Generalized Techniques in Numerical Integration

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1 Generlized Techniques in Numericl Integrtion. p. 1/29 Generlized Techniques in Numericl Integrtion Richrd M. Slevinsky nd Hssn Sfouhi Mthemticl Section Cmpus Sint-Jen, University of Albert Approximtion nd Extrpoltion of Convergent nd Divergent Sequences nd Series CIRM Luminy September 28 October 2, 29

2 Generlized Techniques in Numericl Integrtion. p. 2/29 The Pln Introduction A Mthemticl Tool Formule for Higher Order Derivtives PART I Ongoing Reserch The generl Ide Exmple of Applictions nd Numericl Results PART II Almost Completed An Algorithm for the G (1) n Trnsformtion Computing the Incomplete Bessel Functions

3 Generlized Techniques in Numericl Integrtion. p. 3/29 Introduction The Euler series rising from integrting the Euler integrl by prts: e t dt = e x ( 1) l l! e t t x x l +( 1)n n! dt tn+1 x e x x l= ( 1) l l! xl, x. l= Integrtion by prts by xdx led to: ( ) d λ ( ) sin(x) g(x)j λ (x)dx = g(x) ( 1) λ x λ dx xdx x [ ] ( ) = ( 1) λ x λ 1 d λ ( ) sin(x) g(x) xdx. xdx x x

4 Generlized Techniques in Numericl Integrtion. p. 4/29 Introduction [ ] ( = ( 1) λ x λ 1 d g(x) = +( 1) λ 1 λ 1 ( d ( 1) λ+l xdx l= + Leding t: ( d xdx ) λ 1 ( ) sin(x) xdx x ) [ ] ( x λ 1 d g(x) ) l [ ] ( x λ 1 d g(x) ( ) d λ [ ] ( x λ 1 sin(x) g(x) xdx x ) λ 1 ( ) sin(x) xdx xdx x ) λ 1 l ( ) sin(x) xdx x ) xdx. g(x) j λ (vx) dx = 1 v λ+1 [ ( ) d λ ( x g(x)) ] λ 1 sin(vx) dx. xdx

5 Introduction Semi-infinite sphericl Bessel integrls in moleculr integrls: ˆk g(x) = x n x n+ 1[Rγ(s,x)] n 2 [γ(s,x)] n, ˆkn+ 1(z) = zn (n+j)! γ 2 e z j!(n j)! γ(s,x) = j= (1 s)ζ 2 i +sζ2 j +s(1 s)x2. s [,1]. 1 (2z) j.4 Integrnd with sphericl Bessel 4 Integrnd with sine function How cn we use this technique for ny b f(x)dx? Generlized Techniques in Numericl Integrtion. p. 5/29

6 Generlized Techniques in Numericl Integrtion. p. 6/29 Higher Order Derivtives Let us determine the k th derivtives of G 1 (x) = x 3 f(x 2 ): ( ) d G 1 (x) = 3x 2 f(x 2 )+2x 4 f (x 2 ). ( ) dx d 2 G 1 (x) = 6xf(x 2 )+(6x 3 +8x 3 )f (x 2 )+4x 5 f (x 2 ). dx How bout this: ( ) d (x 3 G 1 (x)) = 2f (x 2 ) = xdx ( ) d k (x 3 G 1 (x)) = 2 k f (k) (x 2 ). xdx For G 2 (x) = x 2 f (ln(x)) : ( ) d k x 1 (x 2 G 2 (x)) = f (k) (ln(x)). dx Cn we express i =,1,...,k? ( ) d k G(x) in terms of dx ( ) d i x m (x n G(x)) for dx

7 Higher Order Derivtives The Slevinsky-Sfouhi formule [Slevinsky nd Sfouhi, 29]: ( ) d k Theorem Let G(x) be k th differentible with x m (x n G(x)) dx well-defined. The Slevinsky-Sfouhi formul I for (α,β,m,n) is given by: ( ) d k x α (x β G(x))= dx k i= ( ) d i A i k xn β+i(m+1) k(α+1) x m (x n G(x)), dx with coefficients [N = (n β (k 1)(α+1))]: 1 for i = k A i k = N A k 1 for i =, k > (N +i(m+1))a i k 1 +Ai 1 k 1 for < i < k. A i k = i j= ( 1) i j (n β +j(m+1) (k 1)(α+1)) k,α+1 (m+1) i j!(i j)! The Slevinsky-Sfouhi formul II:(α, β, m, n) = (,, 1, )., m 1. Generlized Techniques in Numericl Integrtion. p. 7/29

