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1 World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd Interntionl Science Index Mthemticl nd Computtionl Sciences Vol:5 No:7 011 wset.org/puliction/8083 Astrct We present new qudrture rule sed on the spline interpoltion long with the error nlysis. Moreover some error estimtes for the reminder when the integrnd is either Lipschitzin function function of ounded vrition or function whose derivtive elongs to L p re given. We lso give some exmples to show tht prcticlly the spline rule is etter thn the trpezoidl rule. Keywords Qudrture Spline interpoltion Trpezoidl rule Numericl integrtion Error nlysis. I. INTRODUCTION QUADRATURE is n old-fshioned word tht refers to the numericl pproximtion of definite integrls. There re severl resons for crrying out numericl integrtion. The integrnd fx my e known only t certin points such s the one otined y smpling. The formul for integrnd my e known ut it my e difficult or impossile to find n ntiderivtive which is n elementry function. An exmple of such n integrnd is fx e x the ntiderivtive of which cnnot e written in n elementry form. It my e possile to find n ntiderivtive symoliclly ut it my e esier to compute numericl pproximtion thn to compute the ntiderivtive. Tht my e the cse if the ntiderivtive is given s n infinite series or product or if its evlution requires specil function which is not ville. Numericl integrtion methods cn generlly e descried s comining evlutions of the integrnd to get n pproximtion to the integrl. An importnt prt of the nlysis of ny numericl integrtion method is to study the ehvior of the pproximtion error s function of the numer of integrnd evlutions. A method which yields smll error for smll numer of evlutions is usully considered superior. Reducing the numer of evlutions of the integrnd reduces the numer of rithmetic opertions involved nd therefore reduces the totl round-off error. Also ech evlution tkes time nd the integrnd my e ritrrily complicted. This pper is orgnized s follows. In section we present new three-point qudrture rule which is sed on the spline interpoltion while section 3 is devoted to error nlysis. Section 4 provides n estimte for the reminder when f is Lipschitzin function while the fifth section dels with n upper ound for the reminder in spline inequlity for the clss of functions of ounded vrition. Section 6 gives n inequlity for functions whose derivtives elong to L p. In section 7 some exmples re presented to show tht the new qudrture H. Tghvfrd is with the Deprtment of Mthemtics Mlek-Ashtr University of Technology Isfhn Irn. e-mil: tghvfrd@gmil.com. rule is etter thn the trpezoidl rule. Finlly conclusions in section 8 close the pper. II. NEW QUADRATURE RULE A lrge clss of qudrture rules cn e derived y constructing interpolting functions which re esy to integrte. Typiclly these interpolting functions re polynomils. We know tht cuic spline functions re populr spline functions for vriety of resons. They re smooth functions with which to fit dt nd when used for interpoltion they do not hve the oscilltory ehvior which is the chrcteristic of high-degree polynomil interpoltion. Let Δ{x i i n} e fixed prtition of the intervl ] y knots x 0 < x 1 < x < < x n nd y i fx i ;i n e n 1 prescried rel numer. In ddition let h i1 : x i1 x i ; i n1; then the cuic spline function S Δ x which interpoltes the vlues of the function f t the knots x 0 x 1... x n Δ nd stisfies S Δ S Δ 0is redily chrcterized y their moments nd these moments of interpolting cuic spline function cn e clculted s the solution of system of liner equtions. We cn otin the following representtion of the cuic spline function in terms of its moments 4]: S Δ x α i β i x i γ i x i δ i x i 3 1 for x x i x i1 ]i n 1 where α i y i γ i M i1 M i 6h i1 β i y i1 y i M i M i1 h i1 6 h i1 M 0 M n 0 M i S Δx i ; i n nd the moments M i for i n otin the following system 0 0 M 0 d 0 μ 1 λ 1 0 M 1 d 1 0 μ λ 0 M d μ n1 λ n such tht for i n 1 M n λ i h i1 h i h i1 μ i 1 λ i d n Interntionl Scholrly nd Scientific Reserch & Innovtion scholr.wset.org/ /8083
