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REVT]E D'ANÄLYSE NI]MÉRIQUE ET DE TIIÉORIE DE L'APPROXIMÄTIONTome XXVI, N* l_2, 1997, pp. ljg_lï3ON AN APPROXIMATION FORMULAION PAVALOIU1. INTRODUCTIONThis Note contains some remarks concerning an approximation formula forfunctions, which is a generalization of some interpolation formulae given in [2]and [4], Inparticular, we shall show that only one of the formulae õrurir type,mentioned in [4], has a maximal degree of exactness. Some particular cases ofsuch formulae were also mentioned in 14,p. 1631.Denote by 1" the closed interval determined by two distinct points xs, x inR, For a (2n + l){imes derivable function .f : I, -+ R ancl n c N, consiáer theclass G of functions givenbyI(1,1) c = tr:g(r) = f(rr)+(, - r,)2,',=,o,f,(*o+ b,(t - xo)),a¡,b¡ eF.,i = þ,t e I*jConsider the following problem: Find a function g e G such that(r.2) ,f(,)þ)=g(,)(ro), i=7,rn.In [4] this problem was solved in some particular cases. We shall show that,for m: 2n, thís problem has a unique solution and we shall give a representationfor the remainder.2. DE,TERMINATION OF THE APPT{.OXIMÄTING FUNCTIONFor m : 2n,weare looking for a function g in G verifying conditions (1.2)and having amaximal degree of approximation.It is easily seen that conditions (1,2) lead to the follov¿ing systern, having thereal numbers ao bu i -= irn, as unknowns:1991 AMS Subject ClassiÍication: 65D05, 65D102 J On an Approximation Fornrula181180 Ion l'ávãloiu(2,r) Zo,ø! = I I (k +I), k = 0,1,'.,,2n'- I't=lConsidor now acontinrrcus function <p:fO, t] + R. and letjjq(')¿t =f o,*(0,)+ n[q]2nf'(*o+b,(r-*o)) =Ij=1./(r)(ro b!-t(r-"0)i-t +r,(x),(3.2)where(3.3)(; - t)r,,(*) =¡(zn+t) 0,)ü^(* - *o)'",(22) (zn)rbe a quadrature formula, having {¿,}i ^ knots and {ø,}i' as coefficients. Askingthat ,R[q/,] = 0for qo(r) = *,h = 0,2n - 1, formula(2'2) becomes the classicalGauss quaclrature formula,Ontheotherhand, theconditions n[or1-= 0, for qt(t) : tk ,k : O,zn -1,lead again to the system (2.1), implying that å, must be the roots of the Legenclrepolynomiai wn of degree n, i.e., the roots of the equationand 0, is a number contained in the open interval determined by xo and xo ++ b,(x - ro), 1 < i < n,From (3.2) we obtain the equalities(3.4) f (*) - f(rr) - (* - ,òf'(x, + b,(x - 'o)) =zn ,.(¡)r- I= f(*)- fQò Effi b¡i-t(x -"0)' -r¡(x)(x -ro), i =t,n'. Multiplying equaliti.r (3r9_ by ap taking into account solutions (2'1) andsumming up, we obtain 'b.i. l.(3.5) rG) -s(') = r(*) -i4P(' -,u)' -t,"r,Q)(' -'o)j=0 J i i=lNow, using (3.3) and Lagrangeform of the remainder in the Taylor formula,we get(23) w,(Ì):= &#lrr, - Ð'] = nThe coefficients a, are givenby the following formula("t)o(z'4) o' = leùq'r,çì -'u,ffi,¡f i = r'n '(see [1], p.261).Now, it is clear that the following theorem holds: (3.6) f(*)-s(,) =t## -2'l,7 ,(2,+r)10,)(zn)r x r2n*l- xo) 2Tnponev 2.1. If f : I * -+ R. is ø (Zn + t)-times deriyable function on Iothen there exísts only onefunction g e G verifying conditions (l2) for m:Zn.The parameters {a,}i-_, are given byformula Q,4),where {b,}i=, are theroots oJequation (2,3).where ïì € 1,.Setting'q(l) = t,n iÍt Q,2) andtaking into account the form of the remainderterm in the Gauss quadrature formula ll,p' 2591, we gettl3. DETERMINATION OF THD REMAII'{DER Zo,bl'*Consider the approximation formula(3.r) f(*) = g(x) + rlf),where g e G is a function veriffing (2.I) andrffi is the remainderIn the conditions of Theorcm2.I, it follows thatimpþingn ç2"¡rf'-l"tloZo,t" =(3.7)i=1i=1 (zn +t)l(zn)!182 Ion Pävãloiu 4 5 On an Approximation Formula 183Suppose now that the (2n + l)-order derivative of/ is bounded on 1" and let(3,S) M2,+1 = tuPl¡(".t)11¡l't eI,lTaking into account relations (3,6) and (3,7), one obtains the followingdelimitation for rl[](3.e) l,lr] = ,!"*lr, ?-1]'+ [rt]o ' 12u+t= (2', + ry - leùf- l'- ¿olREFEREN CE Sl. C. I. Berezin and N. Jidkov, Metody vychisleny, Finnatgiz, Moscow (1962).2. P. M. Humol and C. L. Seobock Jr., A generalization of Taylor's expansíon, Amor. Math.Monthly 56 (1949), 243-24'7.3. A. Lupaç, Calculul valorilor unor funclìí elementare, Gazeta Matomaticã (Ser. A) VII, .l(1986), 15-26.4. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., EnglowoodCliffs, N.J., 1964.Recoived March 15, 1996 Institutul de Calcul "Tiberiu Popovicíu"Str. G. BilaScu, nr. 37C.P. 68, O.P. I 3400 Clui-NapocøRonânia4. PARTICT]LAR CASESa) n: l. In this ca.se, br: l/2, ar: I ands(*) = /'('o) * (, - *o).t'',(-,. +U- ,,))From (3.9) we getvþ)-g(')l < Tl.-'ol"where M3 = stJp,rr,l,r"'(l)1,b) n:2, In this case, å, = +, b2 = +,et = az = L^os(,) = r(,ù * :e- "{r [- .t*r,-',)) * r'(* .yfr-',i]One also obtains the evaluationlr(')-g(')< ffiV-'ol',where M5 = sûp¡.r,lf(t){t)lRemark. Approximation formula of the type considered in this Note could beuseful for the approximate calculation of the values of some functions havingrational functions as derivatives,

On an approximation formula

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On an approximation formula

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