Mechanical Quadrature Near a Singularity

Transcription

1 MECHANICAL QUADRATURE NEAR A SINGULARITY 215 Mechaical Quadrature Near a Sigularity The purpose of this ote is to preset coefficiets to facilitate computatio of itegrals of the type I x~^fix)dx. If the itegrad x~»/(x) has o sigularity at the origi, the coefficiets preseted are still applicable, though i this case other well-kow procedures for umerical itegratio might be more appropriate. I this regard, it should be oted that suitable modificatios of fix) permit oe to employ the coefficiets i evaluatig xifix)dx. I applied problems, the aalytical form of fix) is usually kow though it may be rather complicated i form. The sigularity may be removed by imposig the trasformatio x = y2 or by fractioal itegratio after which ay oe of a umber of umerical procedures ca be applied. The former approach is ot satisfactory sice the upper limit of itegratio as a rule becomes irratioal ad so complicates tabulatio of the resultig itegrad. Itegratio by parts requires the computatio of fix). If fix) is quite complicated, its derivative is usually eve more so, ad agai a excessive amout of tabulatio is required. A third possibility is to expad fix) ito a power series about the origi. However, i may cases this is ot easy owig to the complexity of fix). Furthermore, the series may be slowly coverget. A more satisfactory procedure is to write the Lagragia polyomial which precisely fits fix) at equally spaced poits over the iterval (0, b). The a umerical itegratio formula is easily evolved by calculatig the momets I x~^xmdx, m = 0, 1, 2, -. It is /r = firh), r = 0, 1,, ad write x~ifix)dx. Let (1) A = 2(«A)* ( fr7rm \ A,"1 + R, (èm-m) where R is the remaider term. Followig the procedure described above, the exact coefficiets yrm ad D are give i Table I for «= 1(1)10. It is clear that the (m + 1) poit formula so derived is exact if fix) is a polyomial of the th degree. If fix) is ot of this form, it is of iterest to examie the remaider. Evaluatio of the remaider term i liear methods of approximatio has bee discussed by Mile.12 Followig his otatio, the remaider is give by (2) R = fh f<»+»is)gis)ds. The aalytical form of Gis) is easily deduced from the approximatig formula. For the ( + 1) poit formula (3) \Gis) = I ar*(x - s)"dx - 2(mä)» (ra - s)"yrm/d, J s r_

2 216 MECHANICAL QUADRATURE NEAR A SINGULARITY where irh s) = irh - s) if rh > s W =0 if rh < s. The formula for A is exact if fix) = (a; s) ad so Jmh *-*(* - s)»dx - 2(mA)» irh - s)"ytm/d. 0 r=0 Suppose 0 < s < h. Subtractig (5) from (3) ad usig (4), a straightforward reductio gives (6) \Gis) = 2(-l)»(«A)lT <»>5» - ^.g....(2w + t) ^ 0 < s < h. Deote the right had side of (5) by Gi(s). Let Hris) = 2ih)\rh - syyt^/d; (^ Gris) = Gr_!(j) + Hr-!ÍS), f = 2, 3,, M. The (8) m!g(s) = Gr(s) if (r - 1)A < 5 < rh, r = 1, 2, -,. We exted some termiology by Mile ad say that Gis) is defiite if it does ot chage sig i the iterval of (0, h) ; otherwise, it is idefiite. If Gis) is defiite, applicatio of the mea value theorem to (2) gives XaJt Gis)ds, where 0 < 6 < h. I ay evet, bouds for the error are give by ( h \R\ <M\ \G(s)\ds or \R\ < MKh, (10) M = Max /(B+1)(*)I. K = Max G(î). 0< j< h 0< j< h If Gis) is idefiite, it must vaish at least oce i the ope iterval (0, ma) (observe that G(0) = G(mA) = 0). Suppose is the oe ad oly oe such poit. The the remaider may be composed of two parts. X r«h Gis)ds + /< +»>(*,) J Gis)ds where 0 < 0i < < B2 < h. The extesio of this argumet for Gis) vaishig at more tha oe poit i the ope iterval of itegratio is obvious. Gis)ds may be evaluated directly from the approximatig formula without a kowledge of Gis). I the preset istace / Ä (12) I Gis)ds = 2(mA)»â"+1{m"+1/(2m + 3) - r^y^/dj/i + 1)! J0 Employig the remaider term for the polyomial approximatio to fix), r-l

3 MECHANICAL QUADRATURE NEAR A SINGULARITY 217 a expressio equivalet to (12) is / A t*h (13) Gis)ds = I x*ix - h)ix - 2h) (x - h)dx/i + 1)! Thus, if Gis) is defiite, computatio of a rigorous error term is cosiderably simplified. As to the defiiteess of Gis), o geeral theorem is available. But employig (8), we ca compute G(s) for each formula ad ascertai i a heuristic fashio if it is defiite. I this study, Gis) is defiite for all the eve poit formulas. For the odd poit formulas, Gis) vaishes at oe poit i the ope iterval (0, h). For each itegratio formula give i Gis)ds is preseted i Table II. For the X{ Çh Gis)ds ad I Gis)ds are also tabulated; the first mostly to 5D, the latter to 6D. The coefficiets yr(b) were checked by verifyig that (1) is exact for Gis)ds were obtaied employig (12) ad checked by (13) ad itegratio of (8). The fuctio G(s) TABLE I Values of -y,00 ad D I each colum headed by re, the first coefficiet is 7o<"\ the secod is 7i<*>, etc. The last umber is the value of Z> = 2 S v.<«>. y," r-0»= 1 (2-poit) 2 1 = 7 (8-poit) » = 2 (3-poit) = 3 (4-poit) re = 8 (9-poit) » = 4 (5-poit) re = 9 (10-poit) = 5 (6-poit) re = 6 (7-poit) »- 10 (11-poit)

