Key words. Numerical quadrature, piecewise polynomial, convergence rate, trapezoidal rule, midpoint rule, Simpson s rule, spectral accuracy.

Transcription

1 O SPECTRA ACCURACY OF QUADRATURE FORMUAE BASED O PIECEWISE POYOMIA ITERPOATIO A KURGAOV AD S TSYKOV Abstrct It is well-known tt te trpezoidl rule, wile being only second-order ccurte in generl, improves to spectrl ccurcy if pplied to te integrtion of smoot periodic function over n entire period on uniform grid More precisely, for te function tt s squre integrble derivtive of order r te convergence rte is o r/, were is number of grid nodes Accordingly, for C -function te trpezoidl qudrture converges wit te rte fster tn ny polynomil In tis pper, we prove tt te sme property olds for ll qudrture formule obtined by integrting fixed degree piecewise polynomil interpoltions of smoot integrnd, suc s te midpoint rule, Simpson s rule, etc Key words umericl qudrture, piecewise polynomil, convergence rte, trpezoidl rule, midpoint rule, Simpson s rule, spectrl ccurcy et f fx be function defined in te intervl x b, b To pproximte te vlue of its integrl b fx dx, let us introduce prtition of te intervl [, b] into equl subintervls of size / Te prtition is rendered by te nodes of te grid: x j + j, j, 1,,, so tt x nd x b On ec subintervl [x j, x j+1 ], j, 1,, 1, we pproximte te function f by te interpolting polynomil of degree 1: fx Q 1 x, f, x j, x j+1 f j x j+1 x x x j + f j+1, 1 were f j fx j, j, 1,, Ten, by replcing fx wit te corresponding liner interpolnt 1 on ec subintervl, one obtins te trpezoidl qudrture rule: b x j+1 fx dx Q 1 x, f, x j, x j+1 dx j x j f + f 1 + f + + f + f j f j + f j+1 It is well-known tt if te second derivtive f x is bounded on [, b], ten te interpoltion error in formul 1 is: were ξ ξx [x j, x j+1 ] Terefore, fx Q 1 x, f, x j, x j+1 f ξ x x j x x j+1, 3! fx Q 1 x, f, x j, x j mx f x 1 x j x x j+1 8 mx f x x b Mtemtics Deprtment, Tulne University, ew Orlens, A 7118, USA E-mil: kurgnov@mttulneedu, UR: ttp://wwwmttulneedu/ kurgnov Reserc of tis utor ws supported in prt by te SF Grnt # DMS Deprtment of Mtemtics nd Center for Reserc in Scientific Computtion, ort Crolin Stte University, Box 85, Rleig, C 7695, USA E-mil: tsynkov@mtncsuedu, UR: ttp://wwwmtncsuedu/ stsynkov Reserc of tis utor ws supported in prt by te US Air Force Grnt # FA

2 On Spectrl Accurcy of Qudrture Formule Bsed on Piecewise Polynomil Interpoltions Consequently, te error of te trpezoidl rule cn be estimted s: b fj fx dx + f x j+1 j j+1 fx Q 1 x, f, x j, x j+1 dx j j x j+1 x j fx Q 1 x, f, x j, x j+1 dx x j 8 mx x b f x 3 8 mx x b f x In oter words, wen, te error of te trpezoidl rule decys t lest s fst s O importnt to empsize tt in generl tis estimte cnnot be improved mely, even if te function fx s bounded derivtives of order iger tn two on [, b], te convergence of te trpezoidl rule will still remin qudrtic Tis immeditely follows from te fct tt even for smooter functions te interpoltion error is given by te sme formul 3 However, tere is one prticulr cse wen te trpezoidl rule provides muc better ccurcy of pproximtion Tis is te cse of smoot nd -periodic function f, for wic te rte of convergence of te trpezoidl qudrture will utomticlly djust to te degree of regulrity of f In te literture, tis result is commonly referred to s stndrd; proof, for instnce, cn be found in [, Section 9] or in [3, Section 41] It is Moreover, tis result s fr-recing implictions in scientific computtion; it fcilittes construction of efficient numericl metods, for exmple, in te scttering teory, see [1] Since one cn ctully come cross severl sligtly different versions of tis result, we formulte it s teorem nd provide proof tt follows [3] nd is bsed on simpler rgument tn te one used in [] Teorem 1 et f fx be n -periodic function nd ssume tt its derivtive of order r is squre integrble: f r Ten te error of te trpezoidl qudrture rule cn be estimted s: + fj fx dx + f j+1 ζ, 4 r/ were ζ s j Proof We first note tt since te function f is integrted over full period, we my ssume witout loss of generlity tt We ten represent f s te sum of its Fourier series: were fx S x + R x, 5 S x + n cos πnx n1 + n1 b n sin πnx 6 is te prtil sum of order 1 nd R x n n cos πnx + n b n sin πnx 7 is te corresponding reminder Te coefficients n nd b n of te Fourier series of f re given by te formule: n fx cos πnx dx, b n fx sin πnx dx 8

