf. [8, 5.1]). At the ed, as a example, we metio oly a quadrature rule with respect to the Chebyshev weight of the secod kid of the precisio d = 15, 1 f z) 1 z 2 dz 131π 5π f ) f )++ 2 A k f 4 r k )+ f 4 r k ) ] [ + B k f i 4 r k )+ f i 4 r k ) ]}, 148

5 TABLE 2. The values of σ k, k = 1,...,, for = 1,2,3,4 for the Chebyshev weight of the first ad secod kid weight Chebyshev I γ =, α = 1/2) Chebyshev II γ =, α = 1/2) ν σ 1 σ 2 σ 3 σ 4 σ 1 σ 2 σ 3 σ 4 1 3/8 1/8 1, 2 5/8 5/ /48 7/16 2 7/8 7/128 7/12 7/384 1, 2 21/2 21/128 3/4 9/ /28 33/128 99/112 33/ /224 99/256 33/496 33/32 55/256 11/496 1, 2 143/96 143/ / /12 429/ / /8 1287/ / /88 117/256 65/ /22 117/128 65/512 39/ /44 39/64 65/124 13/ , 2 85/44 221/ / / /52 51/64 119/124 85/ / /74 323/ / / / / / where r 1,2 = 99 ± 3333 ) /224, ad the correspodig umerical values of the coefficiets A 1,A 2,B 1,B 2 ) are ) , , , Applyig this rule to the itegral I = 1 ez 1 z 2 dz = πi 1 1), where I 1 z) is the modified Bessel fuctio of the first kid, we obtai the approximative value , with the relative error ACKNOWLEDGMENTS The authors were supported i part by the Serbia Miistry of Educatio ad Sciece Project: Approximatio of itegral ad differetial operators ad applicatios, grat umber #17415). REFERENCES 1. M. Acharya, B.P. Acharya, S. Pati, Numerical evaluatio of itegrals of aalytic fuctios, Iterat. J. Comput. Math. 87, No. 12, ). 2. G. Birkhoff, D. M. Youg, Numerical quadrature of aalytic ad harmoic fuctios, J. Math. Phys. 29, ). 3. P. J. Davis, P. Rabiowitz, Methods of Numerical Itegratio, New York: Academic Press, F. Lether, O Birkhoff-Youg quadrature of aalytic fuctios, J. Comput. Appl. Math. 2, ). 5. J.N. Lyess, Quadrature methods based o complex fuctio values, Math. Comp. 23, ). 6. J. N. Lyess, L. M. Delves, O umerical cotour itegratio roud a closed cotour, Math. Comp. 21, ). 7. J.N. Lyess, C.B. Moler, Numerical differetiatio of aalytic fuctios, SIAM J. Numer. Aal. 4, ). 8. G. Mastroiai, G.V. Milovaović, Iterpolatio Processes Basic Theory ad Applicatios, Spriger Moographs i Mathematics, Berli Heidelberg: Spriger Verlag, M.T. McGregor, O a modified Birkhoff-Youg quadrature formula for aalytic fuctios, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat. 3, ). 1. G.V. Milovaović, Numerical quadratures ad orthogoal polyomials, Stud. Uiv. Babeş-Bolyai Math. 56, ). 11. G.V. Milovaović, Geeralized quadrature formulae for aalytic fuctios, Appl. Math. Comput. 218, ). 12. G.V. Milovaović, A.S. Cvetković, M. Staić, A geeralized Birkhoff-Youg-Chebyshev quadrature formula for aalytic fuctios, Appl. Math. Comput. 218, ). 13. J. B. Scarborough, Numerical Mathematical Aalysis, Baltimore: The Johs Hopkis Press, G. V. Milovaović, R. Ž. D- ord - ević, O a geeralizatio of modified Birkhoff-Youg quadrature formula, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat. Fiz., No. 735 No. 762, ). 15. D. D-. Tošić, A modificatio of the Birkhoff-Youg quadrature formula for aalytic fuctios, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat. Fiz., No. 62 No. 633, ). 16. D. M. Youg, A error boud for the umerical quadrature of aalytic fuctios, J. Math. Phys. 31, ). 149