2000 Mathematical Subject Classification: 65D32
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1 Generl Mthemtics Vol. 11 No. 4 ( On the Tricomi s qudrture formul Dumitru Acu Dedicted to Professor Gheorghe Micul on his 60 th birthdy Abstrct In this pper we obtin new results concerning the Tricomi s qudrture formul Mthemticl Subect Clssifiction: 65D2 In [10] F. Tricomi introduced interesting qudrture rule nlogous to fmous Richrdson s method [5]. The pper [8] [9] [1] complete with new results the Tricomi s pper. In this rticle we present some new results concerning the generliztion of the Tricomi s qudrture formul given in [1]. 1. Let f be function from AC n 1 [ b] - the clss of functions f whose (n 1-th derivtive f (n 1 is bsolutely continuous in [ b]. 5
2 6 Dumitru Acu (1 We consider the elementry qudrture formul b f(xdx = n 1 k=0 m i=1 A ki f (k (x i R(f with the lgebricl degree of exctness n 1 nd with the reminder given by (2 b R(f = f (n (ξ where φ is the influence function ([7]. φ(xdx ξ ( b We divide the intervl [ b] into certin number p prtil intervls [u 1 u ] = 1 p defined by the points ( = u 0 < u 1 <... < u p 1 < u p = b nd we put d = u u 1 = 1 p. We evlute the integrl b I = f(xdx using the generlized qudrture formul generted to the elementry formul (1 (see [7]. We obtin (4 I = S p R p with (5 S p = nd the reminder (6 p n 1 m k=0 i=1 R p = T [ p ( k1 ( d A kif (k x u 1 d ( ] n1 d f (n (ξ 1 ξ 1 [ b
3 On the Tricomi s qudrture formul 7 where (7 T = b φ(xdx. Now we divide the intervl [ b] into certin number q q > p prtil intervls [v 1 v ] = 1 q defined by the points (8 = v 0 < v 1 <... < v q 1 < v q = b with c = v v 1 = 1 q. We evlute the integrl I using the generlized qudrture formul generted to the elementry formul (1 nd the prtition (8. We find: (9 I = S q R q with (10 S q = nd the reminder (11 (12 q n 1 m k=0 i=1 R q = T [ q ( k1 ( c A kif (k x i v 1 c ( ] n1 c f (n (ξ 2 ξ 2 ( b. If the function f is polynomil of degree n then f (n (ξ 1 = f (n (ξ 2. In this sitution if we put ( D = p n1 d nd C = p then from (4 (6 (9 (11 it results ( c n1 (1 T f (n (ξ 1 = T f (n (ξ 2 = S q S p D C.
4 8 Dumitru Acu Using (9 (11 şi (1 we get (14 I = S q tn 1 t n (S q S p where t = n C/D. Now for n rbitrry function f AC n 1 [ b] we consider the qudrture formul (15 I = S q where S q S p nd t re the bove. From (15 (4 nd (9 we hve tn 1 t n (S q S p R pq R pq = 1 1 t n (R q t n R p whence using (6 nd (11 we find the following forms for the reminder of the formul (15: (16 R pq = T C (n n [f(ξ 1 t 2 f (n (ξ 1 ] (17 (18 R pq = T Dtn (n n [f(ξ 1 t 2 f (n (ξ 1 ] R pq = T CD (n [f(ξ D C 2 f (n (ξ 1 ] ξ 1 ξ 2 ( b In view of (16 (17 nd (18 it results (19 T C 1 t n Ω(n (20 T Dtn 1 t n Ω(n
5 On the Tricomi s qudrture formul 9 (21 T CD D C Ωn where Ω (n is the oscilltion of the function f (n on [ b]. If we tke into ccount tht for T usully hve reltion by form b T = φ(xdx = α n1 where α is constnt then from (9 - (21 we hve (22 (2 (24 (25 2. Prticulr cses. 2.1 If we consider then t = p/q. α α p α d n1 d n1 C n1 Ω (n 1 t n d n1 Ω (n 1 t n ( q n C n1 c n1 d 1 = d 2 =... = d b = p nd = ω p c 1 = c 2 =... = c q = q = ω q Ω (n.