2 World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 Interntionl Science Index Mthemticl nd Computtionl Sciences Vol:5 No:7 011 wset.org/puliction/ yi1 y i d i h i h i1 h i1 y i y i1 h i nd d 0 d n 0. By integrting on oth side of 1 on x i x i1 ] we otin the formul xi1 S Δ x h i1 x i y i y i1 h3 i1 4 M i M i1 for i n 1. Now let us suppose tht Δ { } is uniform prtition of the intervl ]. Then y solving the system M 0 M 1 d 0 d M with d 0 d 0 M 0 M 0nd d 1 3 h f f] we otin f M 1 3 f f h f]. On the other hnd from we hve Therefore nd S Δ x S Δ x fx h h3 d S Δ x S Δ x ff ] f 4 M 0 M 1 M ] h 3f10f ] 8 3f. 3f10f ] 3f 3f10f ] 3f which is the desired three-point qudrture rule tht henceforth we cll it the spline rule. Let x 0 <x 1 <x < <x n1 <x n e the nodes of the prtition σ n of the intervl ]. The spline qudrture formul of the integrl fx on the prtition σ n hs the form n1 h i fx 3fx i10f x i x i1 i0 3fx i1 ] Rfσ n 3 where h i x i1 x i for i n 1. III. ERROR ANALYSIS In this section we nlyze the error of the spline rule i.e. εf fxd 3f10f ] 3f. The stndrd derivtion of qudrture error formuls is sed on determining the clss of polynomils for which these formuls produce exct results. The next definition is used to fcilitte the discussion of this derivtion 6]. Definition 1: The degree of ccurcy or precision of qudrture formul is the lrgest positive integer n such tht the formul is exct for x k for ech k n. Remrk 1: Definition 1 implies tht the spline rule hs the degree of precision one. Definition : For qudrture rule Lf fx with error εf fxdlf Peno s kernel of the degree n is defined y K n t 1 n! ε tn where t n if x t t n 0 if x < t. We hve the following theorem from 5] which is known s Peno s kernel theorem: Theorem 1: Let Lf fx e qudrture rule such tht for ech polynomil p of the degree n εp 0.If f C n1 ] then εf f n1 tk n tdt. Now let f nd K 1 e integrle functions on the intervl ] where f C ] nd K 1 t is Peno s kernel of the degree 1 such tht it doesn t chnge sign on ]. Vi theorem 1 wehve εf f tk 1 tdt. 4 Indeed ccording to the men vlue theorem for integrls there exists ζ such tht f tk 1 tdt f ζ K 1 tdt 5 On the other hnd since the spline rule is ccurte for polynomils of the degree 1 simple clcultion shows tht εx Therefore from 4 5 nd 6 one otins K 1 tdt. 6 εx f ζ εx 3 f ζ ζ. 19 Therefore the following theorem hs een proved: Theorem : Let f C ]. If Peno s kernel K 1 t doesn t chnge sign on ] then there exists ζ such tht 3 εf f ζ 19 where εf is the spline rule error. For uniform prtition of the intervl ] the following corollry is otined: Corollry 1: Under the ssumptions of Theorem for uniform prtition σ n of ] the reminder term Rfσ n in the composite spline rule 3 stisfies Rfσ n 19 h f ζ. Interntionl Scholrly nd Scientific Reserch & Innovtion scholr.wset.org/ /8083
3 World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 Interntionl Science Index Mthemticl nd Computtionl Sciences Vol:5 No:7 011 wset.org/puliction/8083 or Rfσ n 19 h f nd this conclusion shows tht the composite spline rule error decreses with the minimum rte of h. IV. SPLINE INEQUALITY FOR LIPSCHITZIAN FUNCTIONS In this section we present n inequlity of spline type for L-Lipschitzin functions nd show tht the error estimte in the spline rule is less thn the one in Simpson s rule for L- Lipschitzin functions. Definition 3: A rel function f : ] R is clled n L-Lipschitzin function if there exist constnt L>0 such tht f fy L y x y ]. Drgomir 1] proved the following inequlity of Simpson s type for L-Lipschitzin functions: Theorem 3: Let f : ] R e n L-Lipschitzin function on ]. Then fxd 3 ff f ] 5 36 L. 7 Also we hve the following result from 1]. Lemm 1: If p : ] R is Riemnn integrle function on ] nd v : ] R is L-Lipschitzin function on ] then pxdvx L Let sx e function defined s sx 133 x 313 x ] px. 8 Using integrtion y prts for Riemnn-Stieltjes integrl we otin sxdfx 3f10f ] 3f fx. In fct sxdfx fx fx dfx dfx ] ] 3f10f fx ] 3f fx. 9 Now let f : ] R e n L-Lipschitzin function on ]. Using lemm 1 one otins sxdfx L sx. On the other hnd sx x x Thus we hve proved the following theorem: Theorem 4: Let f : ] R e n L-Lipschitzin function on ]. Then the following inequlity holds: fxd 3f10f ] 3f L. 10 Corollry : Suppose tht f : ] R is differentile function such tht f L ] i.e. then f : fxd sup f x < x ] 3f10f ] 3f f. Interntionl Scholrly nd Scientific Reserch & Innovtion scholr.wset.org/ /8083
