A New Fluctuation Expansion Based Method for the Univariate Numerical Integration Under Gaussian Weights

Transcription

1 Proceedings of the 8th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp68-73 A New Fluctution Expnsion Bsed Method for the Univrite Numericl Integrtion Under Gussin Weights METİN DEMİRALP Informtics Institute, İstnbul Technicl University İTÜ Bilişim Enstitüsü, Ayzğ Yerleşkesi, Mslk, 34469, İstnbul, Türkiye (Turkey TURKEY (TÜRKİYE demirlp Abstrct: - This pper presents new method bsed on quite recently proposed fluctution expnsion for the evlution of certin opertors expecttion vlues over Hilbert spces. The fluctution expnsion hs been constructed with the id of projection opertor which projects to one dimensionl subspce of the Hilbert spce under considertion. We, now, extend this ide to the utiliztion of projections to multidimensionl subspce of the sme Hilbert spce. We tke univrite integrl under Gussin weight (tht is, bell like shped function nd keep only zeroth order terms which contin no fluctution functions. After some mtrix lgebric mnipultions we obtin n interpoltion formul s liner combintion of the integrl s kernel function s vlues t the eigenvlues of the mtrix which is upperleftmost trunction from the mtrix representtion of the independent vrible Key-Words: - Fluctution Expnsion, Gussin Weight, Numericl Integrtion, Expecttion Vlues, Quntum Mechnics Introduction We hve recently offered new method for the numericl evlution of univrite integrls[-4]. The integrnd of the integrl hs been considered s the product of two given functions, one of which is specified s weight function. Tht is, I dxw(x f (x ( where W(x stnds for the weight function nd ll entities re ssumed to be rel vlued for simplicity. Since the weight function cn vnish only t finite number of the points of the integrtion domin nd remins positive elsewhere in the sme domin by definition, we hve considered its positive squre root s wve function of the quntum mechnics nd hve written W(x ψ(x ( where ψ(x denotes the so-clled wve function. The weight function s integrl over the domin hs been ssumed to be for providing consistency to probbilistic issues since we need to use those tools for the employment of expecttion or men vlue concepts. Tht is, dxw(x dxψ(x =. (3 Then the integrl hs been interpreted s the expecttion vlue of the function f (x with respect to the wve function mentioned bove through the following formul f (x dxw(xψ(x f (xψ(x (4 We hve ssumed tht f (x is continuous everywhere in domin including the integrtion intervl in the integrtion vrible s complex plne nd squre integrble over the intervl [, b ] to provide the utiliztion of Hilbert spce concepts. This enbled us to expnd f (x into Tylor series t point, sy c, in the integrtion intervl s follows f (x = k! f (k (c(x c k (5 where superscript (k stnds for the k th derivtive. The next step hs been the replcement of (4 with the following eqution by using (5 f (x = k! f (k (c (x c k. (6 After this point we hve defined the following opertors Ig(x g(x, ( P ψ g(x dxψ(xg(x ψ(x (7

2 Proceedings of the 8th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp68-73 where I nd P ψ denote the unit opertor nd the projection opertor which projects to the subspce spnned by ψ(x in the spce of squre integrble functions over the intervl [, b ]. g(x represents ny function chosen from the set of functions which re squre integrble over the intervl [, b ]. To mke concrete nlogy to the quntum mechnics we hve ssumed tht this spce is Hilbert spce, in other words, distnce between two points, the norm of ny element (vector of the spce nd the ngle between two vectors, re ll defined nd, beyond these, the spce is ssumed to be complete. The complementry compnion of P ψ is, of course, simply I P ψ. By keeping this fct in mind we hve written the following equlity (x c k = (x c {[ P ψ + ( I P ψ ] (x c } k (8 which is required nd remins vlid for ll positive integer k vlues. Its right hnd side should be ssumed to be when k vnishes. In the cses where the wve function ψ(x is lmost shrply loclized round single point in