Bulletin T.CXLV de l'Acad´emieserbe des sciences et des arts − 2013 Classe des Sciences math´ematiqueset naturelles Sciences math´ematiques, No 38 FAMILIES OF EULER-MACLAURIN FORMULAE FOR COMPOSITE GAUSS-LEGENDRE AND LOBATTO QUADRATURES G. V. MILOVANOVIC´ (Presented at the 4th Meeting, held on May 31, 2013) A b s t r a c t. Beside a short account on the Euler-Maclaurin for- mula, using a recent progress in variable-precision arithmetic and symbolic computation and the corresponding Mathematica package Orthogonal- Polynomials [Facta Univ. Ser. Math. Inform. 19 (2004), 17 − 36; Math. Balkanica 26 (2012), 169−184], we obtain extensions of the Euler-Maclaurin formula for the composite (shifted) Gauss-Legendre formula, as well as for its Lobatto modification. Some special cases are also presented. AMS Mathematics Subject Classification (2000): 65D30, 65D32, 40C15 Key Words: Gauss-Legendre quadrature, Lobatto quadrature, Euler- Maclaurin formulae, Bernoulli numbers, Bernoulli polynomials, orthogonal polynomials, quadrature sum, remainder term. 1. Introduction The Euler-Maclaurin summation formula plays an important role in the broad area of numerical analysis, analytic number theory, and the theory 64 G. V. Milovanovi´c of asymptotic expansions, as well as in many applications in other fields. In connection with the so-called Basel problem (or in modern terminology, with determining ζ(2)), in 1732 Leonhard Euler discovered this formula, Z Xn n 1 f(k) = f(x) dx + (f(0) + f(n)) 2 k=0 0 h i Xr B + 2j f (2j−1)(n) − f (2j−1)(0) + E (f); (1.1) (2j)! r j=1 2r which holds for any n; r 2 N and f 2 C [0; n], where B2j are Bernoulli numbers (B0 = 1, B1 = −1=2, B2 = 1=6, B3 = 0, B4 = −1=30, :::). This formula was also found independently by Maclaurin. While in Euler's case the formula (1.1) was applied for computing slowly converging infinite series, in the second one Maclaurin used it to calculate integrals. A history of this formula was given by Barnes [3], and some details can be found in [19], [1], [11], [12], [5]. Bernoulli numbers Bn, n = 0; 1;:::, can be expressed as values at zero of the corresponding Bernoulli polynomials, which are defined by the gener- ating function 1 text X B (x)tj = j : et − 1 j! j=0 Bernoulli polynomials play a similar role in numerical analysis and approx- imation theory like orthogonal polynomials. First few polynomials are 1 1 3x2 x B (x) = 1;B (x) = x − ;B (x) = x2 − x + ;B (x) = x3 − + ; 0 1 2 2 6 3 2 2 1 5x4 5x3 x B (x) = x4 − 2x3 + x2 − ;B (x) = x5 − + − ; etc. 4 30 5 2 3 6 Some interesting properties of these polynomials are Z 1 0 − − n 2 N Bn(x) = nBn−1(x);Bn(1 x) = ( 1) Bn(x); Bn(x) dx = 0 (n ): 0 The error term Er(f) in (1.1) can be expressed in the form (cf. [5]) 1 Z +X n ei2πkt + e−i2πkt E (f) = (−1)r f (2r)(x) dx; r (2πk)2r k=1 0 Families of Euler-Maclaurin formulae 65 or in the form Z n − b c B2r(x x ) (2r) Er(f) = − f (x) dx; (1.2) 0 (2r)! where bxc denotes the largest integer that is not greater than x. Supposing f 2 C2r+1[0; n], after an integration by parts in (1.2) and recalling that the odd Bernoulli numbers are zero, we get (cf. [14, p. 455]) Z n − b c B2r+1(x x ) (2r+1) Er(f) = f (x) dx: (1.3) 0 (2r + 1)! If f 2 C2r+2[0; n], using Darboux's formula one can obtain (1.1), with Z ( − ) 1 1 nX1 E (f) = [B − B (x)] f (2r+2)(k + x) dx (1.4) r (2r + 2)! 2r+2 2r+2 0 k=0 (cf. Whittaker & Watson [26, p. 128]). This expression for Er(f) can be also derived from (1.3), writting it in the form Z ( − ) 1 B (x) nX1 E (f) = 2r+1 f (2r+1)(k + x) dx r (2r + 1)! 0 k=0 Z ( − ) 1 B0 (x) nX1 = 2r+2 f (2r+1)(k + x) dx; (2r + 2)! 0 k=0 and then by an integration by parts, the last expression becomes " # ( − ) 1 Z ( − ) B (x) nX1 1B (x) nX1 2r+2 f (2r+1)(k + x) − 2r+2 f (2r+2)(k + x) dx: (2r + 2)! 