3 On Spectrl Accurcy of Qudrture Formule Bsed on Piecewise Polynomil Interpoltions 3 otice tt ccording to formule 6 nd 8, te following equlity olds: We now integrte equlity 5 over [, ]: fx dx S x dx S x dx + fx dx 9 R x dx 1 nd pply te trpezoidl qudrture rule to te rigt-nd side RHS of 1 For te first integrl tere, tis cn be done vi te term-by-term integrtion of 6 We strt wit te constnt component n, for wic, tking into ccount 9, we immeditely get: j / + / S x dx fx dx For ll oter terms n 1,,, 1 we exploit periodicity wit te period, use te definition of te grid size /, nd obtin: j n n j Similrly, one cn sow tt cos πnj e i πnj j + e bn + n πnj i πnj + 1 cos sin πnj n + b n 1 e i πn 1 e i πn n + j cos πnj πn 1 e i πnj + 1 sin πn 1 e i Altogeter we conclude tt te trpezoidl rule is exct for te prtil sum S given by formul 6, nmely: j S x j + S x j+1 S x dx 11 We lso note tt coosing te prtil sum of order 1, wile te number of grid cells being, is not ccidentl From te previous derivtion is is esy to see tt equlity 11 would no longer old lredy if we were to tke S x insted of S x tere ext, we need to pply te trpezoidl rule to te reminder of te series R given by formul 7 First we recll tt te mgnitude of te reminder or, equivlently, te rte of convergence of te Fourier series, is determined by te smootness of te function f More precisely, one cn sow tt for function f suc tt f r te following estimte olds: sup x R x ζ, 1 r/

4 On Spectrl Accurcy of Qudrture Formule Bsed on Piecewise Polynomil Interpoltions 4 were ζ s Terefore, R x j j + R x j+1 sup R x x Finlly, we combine formule 11 nd 13 to conclude wit: fj fx dx + f j+1 j S x j + S x j+1 + R x j j R x j + R x j+1 ζ, r/ j wic completes te proof of te teorem + R x j+1 ζ 13 r/ Remrk If f C, ten Teorem 1 implies spectrl convergence of te trpezoidl qudrture rule In oter words, te rte of decy of te error will be fster tn ny polynomil, tt is, fster tn O r for ny r > et us now consider te entire fmily of pproximte qudrtures obtined, like te trpezoidl rule, by integrting piecewise polynomil interpolnts of given fixed degree et P M, were P nd M re positive integers Ten, we prtition te originl grid of cells into P clusters of M cells ec, nd witin every cluster pproximte te function f s follows: fx Q d x, f, x pm, x pm+1,, x p+1m, x [x pm, x p+1m ], p, 1,, P 1, 14 were Q d x, f, x pm, x pm+1,, x p+1m is unique lgebric interpolting polynomil of degree d M tt coincides wit te function fx t te nodes x j, j pm, pm + 1,, p + 1M By integrting te interpolting polynomils 14 over te corresponding cell clusters we rrive t te following pproximte qudrture formul: b fx dx P p P p x p+1m x pm Q d x, f, x pm, x pm+1,, x p+1m dx c f pm + c 1 f pm c M f p+1m, 15 were te coefficients c, c 1,, c M, re uniquely defined for given M For exmple, for te previously nlyzed trpezoidl rule we ve P, M 1, c 1/, nd c 1 1/ For te well-known Simpson s rule we ve P /, M, c 1/3, c 1 4/3, nd c 1/3 An obvious importnt property of te qudrture coefficients follows from pplying formul 15 to fx const: c + c c M M 16 Typiclly, te ccurcy of te qudrture formule 15 is O M+1, provided tt te function f s bounded derivtive of order M + 1 Sometimes, symmetries my led to somewt better ccurcy For