the intervl of integrtion (s well-known exmple of shrply loclized functions we cn ddress to the delt function of Dirc we cn conjecture tht P ψ overdomintes its complement, tht is, behves like or goes to, unit opertor s cn be shown vi n nlysis bsed on distribution theoreticl tools[5]. Hence, we cn rewrite (8 s the following pproximtion formul (x c k (x c { P ψ (x c } k = (x c k (9 where the equlity between the leftmost nd rightmost terms holds for ll nonnegtive vlues of k. This equlity cn be used to construct the following pproximte equlity for the expecttion vlue of f (x f (x f ( x (0 where the error terms contin the expecttion vlue of x nd the following entities ϕ k (c (x c {[ ] } k I P ψ (x c ( where the integer prmeter k vries between (inclusive nd infinity. Amongst these functions, ϕ (c is directly relted to the stndrt devition becuse of its following reduced form ϕ (c = x x ( The wve function depends on not only spce coordintes but lso on time in quntum dynmics where the evolution of the system in time is t the focus. Hence, in the cse of the quntum dynmicl problems, the integrl I of ( becomes prmetric integrl since its kernel nd therefore its weight function depends on time prmeter. This is reflected s time dependence in the wve function. Hence, ϕ k entities bove lso become time dependent. This results in time dependent, or in quntum dynmicl terminology, temporlly fluctuting function. Hence, despite the nonexistence of time dependence here, we cll these entities fluctution functions. In ϕ k, k chrcterizes the number of the ppernces of the opertor [ I P ψ ] which is responsible for the fluctution (or devition for the limited cse here in the error term of (0 nd the rgument c, which gives function structure to ϕs, denotes the focus of the expnsion. Therefore, we explicitly cll ϕ k (c k-th Order Fluctution Function t the Point c. Although we hve ssumed tht k does not vnish we cn extend the definition of ϕ k (c to cover k = 0 where the fluctution function becomes the difference between the expecttion vlue of x nd c. The only wy to mke this fluctution zero is to tke c = x. All these men tht (0 is zeroth order fluctution expnsion round the point where x tkes its expecttion vlue. We do not intend to give full detils of the complete nlysis presented in our recent works since we re intending to use only zeroth order fluctution expnsion, tht is, the expression obtined by ignoring the first nd higher order terms (the terms contining t lest first power of [ I P ψ ] in the fluctution expnsion of the expecttion vlue of the function f (x. We use the zeroth order pproximtion under new projection opertor which projects to subspce spnned by not only single function but set of orthogonl functions. We lso mke specifictions bout the integrl limits nd the weight function here. Pper is orgnized s follows. The second section presents the formultion of the new method proposed here. Third section is bout the clcultion of certin universl constnts. Fourth section contins certin numericl comprisons. Fifth section finlizes the pper by giving concluding remrks. Formultion of the Method Let us specify the intervl s (, nd the weight function in ( s follows W(x, t, u = πt e (x u t (3 where t is positive prmeter nd u stnds for rel prmeter. (3 defines the wve function s follows

3 Proceedings of the 8th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp68-73 ψ(x, t, u = e (x u π t (4 4 t We cn now rewrite ( s I(t, u = dxψ(x, t, u f (xψ(x, t, u (5 then consider the Hilbert spce spnned by the functions which re squre integrble under the weight function W(x. The nonnegtive integer powers of x form bsis set for this spce since ny function in this Hilbert spce cn be expressed s liner combintion of these functions s long s it is continous everywhere except infinity. Grm-Schmidt orthonormliztion of this set produces the bsis set whose elements re given through the following equlity ( φ n (x, t, u A n (te (x u (x u t H n (6 t where the positive integer n strts from nd runs up to infinity nd the symbol H n stnds for the Hermite polynomils[6]. The normliztion