6 40 Dumitru Acu In this cse the formul (15 presents generliztion of the results from [8] for the qudrture formuls which they contin the vlues of the derivtive the function f on the nodes. For the reminder from (2 we found (26 α pω n1 p t n 1 t n Ω(n. For q = 2p we obtin generliztion of the Tricomi s results. 2.2 If (25 re stisfied nd the elementry qudrture formul (1 contins only the vlues of the function f on the nodes then (15 nd (22 - (24 give the results by [8]. For q = 2p we obtin the Tricomi s results ([10]. Exmples. In this section we present some exmples of qudrture formul by Tricomi s type obtined by the prticulrizing of the elementry rule (1..1. Assume tht (1 is the rectngulr formul ([7]. We find the following Tricomi s qudrture formul: t2 I = S q 1 t 2 (S q S p R pq with S p = S q = p q d ( d 1 d 2... d 1 d 2 c ( c 1 c 2... c 1 c 2 ( q t = c / ( q d
7 On the Tricomi s qudrture formul 41 1 d t t 2 Ω(2 where Ω (2 is the oscilltion of the function f on [ b]. From here for d 1 =... = d p = b b = ω p c 1 = c 2 =... = c q = b q we find the results by [8]..2 We consider (1 the Simpson s formul. It results the following Tricomi s qudrture formul = ω q with I = S q S p = d p 1 2 f( p t4 1 t 4 (S q S p R pq d d 1 ( d 1... d 6 ( 2d f d 1... d 1 d S q = c q f( q d p 6 f(b c c 1 f ( c 1... c 6 2c ( f c 1... c 1 c ( q t = 4 / c 5 d 5 d 5 C q 6 f(b t 4 1 t 4 Ω(4 where Ω (4 is the oscilltion of the function f (4 on [ b]. From here for d 1 = d 2 =... = d p = 2 b 2p = 2ω 2p nd c 1 = c 2 =... = 2( c q = 2q we obtin the results by [8]. If q = 2p then we find the cse studied by Tricomi s ([10].
8 42 Dumitru Acu []. with.. Finlly we ssume tht (1 is Newton s qudrture formul ([2] We obtin the following Tricomi s qudrture formul: t4 I = S q 1 t 4 (S q S p R pq S p = d p 1 i 8 f( d p 8 f(b p p d 8 f d 8 f S q = c q 1 i 8 f( c q 8 f(b q q c 8 f d d 1 f( d 1... d 8 ( d 1... d 1 d ( d 1... d 1 2d c 8 f ( c c 1 f( c 1... c 8 c 1... c 1 c ( c 1... c 1 2c ( q t = c 5 / d 5 d 5 t 4 1 t 4 Ω(4 where Ω (4 is the oscilltion of the function f (4 on the [ b].
9 On the Tricomi s qudrture formul 4 References [1] Acu D. Generlizre procedeului de integrre numerică lui Tricomi Buletinul Ştiinţific l Institutului de Învăţământ Superior din Sibiu Serie tb. mt. vol. VI (in Romnin. [2] Acu D. Formule combinte de cudrtură Studi Univ. Bbeş - Bolyi Mthemtic XXVI (in Romnin. [] Acu D. O generlizre formulei de qudrtură lui Newton St. Cerc. Mt (in Romnin. [4] Acu D. O formulă combintă de cudrtură St. Cerc. Mt (in Romnin. [5] Comn Gh. O generlizre formulei de cudrtură trpezelor şi formulei lui Simpson Studi Univ. Bbeş - Bolyi ser. Mth. Phys (in Romnin. [6] Engels H. Numericl Qudrture nd Cubture Acdemic Press 1980 [7] Ghizzetti A. Ossicioni A. Qudrture formule Acd.-Verlg-Berlin 1970 [8] Icob A. Generlizre unor formule de cudrtură lui Tricomi St. Cerc. Mt (in Romnin. [9] Micul M. Micul Gh. Sur l formule de qudrture de l Tricomi Bull. Mth. de l Soc. Sci. Mth. de l R.S.R. 12 ( (in Romnin.
10 44 Dumitru Acu [10] Tricomi F. Sul resto delle formule di qudrture numeric migliorte con metode extrpolzione Boll. dell Unione Mth. Itl (in Romnin. Deprtment of Mthemtics Lucin Blg University of Sibiu Str. Dr. I. Rţiu nr Sibiu Romni. E-mil ddress:
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