8 Generlized Techniques in Numericl Integrtion. p. 8/29 Prt I The Generlized S n Trnsformtion & The Stircse Algorithm

9 Generlized Techniques in Numericl Integrtion. p. 9/29 The generlizeds n Letf(x) be integrble on [,b], i.e. b f(x)dx exists. We write: b f(x)dx = b G (x)h (x)w(x)dx, for some weight function w(x), whose choice depends onf(x). If f(x) C n [,b], then b f(x)dx hs the equivlent representtion, which we obtin fter n integrtion by prts byw(x)dx: b f(x)dx = where: G l (x) = ( 1) l ( n 1 l= G l (x)h l+1 (x) b + b ) d l ( g(x) nd H l (x) = w(x) dx G n (x)h n (x)w(x)dx d w(x) dx ) l h(x).

10 Generlized Techniques in Numericl Integrtion. p. 1/29 The Stircse Algorithm Approximtions to b b f(x) dx tke the following form: For < x < b, initilize: S = x G (x)h (x)w(x)dx, b b G (x)h (x)w(x)dx = G (x)h 1 (x) + G 1 (x)h 1 (x)w(x)dx. x x x b x1 S 1 = S +G (x)h 1 (x) G 1 (x)h 1 (x)w(x)dx, x < x 1 < b. x x + For the sequence {x l } n l=1 stisfying < x l 1 < x l < b, define: b xl S l = S l 1 + G l 1 (x)h l (x) G l (x)h l (x)w(x)dx. x l 1 The pproximtions to b x l 1 + f(x)dx form the sequence {S l } n l=.

11 Generlized Techniques in Numericl Integrtion. p. 11/29 Bessel Integrl The integrl tht follows ppered in Numericl Recipes: I 1 = b x x 2 +1 J (x)dx = K (1). By choosing w(x) = x, we hve G (x) = 1 x 2 +1 ndh (x) = J (x). G l (x) = 2 l l! (x 2 +1) l+1 nd H l (x) = x l J l (x). The integrl then hs the equivlent representtions: n 1 2 l l!x l+1 I 1 = (x 2 +1) l+1 J l+1(x) +2 n x n+1 n! (x 2 +1) n+1 J n(x)dx l= All the boundry terms vnish nd consequently: x x 2 +1 J (x)dx = 2 n n! x n+1 (x 2 +1) n+1 J n(x)dx = K (1).

12 Generlized Techniques in Numericl Integrtion. p. 12/29 Bessel Integrl - Results Tble 1: I 1 = x l = 2π(l+1). l Sl l Sl

13 Generlized Techniques in Numericl Integrtion. p. 13/29 Fresnel Integrls The integrls re given by: I 2 (,v) = sin(vx 2 )dx nd Ĩ 2 (,v) = cos(vx 2 )dx. By choosing w(x) = x, we hve G (x) = 1 x nd H (x) = sin(vx 2 ). G l (x) = (2l)! 2 l l!x 2l+1 nd H l (x) = sin(vx2 lπ/2) (2v) l. The integrl I 2 (,v) then hs the equivlent representtions: n 1 2(2l)! sin(vx 2 (l+1)π 2 ) (4v) l+1 l! x 2l+1 + (2n)! sin(vx 2 nπ 2 ) (4v) n n! x 2n dx l=

14 Generlized Techniques in Numericl Integrtion. p. 14/29 Fresnel Integrls - Results Tble 2: I 2 (,1) = x l = 2π(l+1)/v. l Sl l Sl The integrl Ĩ2(,v) then hs the equivlent representtions: n 1 2 (2l)! cos(v x 2 (l+1)π 2 ) (4v) l+1 l! x 2l+1 + (2n)! cos(vx 2 nπ 2 ) (4v) n n! x 2n dx l=

15 Generlized Techniques in Numericl Integrtion. p. 15/29 The Twisted Til The integrl is proposed in the book "The SIAM 1-Digit Chllenge": I 3 = 1 t 1 cos ( t 1 ln(t) ) dt = w(x) = (1+x)e x G (x) = cos(xe x )dx, (x = ln(t)). 1 (1+x)e x nd H (x) = cos(xe x ). ( ) d l 1 G l (x)= (1+x)e x dx (1+x)e x nd H l(x) = cos The generl form of G l (x) is: G l (x) = e (l+1)x (1+x) 2l+1 p l(x) ( xe x lπ 2 p (x) = 1 p 1 (x) = 2+x p 2 (x) = 9+8x+2x 2 p 3 (x) = 64+79x+36x 2 +6x 3... ).