constnt A n is explicitly given s follows A n (t π 4 n (7 (n! t Let us define the following projection opertor P n g(x n ( dxφ k (x, t, ug(x φ k (x, t, u (8 k= where g(x is ny function chosen from the Hilbert spce spnned by φ k (x, t, u, (k =,,... nd n stnds for positive integer. This opertor pprently trunctes the representtion of ny given function in the Hilbert spce mentioned bove to finite liner combintion of the orthonorml bsis functions defined bove. In other words, it projects ny given function in the Hilbert spce spnned by φ k (x, t, u, (k =,,... functions to the subspce spnned by first n bsis functions, tht is, φ k (x, t, u, ( k n. This multidimensionlity of the subspce onto which the projection opertor trnsforms is the bsic extension of the fluctution expnsion here. We cn now proceed by using this opertor s first pproximtion to unit opertor. Since we cn write (x c m = I {(x ci} m (9 we cn get the following pproximtion by stying t the zeroth order trunction of fluctution expnsion vi P n (x c m Pn {(x cpn} m (0 where n nd m stnd for positive nd nonegtive integer respectively. Since creful glnce t the definition of the wve function shows tht we cn write ψ(x, t, u = φ (x, t, u ( (x c m {(x cp n } m (x c ( where we hve used the fct tht the ction of P n on φ (x, t, u is gin φ (x, t, u becuse it is sme s the unit opertor on the spce spnned by φ (x, t, u. If we define n (n n type mtrix X n (t, u whose elements re given through the following equlities X (n jk (t, u dxφ j (x, t, uxφ k (x, t, u (3 where j nd k re positive integers vrying between nd n inclusive, then, we cn rewrite ( s the following qudrtic form (x c m e (n T (Xn (t, u ci n m e (n (4 where I n nd e (n represent n dimensionl unit mtrix nd the first crtesin unit vector, whose only nonzero element is nd locted t the first position, in n th dimensionl Euclid spce. Eqution (4 enbles us to write f (x c m! f (m (ce (n T (Xn (t, u ci n m e (n nd therefore m=0 (5 f (x c e (n T f (Xn (t, u e (n (6 s long s f (x converges everywhere except perhps infinity in the complex plne of x. Now, for further simplifiction, we cn del with the structure of the mtrix X n (t, u nd write ( x u dxφ j (x, t, u φ k (x, t, u t π j+k = dξe ξ H j (ξξh k (ξ (7 ( j!(k!

4 Proceedings of the 8th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp68-73 where the integers j nd k strt from nd run up to nd including n. The right hnd side of this equlity does not depend on t nd u s cn be noticed immeditely. This independence urges us to define n (n n type mtrix Ξ n whose elements re defined through the following equlity ( x u Ξ jk dxφ j (x, t, u φ k (x, t, u (8 t where j nd k stnd for positive integers less thn or equl to n. We cn now express X n (t, u in terms of Ξ n fter creful look t the structures of those mtrices s given below X n (t, u = ui n + tξ n (9 which enbles us to rewrite (6 s f (x c e (n T f (uin + tξ n e (n (30 nd therefore s I(t, u e (n T f (uin + tξ n e (n (3 To proceed towrds the ultimte form of our pproximtion formul we need to explicitly express the kernel mtrix of the qudrtic form bove. To this end we cn use the Cyley Hmilton Theorem nd write ( k prmeters re unknown yet n f (ui n + tξ n = k Ξ k n (3 k= If we postmultiply both sides of this eqution by the j th eigenvector of Ξ n nd denote its corresponding eigenvlue by ξ j,n then we cn write f ( u + tξ j,n = n k= k ξ k j,n, j n (33 which cn be put into the following mtrix form where nd V n n = f n (34 T n [... n ] (35 f T n [ f (u + ξ,n... f (u + ξ n,n ] (36 ξ,n ξ,n ξ n,n ξ,n ξ,n ξ n,n V n ξ n,n ξn,n ξn,n n (37 Now, (3 nd (3 led us to write where nd q k,n e (n I(t, u = q T n n (38 q n [ q,n... q n,n ] T Ξ k n (39 e (n, k n (40 Eqution (38 cn be combined with (34 to get the following eqution s long s V n is invertble. If we define I(t, u = q T n V n f n (4 w T n = qt n V n (4 where the elements of the vector w n re denoted by w,n,..., w n,n respectively then we cn get the ultimte form of our pproximtion formul s follows I(t, u n w k,n f (u + tξ k,n (43 k= We cll w,n,..., w n,n vlues weights nd ξ,n,..., ξ n,n vlues nodes within n nlogy to the Guss qudrtures. 