On the variance of the Gaussian quadrature rule

Denote by Pn =1 a f (x ) the Gaussian quadrature rule for the integral R 1 1 f (x) dx. We give a simple explicit expression for the \variance" Pn =1 a2 . The method can be used to obtain similar results for the Lobatto rule. 1 1. The result A quadrature rule (on [ 1; 1]) is a functional Qn on C [ 1; 1] of the form Qn [f ] = n X =1 a f (x ) ; a 2 R (1) 1 x1 < x2 < : : : < xn 1 : The \variance of Qn" VarQn := n X =1 a2 is of interest in the statistical theory of error propagation, for a survey cf. Forster [3]. Among all quadrature rules the Gaussian rule QGn (to be de ned below) is the most interesting and the most useful rule. Forster and Petras [4] proved 2 2n+ 1 1 1 (2n + 1)2 < VarQGn < 2 2n+ 1 ; (2) n = 3; 4; : : : : Our aim is the proof of Theorem 1 VarQGn = 6 2n+ 1 1 1 (2n 1) (2n+ 3) n X =1 1 2 + 12 (2n 1) (2n+ 3) : (3) The relation lim n!1 (2n+ 1)VarQGn = 2 is an immediate consequence, with a little computation we obtain more precisely: The sequence (2n+ 1)VarQGn (n = 4; 5; : : :) tends increasingly to 2, and the sequence (2n+ 1) + (2n+ 1) 1 VarQGn (n = 1; 2; : : :) tends decreasingly to 2, then (2) (and a little more) follows. Our proof of (3) uses only quite elementary properties of the Legendre polynomials, whereas the proof of (2) in [4] depends on the bounds for the coe cients a of QGn , whose proofs are long and di cult. 2. The proof We start with the Legendre polynomial Pn (x) := 1 2nn! d dx n x2 1 n : 2 It is well known (for this and further results on Pn and QGn see Szego [7] or Brass [1]) that Pn has n simple zeros located in the interior of [ 1; 1]. QGn is de ned by using the zeros x1; : : : ; xn as evaluation points in (1) and by de ning a = aG := Z 1 1 Pn (x) (x x )P 0 n (x ) dx : (4) Applying some identities for the Pn, one obtains as a further expression aG = 2 (1 x2 ) [P 0 n (x )]2 (5) (Szego [7] p. 352 or Brass [1] p. 151). The main property of the Gaussian quadrature rule QGn is n X =1 aG p (x ) = Z 1 1 p (x) dx ; (6) holding for any polynomial p of degree 2n 1. The idea of the proof consists in constructing a polynomial p̂ of degree 2n 1 with p̂ (x ) = a = 1; : : : ; n ; then we have VarQGn = n X =1 aG 2 = n X =1 aG p̂ (x ) = Z 1 1 p̂ (x) dx : (7) To this end we introduce the \associated Legendre polynomials" P̂n by P̂n (x) := Z 1 1 Pn (y) Pn (x) y x dy : We obtain from (4) aG = P̂n (x ) P 0 n (x ) and by applying (5) aG 2 = " P̂n (x ) P 0 n (x )#2 = aG (1 x2 ) hP̂n (x )i2 2 hence we have aG = (1 x2 ) hP̂n (x )i2 2 ~ p (x ) : The polynomial ~ p has the degree 2n and its main coe cient is ( 2) times the main coe cient of P 2 n , so we may choose p̂ (x) := ~ p (x) + 2P 2 n (x) 3 and we obtain according to (7)VarQGn = Z 11 ~p (x) dx+ 2 Z 11 P 2n (x) dx :The relationZ 11 Pi (x)Pj (x) dx = ( 0i 6= j22j+1 i = j(8)is fundamental in the theory of Legendre polynomials. So we arrive atVarQGn =12 Z 11 1 x2hP̂n (x)i2 dx+ 42n+ 1 :(9)The crucial tool in our proof is the following identity of Christo elP̂n (x) = bn 12 cX=0 c Pn 2 1 (x) with c := 2 2n 4 1(2 + 1) (n ) :(10)A proof can be found in Hobson ([5] p. 53), the simplest proof is by induction using therecurrence relations (n+ 1)Pn+1 (x) = (2n+ 1)xPn (x) nPn 1 (x)(11)P0 (x) = 1 ; P1 (x) = xand (easy consequence of (11))(n+ 1) P̂n+1 (x) = (2n+ 1)xP̂n (x) nP̂n 1 (x)P̂0 (x) = 0 ; P̂1 (x) = 2 :If we introduce (10) in (9) and apply (8), we getVarQGn = 12 bn 12 cX=0 c2 Z 11 1 x2 P 2n 2 1 (x) dx+ bn 12 cX=1 c 1c Z 11 1 x2 Pn 2 1 (x)Pn 2 +1 (x) dx+ 42n+ 1 :The integrals can be determined using (11) and (8). We obtainZ 11 1 x2 P 2j (x) dx = 12j + 1 11(2j 1) (2j + 3) ;Z 11 1 x2 Pj 1 (x)Pj+1 (x) dx =2j (j + 1)(2j 1) (2j + 1) (2j + 3) :4 Hence we haveVarQGn =12 bn 12 cX=0 4 (2n 4 1)2(2 + 1)2 (n )212n 4 1 11(2n 4 3) (2n 4 + 1)8 bn 12 cX=1 2n 4 1(2 + 1) (n ) 2n 4 + 3(2 1) (n + 1)(n 2 ) (n 2 + 1)(2n 4 1) (2n 4 + 1) (2n 4 + 3)+ 42n+ 1 :In the last step we have to simplify this expression by partial fraction expansion of thesummands. After some work we obtain the theorem.3. Remarks(i) Chawla and Ramakrishnan [2] were the rst to give an explicit expression for thevariance of a quadrature rule in a nontrivial case. Their result concerns the Polyaquadrature rule and their method can be applied to the rules of Filippi and Clen-shaw/Curtis (for the de nitions cf. Brass [1]), but it does not seem to be possible toobtain theorem 1 with their method.(ii) Our method can be extended to the Gaussian rules with ultraspherical weight functi-ons. The generalization of the crucial identity (10) is known, see e.g. Lewanowicz [6](3.2). But I had no success in the simpli cation of the obtained expression, and so itseems to be of minor interest.(iii) Our method can be applied to the Lobatto quadrature rule QLon+1 (mainly by replacingPn by Pn+1 Pn 1). We give the result asTheorem 2VarQLon+1 = 62n+ 1 1 +3(2n 1) (2n + 3) nX=1 12+12(2n 1) (2n+ 3)36(2n 1) (2n+ 3)n (n + 1) :References[1] H. Brass: Quadraturverfahren. Vandenhoeck und Ruprecht, Gottingen 1977.5 [2] M.M. Chawla, T.R. Ramakrishnan: Minimum variance approximate formulas.SIAM J. Numer. Anal. 13 (1976), 113-128.[3] K.-J. Forster: Variance in quadrature a survey. In: H. Brass and G. Hammerlin(Eds.): Numerical Integration IV. Birkhauser Verlag, Basel, Boston, Berlin (1993),91-110.[4] K.-J. Forster, K. Petras: On the variance of Gaussian quadrature formulae inthe ultraspherical case. Calcolo 31 (1994), 1-33.[5] E.W. Hobson: The theory of spherical and ellipsoidal harmonics. Chelsea PublishingCompany, New York, second reprint 1965.[6] S. Lewanowicz: Properties of the polynomials associated with the Jacobi polynomi-als. Math. Computation 47 (1986), 669-682.[7] G. Szego: Orthogonal polynomials. American Mathematical Society, Providence,Rhode Island, fourth edition (1975).6