16 Generlized Techniques in Numericl Integrtion. p. 16/29 The Twisted Til The integrl I 3 then hs the equivlent representtions: n 1 p l (x)e (l+1)x cos(x e x (l+1)π 2 ) p n (x)e nx cos(xe x nπ 2 (1 +x) 2l+1 + ) (1 +x) 2n dx l= As specific cse, the generlizeds 1 yields the equivlent representtion: cos(x e x )dx = e x 2 +x (1 +x) 2 sin(x ex )dx y.6.4 y x.2 x

17 Generlized Techniques in Numericl Integrtion. p. 17/29 Twisted Til - Results Tble 3: I 3 = l S l l S l { {x l } n l= = ln(2π(l+2)) ln(ln(2π(l+2)))+ ln(ln(2π(l+2))) ln(2π(l+2)) This sequence is derived from the symptotic expnsion of the LmbertW function defined implicitly by w(x)e w(x) = x. } n. l=

18 Generlized Techniques in Numericl Integrtion. p. 18/29 Airy Functions The Airy functions πai(z) re given by: ( ) x 3 I 4 (,z) = cos 3 +zx w(x) = x 2 +z G (x) = 1 x 2 +z ( G l (x) = G l (x) = nd dx. H (x) = cos ( x 3 3 +zx ). ) d l ( 1 x 3 (x 2 +z)dx x 2 +z nd H l(x) = cos 3 +zx lπ 2 p l (x) (x 2 +z) 2l+1 where p l (x) re polynomils. It cn be shown tht p 2k+1 () = ndp 2k () exist nd since H l () =, the boundry terms vnish t = for z. ).

19 Generlized Techniques in Numericl Integrtion. p. 19/29 Airy Functions The integrl I 4 (,z) then hs the equivlent representtions: ( ) p n (x) x 3 nπ I 4 (,z) = (x 2 cos +z x +z) 2n 3 2 How to determine the functionls G l (x) explicitly? We cn decompose I 4 (,z) s follows: [ ( I 4 (,z) = cos(zx)cos x 3 /3 ) sin(zx)sin ( x 3 /3 )] dx. By choosing the weight function w(x) = x 2, we obtin: ( ) d l G l (x) = ( 1) l x 2 x 2 cos dx sin (zx) nd H l(x) = cos sin dx ( x 3 The Slevinsky-Sfouhi formul I with (α, β, m, n) = (2, 2,, ): l ( G l (x) = ( 1)l x 3l+2 A i cos l (zx)i zx+ iπ ). sin 2 i= 3 lπ 2 ).

20 Generlized Techniques in Numericl Integrtion. p. 2/29 Airy Functions - Results Ultimtely, we derive the explicit form of the trnsformed integrls s: n 1 ( 1) l+1 l ( ) x I 4 (,z) = x 3l+2 A i 3 l (zx)i cos 3 +zx (l+1 i)π/2 l= i= n + ( 1) n A i n(zx) i ( ) x 3 x 3n cos 3 +zx (n i)π/2 dx. i= Tble 4: I 4 (,1) = x l = 3 6π(l+1). l Sl l Sl

21 Generlized Techniques in Numericl Integrtion. p. 21/29 Prt II An Algorithm for TheG (1) n Trnsformtion & Computtion of Incomplete Bessel Functions

22 Generlized Techniques in Numericl Integrtion. p. 22/29 TheG (m) n Trnsformtion Letf(x) be integrble on [, ) nd if: m f(x) = p k (x)f (k) (x) where p k (x) x i k k=1 i= i x i s x, i k k. Levin nd Sidi, 1981: f(t)dt The pproximtion G (m) n { d l dx l G (m) n = x where it is ssumed tht x f(t)dt+ k= m 1 k= f (k) (x)x σ k i= n 1 i= β i,k x i. of f(t)dt is given by [Gry nd Wng, 1992]: } m 1 n 1 f(t)dt+ x σ k β f (k) k,i (x) x i, l mn, d l dx lg(m) n, l >.

23 Generlized Techniques in Numericl Integrtion. p. 23/29 TheG (1) n Trnsformtion By considering the eqution with l = : n 1 G (1) n = F(x)+x σ β,i f(x) x i, F(x) = i= x f(t)dt, nd by isolting the summtion on the RHS, we obtin: G (1) n F(x) x σ f(x) = If we pply ( x 2 d dx ( x 2 d dx n 1 i= ) n (G (1) n F(x) x σ f(x) β,i x i = ) n, we obtin: ) ( x 2 d ) ( G (1) n F(x) dx x σ f(x) = = G (1) n = ) n 1 = i=1 ( ) ( ) x 2 d n F(x) dx x σ f(x) ) n ( ( x 2 d dx 1 x σ f(x) i β,i x i 1. ) = N n(x) D n (x).

