~j=zhax~ 6=0, t,2... ), k=~

Transcription

1 Numerische Mathematik 6, (964) Exactess Coditios i Numerical Quadrature* By HERBERT S. WILl* The itegral b () I--f t(x)dx is usually computed umerically by meas of a formula of the type (2) I = H / (xl) H, / (x,) + E. The weights H... H, ad abcissae x... x,~ are determied by selectig a family ~- of fuctios ](x) ad requirig that (3) E= (l~-~)- We propose here to modify this last restrictio by requirig oly that the average error be as small as possible over the family ~-. It will be see that this coditio leads to extremely simple error estimates i certai cases as well as permittig cosiderable elargemet of the families of fuctios cosidered. We specialize at oce to the iterval (a, b)= (, t) ad the family (4) ~-= {x"}~=o. Geeralizatios will be obvious. The error committed i itegratig x i is j+l A=I ad so we cosider the questio of miimizig (~) w( I..., H,; x... x,) ~- 75r_- ( -- H A xi~. k=l Geometrically speakig we wish to fid the vector of approximate momets ~j=zhax~ 6=, t,2... ), k=~ which is earest to the true momet vector ~J- j+l (/---o,... ) i the usual Euclidea (l S) orm, the iteger beig fixed. * This work supported i part by the Argoe Natioal Laboratory.

2 36 HERBERT S, WILF: Suppose for the preset that this has bee doe, ad that W of equatio (5) assumes its miimum value W~ at H*... H~*; x~'... x*. Now let (6) /(x)-- a,x" = be aalytic i Ix] <t ad ~ o the uit circle; i.e., (7) II/3=-~7 f I/(e')l ~a~ = ~, I..' < ~. = The the error committed i itegratig /(x) is J /(x) dx -k=l Y, H* t(x*) = Za.m -- X m= t k=l <-{L,.,'}'[L{ ' }] k=l Cosequetly, (8) [Errorl =< w2 II/ ad so the same quatity which we sought to miimize o geometrical grouds i (5) also appears as the coefficiet i the error estimate. We emphasize here that IV. is idepedet of / ad so ca be (ad is below) tabulated as a fuctio of. To carry out the miimizatio, the equatios are of the form (9) OW W (~=... ) Ha O x~ log -- k - - ~/* (~ = t... ) X~t I -- X~ k=l I -- X~ X k (o) t t x~ (t - x~) xl log - x~ x, - H~ (t-***~)~ "" ' (geomet.). These are 2 simultaeous trascedetal equatios i the ukows, x... x~, H... H~. A aalytic solutio does ot seem feasible, but these equatios do have a iterestig geometric iterpretatio. I fact if we cosider the Stieltjes trasform of the measure dx, l--xg -- x log l--x

3 Exactess Coditios i Numerical Quadrature 3t 7 ad the approximate Stieltjes trasform (2) F*(x) = ~, Hk k=l I--XX k obtaied by doig the itegral i (t ) by the rule (2), the it is easy to verify that equatios (9), (t) are precisely the assertios that the curves y=f(x), y=f*(x) have secod order cotact at each of the poits x... x~. The same remark holds for more geeral measures. The Stieltjes trasform also eters ito the error estimate (8). Ideed, let us ask whe the sig of equality ca hold there. From the derivatio of (8) it is clear that this will happe if ad oly if a,~=~.,,~+ x~ ~ (re=o,... ) i=l which meas that the fuctio beig itegrated is (3) /(x) =~ _ ~;T~ - _ tt~ x~ ~ x ~ 2 = ~ IF(x) -- F*(x)]. We may say the that the "hardest" of all fuctios to itegrate by this method is the error i the approximatio of (tt) by (2). These remarks give a ew ad simple characterizatio of the miimum value W,,. For if we have the equality sig i (8) with [=F--F* the =w~w~ =w~ because the approximate itegral of F(x)--F*(x) is zero sice that fuctio vaishes at each of the x~. It follows that W~ = -- ~ log ~. k=l I words, W, is the magitude o] the error i itegratig x - log (- x) - by our ]ormula. Solutios to equatios (9), () are show below for =2, 3. Table xj Hj W

4 38 HERBERT S. WILF: Cocerig the order of magitude of W~ we ca show oly that W~---- (log /) (-+ oo), which is probably quite coservative. To see this, ote that sice W~ is the miimum of W(H a... H.; x I... x.) we have i particular we have clearly... ' +l... +t --2{ ' '2 i=o j-~l k=l (~]] ~ 2 x-~ +t h~a=,~ log k +l zr 2 2 f +l t 6 3 ~ l~ t dt+~ o - s ~ = dt+o/ 2) / (log /). I[/ max I/(=) I" < + +l +l,ff (+t) 2 did, xy (+)2 I particular, of course, W~-+ (--~ o~), ad so the sequece of approximate itegrals coverges to the exact itegral for ay [ with I[]H <~. We coclude by makig a few remarks about the use of the method. The pricipal disadvatage of our scheme appears to be that i the error estimate (8) we require kowledge of [[[[[ defied i (7). This may be virtually impossible to obtai for fuctios which are give oly tabularly o (, I). For itegrads give aalytically we are i a much better positio. I fact, the estimate which is obvious from (7) may be sufficiet for most purposes. For istace, i I -= f e x'+x dx /~-+ 7 ezt+z e 2 7wT - ad so the error would be less tha W.~e26 - ~ i this case. Next the estimate (8) will be ufavorable compared to estimates from, say, Gauss-Quadrature i the case of a fuctio with slowly growig derivatives. A fuctio like e* o (, ) is a example of this kid. I the other directio, we may get cosiderably more favorable estimates from the preset method if the itegrad is very large or ifiite ear the right had ed-poit of itegratio. A extreme example of this kid is f I---- log t-x dx x -- ~2 6

5 Exactess Coditios i Numerical Quadrature 3t9 i which o error estimate is available at all from the usual Gauss-quadrature ~2 formulae whereas (8) tells us that the error is surely less tha ~- W~ with the preset method. For a less extreme case, cosider [(x)=e ~. For fixed we would get a Gaussia error estimate of the form C~ ~.2~ea. With the preset method our estimate is f~ ~k~89 e x W~/k~o-k.~ ~" W~ (4r~ 2)88 (/,-> oo). Evidetly if is fixed the for large eough ~ the latter will be superior. (Received March 8, 964) Uiversity of Pesylvaia Departmet of Mathematics Philadelphia 4, Pesylvaia/USA