0 (2r + 2)! k=0 0 k=0 Because of B2r+2(1) = B2r+2(0) = B2r+2, Er(f) can be represented in the form (1.4). r Since (−1) [B2r+2 − B2r+2(x)] ≥ 0 on [0; 1] and Z 1 [B2r+2 − B2r+2(x)] dt = B2r+2; 0 according to the Second Mean Value Theorem for Integrals, there exists η 2 (0; 1) such that ( − ) B nX1 B E (f) = 2r+2 f (2r+2)(k+η) = n 2r+2 f (2r+2)(ξ); 0 < ξ < n: r (2r + 2)! (2r + 2)! k=0 (1.5) 66 G. V. Milovanovi´c Remark 1.1. The Euler-Maclaurin summation formula is implemented in Mathematica as the function NSum with option Method -> Integrate. Practicaly, the Euler-Maclaurin summation formula (1.1) is related with the so-called composite trapezoidal rule, Xn nX−1 00 1 1 T f := f(k) = f(0) + f(k) + f(n): n 2 2 k=0 k=1 Namely, h i Xr B T f − I f = 2j f (2j−1)(n) − f (2j−1)(0) + ET (f); (1.6) n n (2j)! r j=1 R n T where Inf := 0 f(x) dx and the remainder term Er (f) is given by (1.5) if the function f belongs to C2r+2[0; n]. Similarly, for a quadrature sum with values of the function f at the 1 − points x = k + 2 , k = 0; 1; : : : ; n 1, i.e., for the midpoint rule − ( ) nX1 1 M f := f k + ; n 2 k=0 there exists the so-called second Euler-Maclaurin summation formula − h i Xr (21 2j − 1)B M f − I f = 2j f (2j−1)(n) − f (2j−1)(0) + EM (f); (1.7) n n (2j)! r j=1 for which (2−1−2r − 1)B EM (f) = n 2r+2 f (2r+2)(ξ); 0 < ξ < n; r (2r + 2)! when f 2 C2r+2[0; n] (cf. [20, p. 157]). The both formulas, (1.6) and (1.7), can be unified as h i Xr B (τ) Q f − I f = 2j f (2j−1)(n) − f (2j−1)(0) + EQ(f); n n (2j)! r j=1 where τ = 0 for Qn ≡ Tn and τ = 1=2 for Qn ≡ Mn. It is true, because of the fact that [24, p. 765] (see also [7]) ( ) 1 B (0) = B and B = (21−j − 1)B : j j j 2 j Families of Euler-Maclaurin formulae 67 1 If we take a combination of Tnf and Mnf as Qnf = Snf = 3 (Tnf + 2Mnf), which is, in fact, the well-known classical composite Simpson rule, " # − − ( ) 1 1 nX1 nX1 1 1 S f := f(0) + f(k) + 2 f k + + f(n) ; n 3 2 2 2 k=1 k=0 we obtain − h i Xr (41 j − 1)B S f − I f = 2j f (2j−1)(n) − f (2j−1)(0) + ES(f): (1.8) n n 3(2j)! r j=2 Notice that the summation on the right hand-side in the previous equality starts with j = 2, because the term for j = 1 vanishes. For f 2 C2r+2[0; n] it can be proved that there exists ξ 2 (0; n), such that (4−r − 1)B ES(f) = n 2r+2 f (2r+2)(ξ): r 3(2r + 2)! Some periodic analogues of the Euler-Maclaurin formula with applica- tions to number theory have been developed by Berndt and Schoenfeld [4]. In the last section of [4], they showed how the composite Newton-Cotes quadrature formulas (Simpson's parabolic and Simpson's three-eighths rules), as well as various other quadratures (e.g., Weddle's composite rule), can be derived from special cases of their periodic Euler-Maclaurin formula, includ- ing explicit formulas for the remainder term. Also, in the papers [8], [23], [25], the authors considered some generalizations of the Euler-Maclaurin for- mula for some particular Newton-Cotes rules, as well as for 2- and 3-point Gauss-Legendre and Lobatto formulas (see also [2], [9], [16], [17]). A recent progress in variable-precision arithmetic and symbolic compu- tation has enabled a development of symbolic/variable-precision software for orthogonal polynomials and quadratures of Gaussian type, as well as for their many generalizations. The corresponding software is available to- day (Gautschi's package SOPQ in Matlab (cf. [21]) and our Mathematica package OrthogonalPolynomials [6], [22]). Using this advantage in this paper we give extensions of Euler-Maclaurin formulae by replacing Qn by the composite Gauss-Legendre shifted formula, as well as by its Lobatto modification. Several special cases were obtained by using our Mathematica package OrthogonalPolynomials. 68 G. V. Milovanovi´c 2. Euler-Maclaurin formula based on the composite Gauss-Legendre formula G G Let wν = wν and τν = τν , ν = 1; : : : ; m, be weights (Christoffel num- bers) and nodes of the Gauss-Legendre quadrature formula on [0; 1], Z 1 Xm G G G f(x) dx = wν f(τν ) + Rm(f); (2.1) 0 ν=1 where the nodes τν are zeros of the shifted (monic) Legendre polynomial ! −1 2m π (x) = P (2x − 1): m m m − G Degree of its algebraic precision is d = 2m 1, i.e., Rm(f) = 0 for all algebraic polynomials of degree ≤ 2m − 1.