5 On Spectrl Accurcy of Qudrture Formule Bsed on Piecewise Polynomil Interpoltions 5 exmple, te ccurcy of Simpson s rule is, in fct, O 4 rter tn O 3, provided tt f 4 is bounded However, for smoot periodic functions f ll qudrture formule of type 15 ttin spectrl ccurcy wen pplied on full period Tis is te centrl result of our note, wic we present in te following teorem Teorem et f fx be n -periodic function nd ssume tt its derivtive of order r is squre integrble: f r Ten te pproximtion error of ny qudrture formul of type 15 cn be estimted s: + were ζ s fx dx P p c f pm + + c M f p+1m ζ, 17 r/ Proof Te proof of te teorem is bsed on reduction of te qudrture formul 15 to liner combintion of te trpezoidl-type qudrtures, for wic estimte 17 reduces to estimte 4 tt ws proved in Teorem 1 mely, we cn rewrite te qudrture formul 15 s: P p c f pm + c 1 f pm c M f p+1m c f + c + c M f M + c + c M f M + + c + c M f P M + c M f P M M + c m fm + f m+m + f m+m + + f m+p M m1 ext, we introduce te gost nodes of te grid: {x j } for j + 1,, + M 1 or equivlently, j P M + 1,, P M + M 1, nd use te -periodicity of fx to recst te previous equlity s follows: We terefore ve: P p m1 c f pm + c 1 f pm c M f p+1m c + c M f + c + c M f M + + c + c M f P M + c + c M f P M M fm + c m + f m+m + f m+m + + f m+p M + f m+p M P p c f pm + c 1 f pm c M f p+1m c + c M M M m α m P fpm + f M p+1m c m + M m1 p fm+pm P p + f m+p+1m P p fpm+m + f p+1m+m, 18 were M denotes te size of te cell clusters Te RHS of 18 is liner combintion of M trpezoidltype qudrtures constructed on uniform grid of P cells of size M Te coefficients of te liner combintion re: α c + c M /M, α 1 c 1 /M,, α M c M /M

6 On Spectrl Accurcy of Qudrture Formule Bsed on Piecewise Polynomil Interpoltions 6 Moreover, ec qudrture wit m > is trpezoidl rule sifted by m, tt is, formlly it integrtes te function f on te intervl [m, + m] rter tn [, ] However, s te function f is -periodic, we obviously ve: m+ m fx dx fx dx 19 Terefore, we cn pply te sme rgument s used in te proof of Teorem 1 to ec individul trpezoidl formul on te RHS of 18 In doing so, we consider te prtil sum S P nd te reminder R P of te Fourier series of f Tis yields [cf formul 18]: P p c + c M M + M m c f pm + c 1 f pm c M f p+1m α m P p Ten, using equtions 16 nd 19 we obtin: P p M m M fx dx + m1 c m M RP x pm+m m+ m fx dx + R P x p+1m+m c f pm + c 1 f pm c M f p+1m α m P p RP x pm+m + R P x p+1m+m fx dx Finlly, te desired estimte 17 is obtined by pplying te tringle inequlity nd estimting te reminder of te Fourier series s: sup R P x ζ P x P r/ Tis estimte is equivlent to 1 s long s wen refining te grid MP we keep te order of te qudrture formul M fixed M const, wile letting te number of clusters P to increse P Remrk ote tt one of te simplest qudrture formule, known s te midpoint rule, b fx dx f 1/ + f 3/ + + f /, does not formlly belong to te clss 15 It s ccurcy O provided tt f is bounded, nd is obtined by pproximting te function f wit constnt zero-degree polynomil on ec grid cell: fx Q x, f, x j, x j+1 fx j + / : f j+1/, x [x j, x j+1 ] Yet te sme rgument s employed wen proving Teorem clerly pplies to te qudrture formul, wic mens tt te midpoint rule lso cieves spectrl ccurcy wen used for integrting smoot periodic functions

7 On Spectrl Accurcy of Qudrture Formule Bsed on Piecewise Polynomil Interpoltions 7 Exmple As numericl demonstrtion, we clculte te integrl π e sin x dx 1 using Simpson s rule on sequence of uniform grids wit, 4, 8, et us denote by I te pproximte vlue of te integrl 1 on te grid wit cells Te results obtined wit double precision re summrized in Tble 1 Tble 1 Computtion of integrl 1 using Simpson s rule I I I / e e e e REFERECES [1] O P Bruno nd A Kunynsky, Surfce scttering in tree dimensions: n ccelerted ig-order solver, R Soc ond Proc Ser A Mt Pys Eng Sci, 457 1, pp [] P J Dvis nd P Rbinowitz, Metods of umericl Integrtion, Computer Science nd Applied Mtemtics, Acdemic Press Inc, Orlndo, F, second ed, 1984 [3] V S Ryben kii nd S V Tsynkov, A Teoreticl Introduction to umericl Anlysis, Cpmn & Hll/CRC, Boc Rton, F, 7