4 218 MECHANICAL QUADRATURE NEAR A SINGULARITY rh G(s)ds/h*+l(hyi* Error 1-2/15 2 8/ / / / / / / / / TABLE II Coefficiets m f G(s)ds/h"+>i* f"* G(.s)ds/h"+'i* was expaded i a Taylor series about a approximate value of, ad was determied by iversio of this series. Taylor's formula was also used to calculate I Gis)ds. The latter calculatios were performed i duplicate ad are correct to withi oe uit of the last decimal place give. It appears that coefficiets of the kid preseted here were first derived i a thesis by M. Bates.3 This work gives exact coefficiets for xml2fix)dx, m = ± 1, m = 2(1)4. The remaider terms are examied i some detail. I a thesis by M. E. Yougberg,4 similar formulas were give for = 3(1)7. With the exceptio of the seve- ad eight-poit formulas (the eight-poit formula is totally i error), exact coefficiets are also tabulated to eable itegratio over ay sub-iterval defied by the poits at which fix) is tabulated. No estimates of the error are give. I a recet paper, E. L. Kapla6 has derived three- ad five-poit rrih coefficiets for the evaluatio of xm,2fix)dx, m = ± 1. All coefficiets Jh are i decimal form. For three poits, coefficiets are give for rx = 0(1)18. I additio, some coefficiets are preseted so that, usig three ordiates, the itegral over the first two or last two ordiates ca be foud. The fivepoit case is similarly treated, but the values permit itegratio over three ordiates at most. Two sets of coefficiets are tabulated. Oe is exact for cosecutive powers of x as i our case. The other is exact for fix) a eve fuctio. Fially, error terms are give for selected cases havig the former property. I effect, it is assumed that the mea value theorem is applicable ad the error coefficiets are derived by evaluatig equatios of the type as of the right had side of (12). But the author rightly remarks that the results must be used with cautio if r2 rx > 1. As a further aid i evaluatig the remaider, Kapla also gives values which accout for the secod approximated term i the Taylor series expasio of fix). We have examied the Gis) fuctio for = 2, r\ = 4, r2 = 6 ad m = 1 ad fid that it too vaishes i the ope iterval of itegratio. The same is true for the three-poit formula where r-i = 0, r2 2 ad m = 1.

5 ON THE NUMERICAL SOLUTION OF EQUATIONS 219 The author ackowledges with thaks the aid of Dolores Ufford, who assisted i the calculatios. Yudell L. Luke Midwest Research Istitute Kasas City 2, Missouri 1 W. E. Mile, "The remaider i liear methods of approximatio," NBS, J. of Research, v. 43, 1949, p W. E. Mile, Numerical Calculus, p M. Bates, O the Developmet of Some New Formulas for Numerical Itegratio. Staford Uiversity, Jue, M. E. Yougberg, Formulas for Mechaical Quadrature of Irratioal Fuctios. Orego State College, Jue, (The author is idebted to the referee for refereces 3 ad 4.) 6 E. L. Kapla, "Numerical itegratio ear a sigularity," J. Math. Phys., v. 26, April, 1952, p O the Numerical Solutio of Equatios Ivolvig Differetial Operators with Costat Coefficiets 1. The Geeral Liear Differetial Operator. Cosider the differetial equatio of order (1) Ly + Fiy, x) = 0, where the operator L is defied by j» dky Ly = ZPkix), k=o " * ad the fuctios Pk{x) ad Fiy, x) are such that a solutio y ad its first m derivatives exist i 0 < x < X. I the special case whe (1) is liear the solutio ca be completely determied by the well kow method of variatio of parameters whe idepedet solutios of the associated homogeeous equatios are kow. Thus for the case whe Fiy, x) is idepedet of y, the solutio of the o-homogeeous equatio ca be obtaied by mere quadratures, rather tha by laborious stepwise itegratios. It does ot seem to have bee observed, however, that eve whe Fiy, x) ivolves the depedet variable y, the umerical itegratios ca be so arraged that the cotributios to the itegral from the upper limit at each step of the itegratio, at the time whe y is still ukow at the upper limit, drop out. Thus agai the computatio ca be made to ivolve merely quadratures. It is ot ofte that the solutio of the homogeeous equatio ca be simply determied, ad it is perhaps for this reaso that attetio has ot bee give heretofore to the possibility of simplifyig the umerical evaluatio of the solutio by makig use of the solutios to the homogeeous equatio. However, i the case whe the fuctios P*(x) i L are costats, the solutio of the homogeeous equatio is easy to determie. This is particularly true whe the order of the differetial equatio is fairly low. I the istace whe the operator L is of secod order, with costat coefficiets, the method of usig the itegral equatio ofte has decided advatages over the usual methods employed for solvig, differetial equatios.