4 World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 Interntionl Science Index Mthemticl nd Computtionl Sciences Vol:5 No:7 011 wset.org/puliction/8083 Remrk : By compring 7 nd 10 we conclude tht the error estimte in 10 is less thn the one in Remrk 3: We stte tht the constnt 18 which occurred in the inequlity is the est constnt. Nmely for the 1- Lipschitzin function x 133 x 133 fx 133 x 133 x 313 x x 313 we hve Rf Vi Theorem 4 we otin the following estimtion for the reminder Rfσ n. Corollry 3: Under the ssumptions of Theorem 4 the reminder term Rfσ n in the composite spline qudrture formul 3 stisfies Rfσ n 17 n1 18 L h j 11 j0 If σ n is uniform prtition of ] we hve the following corollry: Corollry 4: Under the ssumptions of Theorem 4 for uniform prtition σ n of ] the reminder stisfies the estimtion Rfσ n L n. Remrk 4: If we wnt to pproximte the integrl ftdt y the composite spline rule 3 with n ccurcy less thn ɛ>0 we need t lest n ɛ N points for the division σ n where 17 n ɛ 18 L ] 1 ɛ nd r] denotes the integer prt of r R. V. SPLINE INEQUALITY FOR FUNCTIONS OF BOUNDED VARIATION In this section we present n inequlity of the spline type for functions of ounded vrition nd show tht the error estimte in the spline rule is less thn the one in Simpson s rule for functions of ounded vrition. Drgomir ] proved the following inequlity of Simpson s type for functions of ounded vrition: Theorem 5: Let f : ] R e function of ounded vrition. Then fxd 3 ff f ] 1 3 f 1 where f denotes the totl vrition of f on the intervl ] nd the constnt 1 3 is the est constnt. We hve the following lemm from ]: Lemm : Let p v : ] R e continuous nd of ounded vrition respectively on ]. Then pxdvx mx px v. 13 x ] Let sx e function defined s sx 133 x 313 x ]. With the sme procedure which pssed in the previous section we hve sxdfx 3f10f ] 3f fx Now let f : ] R e function of ounded vrition on ]. Using lemm one otins sxdfx mx sx f. 14 x ] Since s is non-decresing function on nd ] nd s 3 s 5 we deduce tht mx x ] s 5 s 5 sx. 3 Thus we hve proved the following theorem: Theorem 6: Let f : ] R e function of ounded vrition. Then the inequlity fxd 3f10f ] 3f 5 f 15 holds where f denotes the totl vrition of f on the intervl ]. 5 Remrk 5: We stte tht the constnt which occurred in the inequlity is the est constnt. In order to show this let us ssume tht for constnt C>0 the following inequlity holds: fxd 3f10f ] 3f C f. Interntionl Scholrly nd Scientific Reserch & Innovtion scholr.wset.org/ /8083
5 World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 Interntionl Science Index Mthemticl nd Computtionl Sciences Vol:5 No:7 011 wset.org/puliction/8083 Let f : ] R e function defined s 1 x ] fx 1 x. Then we hve fxd 3f10f ] 3f nd f Now using the ove inequlity we get 4C 5 4 which implies tht C 5 5 nd then is the est possile constnt in 15. Corollry 5: Suppose tht f : ] R is differentile function whose derivtive is continuous on nd then f 1 : fxd f x < 3f10f ] 3f 5 f Remrk 6: By compring 1 nd 15 we conclude tht the error estimte in 15 is less thn the one in 1. Vi Theorem 6 we otin the following estimtion for the reminder Rfσ n. Corollry 6: Under the ssumptions of Theorem 6 the reminder term Rfσ n in the composite spline qudrture formul stisfies Rfσ n 5 γh f 0 where γh :mx{h i i n 1}. The cse of uniform prtition of ] re presented in the following corollry: Corollry 7: Under the ssumptions of Theorem 6 for uniform prtition σ n of ] the reminder stisfies the estimtion Rfσ n 5 f. n Remrk 7: If we wnt to pproximte the integrl ftdt y the composite spline rule 3 with n ccurcy less thn ɛ>0 we need t lest n ɛ N points for the