3 Evlution of Universl Constnts To finlize our method wht we need is the evlution of weights nd nodes. Since these entities re directly relted to Ξ n which is free of f (x the weights nd nodes re universl. Hence, once they re evluted for specific n vlue they cn be used in (43 for ny f (x under some constrints like to be continuos. This section is therefore devoted to the evlutions of these universl constnts. To this end we cn strt with the following recursion between consecutive Hermite polynomils H n+ (x = xh n (x nh n (x (44 which is vlid for ll nonnegtive integer vlues of n by ssuming H (x identiclly zero. If we keep in mind the fcts nd dξe x H j (xh k (x = δ j,k j j! π (45 δ j,k j,k = j, j = k,k (46

5 Proceedings of the 8th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp68-73 where δ j,k nd j,k stnd for the Kroenecker s symbol nd ny two-indexed-entity respectively then we cn rrive t the following equlity k k Ξ j,k = δ j, k + + δ j, k (47 where j nd k tke integer vlues between nd n inclusive. Ξ n is pprently symmetric mtrix nd, beyond this, its only nonzero digonls re the upper nd lower djcent neighbors of the min digonl. The elements of these digonls re /,, 3/,..., (n / in downwrd ordering. By using these vlues it is possible to numericlly evlute the eigenvlues of Ξ n. These eigenvlues which re nodes in fct permit us to determine V n nd therefore ll weights. Numericl clcultions cn be relized by using ny softwre which is cpble of doing wht we wnt. Here we hve used MuPAD Computer System Algebr developed by Pderborn University in Germny[7] becuse its cpbility of performing clcultions t ny desired level of ccurcy nd lso its symbolic progrmming fetures. We report the results for 5, 0, nd 5 vlues of n here. They re given below n = 5 ξ,5 = ξ 5,5 = ξ,5 = ξ 4,5 = ξ 3,5 = w,5 = w 5,5 = w,5 = w 4,5 = w 3,5 = n = 0 ξ,0 = ξ 0,0 = ξ,0 = ξ 9,0 = ξ 3,0 = ξ 8,0 = ξ 4,0 = ξ 7,0 = ξ 5,0 = ξ 6,0 = w,0 = w 0,0 = w,0 = w 9,0 = w 3,0 = w 8,0 = w 4,0 = w 7,0 = w 5,0 = w 6,0 = n = 5 ξ,5 = ξ 5,0 = ξ,5 = ξ 4,5 = ξ 3,5 = ξ 3,5 = ξ 4,5 = ξ,5 = ξ 5,5 = ξ,5 = ξ 6,5 = ξ 0,5 = ξ 7,5 = ξ 9,5 = ξ 8,5 = w,5 = w 5,5 = w,5 = w 4,5 = w 3,5 = w 3,5 = w 4,5 = w,5 = w 5,5 = w,5 = w 6,5 = w 0,5 = w 7,5 = w 9,5 = w 8,5 = These result re obtined within 00 deciml digit ccurcy under MuPAD nd only first 0 frctionl digits re reported. the frctionl digits beyond the twentieth one re rounded to 0th frctionl digit. The symmetry in the results is esily noticble. The sum of weights re ll equl to. This mkes meningful to use the word weight for the nming of these entities. 4 Numericl Efficiency We hve pplied our presented method to vrious functions. All of them re encourging nd promising lthough higher vlues of n my be required depending on how continuous f (x is or how the Tylor series expnsion of f (x converges. We do not intend to report nyone of them. Insted, we re going to give the comprison between the exct nd pproximte vlues of the Tylor series expnsion of the integrl s vlue with respect to t. To this end we cn strt with (5 nd obtin the following eqution by n pproprite coordinte trnsformtion I(t, u = dxe x f (u + tx (48 π which cn be rewritten s follows I(t, u = + ( k Γ ( k+ k! π f (k (ut k (49 by expnding f (u + tx into power series of t. On the other hnd, (43 cn be treted in the sme mnner s well. This gives I(t, u f (k (u k! n w j,n ξ k j,n tk (50 which urges us to compre the coefficients of t k in the lst two equlities. In both (49 nd (50 ll odd powers of t vnish. This, in fct, spontneously removes the positivity requirement on t. For n = 5, the coefficient of f (k (ut k in the pproximte expression j=