24 Generlized Techniques in Numericl Integrtion. p. 24/29 Incomplete Bessel functions We begin with: K ν (x,y) = x ν x e t xy/t t ν+1 dt. The integrnds f x,y,ν (t) = f(t) = e t xy/t t ν+1. We hve: f(t) = Progrmming the pproximtion G (1) 1 of G (1) 1 = x νn 1(x) D 1 (x) = t 2 t 2 xy +(ν +1)t f (t). e t xy/t t ν+1 xe x y x 2 xy +(ν +1)x +xν dt, we obtin: x We then extrct the pproximtion to the functions K ν (x,y): G (1) 1 = xe x y x 2 xy +(ν +1)x. e t xy/t t ν+1 dt.

25 Generlized Techniques in Numericl Integrtion. p. 25/29 Incomplete Bessel functions The pproximtions to K ν (x,y) tke the form: G (1) n = x ν Ñn(x) D n (x). The Leibniz product rule nd the Slevinsky-Sfouhi formul I with (α,β,m,n) = ( 2, ν 1,,) led t: ( D n (x) = t 2 d dt ) n ( t ν+1 e t+xy/t) t=x = ( xy) n x ν+1 e x+y n Ñ n (x) = e x y x ν y n r=1 r= ( ) n r ( y) r A i r rx i. i= ( ) n D n r (x,y,ν)(xy) r r r 1 s= ( ) r 1 s y s A i s s( x) i. i=

26 Generlized Techniques in Numericl Integrtion. p. 26/29 Numericl Results Tble 5: Numericl Results. x y ν K ν (x,y) n Error n Error ( 1) 1.57(-1) 21.88(-15) ( ) 7.96(-1) 17.36(-15) (-1) 5.78(-1) 13.31(-15) (-1) 6.21(-11) 9.75(-15) (-1) 7.21(-11) 9.13(-16) (-2) 8.31(-11) 1.52(-17) (-2) 9.66(-11) 11.1(-16) (-2) 1.19(-1) 12.34(-16) (-2) 11.72(-1) 13.11(-15) (-2) 13.62(-12) 14.58(-15) (-4) 16.27(-1) 22.63(-15) (-6) 4.29(-1) 1.17(-15) (-3) 12.62(-1) 27.49(-15)

27 Generlized Techniques in Numericl Integrtion. p. 27/29 Some Conclusions Prt I GenerlizedS n trnsformtion: Expnsions of some chllenging integrls. Integrls representtions with better convergence properties. Generlly pplicble to wide clss of integrls. The stircse lgorithm llow for ccurte numericl evlution. Prt II The G (1) n trnsformtion: The Slevinsky-Sfouhi formul I for higher order derivtives llows for rpid evlution of high-order G (1) n trnsformtions. Accurte computtion of the chllenging incomplete Bessel functions.

28 Generlized Techniques in Numericl Integrtion. p. 28/29 Acknowledgements Finncil support: The Nturl Sciences nd Engineering Reserch Council of Cnd. The Reserch Office of the University of Albert The Reserch Office of Cmpus Sint-Jen. The orgnizing committee: Clude Brezinski, Michel Redivo-Zgli nd Ernst J. Weniger. Thnk you!

29 Generlized Techniques in Numericl Integrtion. p. 29/29 References 1. C. Brezinski nd M. Redivo-Zgli. Extrpoltion Methods: Theory nd Prctice. Edition North-Hollnd, Amsterdm, A. Sidi. Prcticl Extrpoltion Methods: theory nd pplictions. Cmbridge U. P., D. Levin nd A. Sidi. Two new clsses of non-liner trnsformtions for ccelerting the convergence of infinite integrls nd series. Appl. Mth. Comput., 9: , H. Gry nd S. Wng. A new method for pproximting improper integrls. SIAM J. Numer. Anl., 29: , H. Sfouhi. Efficient nd rpid numericl evlution of the two-electron four-center Coulomb integrls using nonliner trnsformtions nd prcticl properties of sine nd Bessel functions. J. Comp. Phys., 176:1 19, R. Slevinsky nd H. Sfouhi. New formule for higher order derivtives nd pplictions. J. Comput. App. Mth., 233:45 419, R. Slevinsky nd H. Sfouhi. The S nd G trnsformtions for computing three-center nucler ttrction integrls. Int. J. Quntum Chem., 19: , E. Weniger. Nonliner sequence trnsformtions for the ccelertion of convergence nd the summtion of divergent series. Comput. Phys. Rep., 1: , R. Slevinsky nd H. Sfouhi. Numericl tretment of twisted til using extrpoltion methods. Numer. Algor., 48:31 316, 28.