division σ n where ] 5 n ɛ f 1 ɛ nd r] denotes the integer prt of r R. VI. SPLINE INEQUALITY IN TERMS OF THE p-norm In this section we present n inequlity of the spline type for functions whose derivtives elong to L p ] nd show tht the error estimte in the spline rule is less thn the one in Simpson s rule for functions whose derivtives elong to L p ]. Drgomir 3] proved the following inequlity of Simpson s type for functions whose derivtives elong to L p ]: Theorem 7: Let f : ] R e n solutely continuous function on ] whose derivtive elongs to L p ]. Then fxd ff f ] 3 q1 ] 1/q 1 11/q f p 1 3q where 1/p 1/q 1p>1. Let sx e function defined s sx 133 x 313 x ]. With the sme procedure which pssed in the third section we hve sxdfx 3f10f ] 3f fx Applying Hölder s inequlity we otin 1/q sxf x sx q f p. On the other hnd sx q q 13 3 q 3 13 q 13 3 x q 13 3 q 3 13 x q q1 5 q1 q1 8q 1 q. Therefore we hve proved the following theorem: Theorem 8: Let f : ] R e n solutely continuous function on ] whose derivtive elongs to L p ]. Then Interntionl Scholrly nd Scientific Reserch & Innovtion scholr.wset.org/ /8083
6 World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 Interntionl Science Index Mthemticl nd Computtionl Sciences Vol:5 No:7 011 wset.org/puliction/8083 TABLE I the inequlity fxd 3f10f ] NUMERICAL RESULTS fx Exct Spline rule Trpezoidl rule 3f cos x q1 5 q1 ] 1/q x 4 4x /q f p 8q 1 x log x holds where 1/p 1/q 1p>1. Remrk 8: By compring 1 nd we conclude tht the error estimte in is less thn the one in 1. Vi Theorem 8 we otin the following estimtion for the reminder Rfσ n. Corollry 8: Under the ssumptions of Theorem 8 the reminder term Rfσ n in the composite spline qudrture formul stisfies logx 1 e x cose x x 3 1 e t 7t /q Rfσ n 1 3 q1 5 q1 ] 1/q n1 REFERENCES h q1 j f p. 31] S. S. Drgomir On Simpson s qudrture formul for Lipschitzin 8q 1 j0 The cse of uniform prtition of ] re presented in the following corollry: Corollry 9: Under the ssumptions of Theorem 8 for uniform prtition σ n of ] the reminder stisfies the estimtion Rfσ n 1 3 q1 5 q1 ] 1/q 11/q f p. n 8q 1 Remrk 9: If we wnt to pproximte the integrl ftdt y the composite spline rule 3 with n ccurcy less thn ɛ>0 we need t lest n ɛ N points for the division σ n where 1 3 q1 5 q1 1/q ] n ɛ 11/q f p 1 ɛ 8q 1 nd r] denotes the integer prt of r R. VII. NUMERICAL RESULTS According to section 3 the error of the spline rule like the error of the trpezoidl rule decreses with the minimum rte of h. Also it ws shown tht the error of the spline rule is less thn the error of the trpezoidl rule. According to these fcts we compre the spline rule with the trpezoidl rule to show tht prcticlly the spline rule is etter thn the trpezoidl rule. In Tle I some exmples re presented to compre the spline rule with the trpezoidl rule on the intervl 0 1] which is divided into ten prtitions. All the progrms hve een written in MATLAB environment. mppings nd pplictions Soochow J. Mthemtics ] S. S. Drgomir On Simpson s qudrture formul for mppings of ounded vrition nd pplictions Tmkng J. Mthemtics ] S. S. Drgomir On Simpson s qudrture formul for differentile mppings whose derivtines elong to L p spces nd pplictions J. KSIAM ] J. Stoer R. Bulrisch Introduction to numericl nlysis Second edition Springer Verlg ] M. B. Allen I. E. Iscson Numericl nlysis for pplied science John Wiley & Sons ] R. L. Burden J. D. Fires Numericl nlysis Seventh edition Brooks/Cole 001. VIII. CONCLUSIONS In this pper new qudrture rule which is derived from cuic spline functions long with the error nlysis ws presented. Moreover some estimtes for the reminder when the integrnd ws Lipschitzin function function of ounded vrition or function whose derivtive elongs to L p were given nd it ws oserved tht these error estimtes were less thn the similr ones for Simpson s rule. Also we presented some exmples to show tht minly the spline rule is etter thn the trpezoidl rule. Interntionl Scholrly nd Scientific Reserch & Innovtion scholr.wset.org/ /8083
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