6 Proceedings of the 8th WSEAS Interntionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 6-8, 005 (pp68-73 (50 devites from its counterprt in the exct expression (49 when k is greter thn or equl to 5. The other coefficients mtch within 9 frctionl deciml digit ccurcy. Indeed the coefficients of f (0 (ut 0 for the exct nd pproximte expressions re roughly nd respectively. In the cse where n = 0 sitution is lmost sme however first 9 coefficients mtch insted of 9 nd mtching is not t the level of 9 but 5 frctionl deciml digit ccurcy. Remrkble devitions strt from the 0. coefficient. Similr behvior is observed for the cse where n = 5. All these men tht the number of the correct digits should be incresed in the weight nd node vlues to get lwys sme precision nd the ccurcy is bout o(t n. We suffice with this discussion here lthough this is rther qulittive nd not precise error estimtion since it is out of the scope of this work. The second importnt thing is the need for incresing high precision in the clcultions s n grows. For the first item, we do not lwys need power series expnsions in fct. Only the vlues of f (x t nodes re required unless we re enforced to use power series for some mthemticl resons. On the other hnd, high ccurcy in the clcultions when it is necessry requires the employment of the multiprecision lgorithms. This cn be done mostly by using symbolic nd/or high performnce computtionl softwres like MuPAD, Mthemtic, REDUCE, Mple, Mcsym nd so on. Acknowledgement: Author is grteful to Turkish Acdemy of Sciences for its generous support nd enthusism incresing motivtion. 5 Concluding Remrks We hve presented new extended form of the recently developed fluctution expnsion method in the evlution of univrite integrls hving Gussin type weight function. The new extension of the fluctution method here is the utiliztion of subspce spnned by not just single but more thn one functions in the Hilbert spce of the squre integrble functions over the domin of the integrl. We do not tke the first nd higher order contributions in the fluctution expnsion. We keep only the terms hving no fluctution functions in the expnsion. At the finl form, we could hve been ble to get Guss qudrture like expression by using mtrix lgebric tools. The prmeter t ppering in the denomintor of the Gussin weight function plys the most importnt role in the nlysis. Its vnishing vlue mkes the weight function Dirc s delt distribution locted t the point where x = u in the intervl. Hence the series expnsion of the integrl in powers of t somehow corresponds to the expnsion of the weight function to liner combintion of Dirc s delt function nd its derivtives. Two items re importnt in the numericl efficiency of the formul constructed here. First one is the pproximte mtch between the pproximtion formul derived here nd the series expnsion of the integrl in powers of t. Exct mtch occurs for finite number of the coefficients of these two entities in scending powers of t nd this finite number depends on the vlue of the method s subspce s dimension n. Denumerbly infinite number of remining terms relted to higher powers of t in pproximte formul devite from their counterprts in the exct formul. References: [] M. Demirlp, A Fluctution Expnsion Method for the Evlution of Function s Expecttion Vlue, ICNAAM, Interntionl Conference on Numericl Anlysis nd Applied Mthemtics Extended Abstrcts, Eds: T.E.Simos, G. Psihoyios, Ch. Tsitours, 005, pp [] M. Demirlp, Determintion ofquntum Expecttion Vlues Vi Fluctution Expnsion, Lecture Series on Computer nd Computtionl Sciences, (Proceedings of ICCMSE - 005, Interntionl Conference of Computtionl Methods in Science nd Engineering, 005, pp [3] N. A. Bykr nd M. Demirlp, Fluctution Expnsion in the Quntum Optiml Control of One Dimensionl Perturbed HrmonicOscilltor, Lecture Series on Computer nd Computtionl Sciences, (Proceedings of ICCMSE - 005, Interntionl Conference of Computtionl Methods in Science nd Engineering, 005, pp [4] E. Merl nd M. Demirlp, Phse Spce Considertion in Quntum Dynmicl Fluctution Expnsion, Lecture Series on Computer nd Computtionl Sciences, (Proceedings of ICCMSE - 005, Interntionl Conference of Computtionl Methods in Science nd Engineering, 005, pp [5] A. H. Zemnin, Distribution Theory nd Trnsform Anlysis: An Introduction to Generlized Functions, With Applictions, Dover Publictions Co., 987. [6] I. S. Grdshteyn nd I. M. Ryzhik, Tble of Integrls, Series, nd Products, Acdemic Press. Inc., 979. [7] Pderborn University, MuPAD, A Computer Algebr System: