On the interval Legendre polynomials

Transcription

1 Journl of Computtionl nd Applied Mthemtics 154 ( On the intervl Legendre polynomils F.Ptrcio, J.A.Ferreir, F.Oliveir Deprtment of Mthemtics, University of Coimbr, Aprtdo 3008, Coimbr 3000, Portugl Received August 001; received in revised form 15 September 00 Abstrct In this pper, the extention to intervl theory of the clssicl Legendre polynomils is considered.the so-clled intervl Legendre polynomils re introduced nd some properties re studied.bsed on these polynomils n intervl minimum squre pproximtion is introduced when continuous nd discrete dt re ten. c 003 Elsevier Science B.V. All rights reserved. 1. Introduction The role of Legendre polynomils on the pproximtion theory of rel functions is well nown.the im of this pper is to study n intervl version of those polynomils.we strt by introducing some bsic denitions.using the recurrence reltion of the denition of the clssicl Legendre polynomils we introduce n intervl version of those functions which we cll intervl Legendre polynomils nd some properties re studied.we introduce minimum squre pproximtion for continuous nd discrete dt bsed on those intervl polynomils.severl exmples showing the behviour of the dened pproximtions re lso presented in this pper.an ppliction of the introduced theory to the computtion of n estimte in the Hertzsprung Russel digrm in Astrophysics is lso included.. Bsic denitions In this section we introduce the bsic denitions introduced by the uthors in [1] nd lso some denitions nown in the intervl nlysis theory. This wor ws supported by Centro de Mtemtic d Universidde de Coimbr. Corresponding uthor /03/$ - see front mtter c 003 Elsevier Science B.V. All rights reserved. PII: S (

2 16 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Denition 1. A rel intervl polynomil degree n is dened by P n (x= A j x n j A 0 =1; A j =[ (1 j ; ( j ] R; j =1;:::;n. (1 According to (1 it is esy to see tht P n (x is fmily of polynomils p n (x= j x n j 0 =1; j A j ;j=1;:::;n. Using the denition of grph of rel function we introduce the grph of rel intervl polynomil. Denition. Let P n (x be rel intervl polynomil.the grph of P n (x is denoted by G(P n nd is given by G(P n ={( x; ỹ R : p n (x P n (x; ỹ = p n ( x}: In the next lemm we chrcterize the grph of rel intervl polynomil P n (x using the grphs of certin rel polynomils.in order to do tht, let q + (x; r +, q (x; r (x be the following rel polynomils q + (x= q +;j x n j ; r + (x= r +;j x n j ; nd q (x= q ;j x n j ; r (x= r ;j x n j ; ( q +;0 = r +;0 = q ;0 = r ;0 =1; q +;j = ( j ; r +;j = (1 j ; j =1;:::;n; q ;j = { ( j ; if n j is even (1 j ; if n j is odd; r ;j = { (1 j ; if n j is even ( j ; if n j is odd: Lemm 1. Let P n (x be the rel intervl polynomil given by (1. The grph of P n is given by G(P n ={(x; y R :(r + (x 6 y 6 q + (x if x 0 or (r (x 6 y 6 q (x if x 0}; q + (x;r + (x nd q (x;r (x re given by (. Let us remember now the denition of the integrl of n intervl function [3,4].

3 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Denition 3. Let F(x be rel intervl function continuous in [; b] nd F(x =[F (x;f + (x]; x [; b].then [ ] F(xdx = F (xdx; F + (xdx : Remr 1. If F(x is rel intervl polynomil, tht is, F(x=P n (x nd [; b] 0 (respectively [; b] 6 0 then [ ] P n (xdx = r + (xdx; q + (xdx ; (respectively P n(xdx =[ r (xdx; q (xdx]. If [; b]=[; 0] [0;b] then [ 0 0 ] [ P n (xdx = r (xdx; q (xdx + r + (xdx; 0 0 ] q + (xdx : Using denition (1 we introduce in the spce of intervl continuous functions dened in [; b] which we denote by C[; b], the following inner product ting vlues in set of ll rel intervls R. Denition 4. Let :; : : C[; b] C[; b] R be dened by F; G = F(xG(xdx; F; G C[; b]: (3 3. Intervl Legendre polynomils In this section we introduce n intervl version of the Legendre polynomils. Denition 5. Let us consider, for ech nturl number, the fmily of intervl polynomils dened by the following recursive formul 1. L 0; (x=[1 1 ; 1+ 1 ],. L 1; (x=[1 1 ; 1+ 1 ]x, 3.for n N, n +1 L n+1; (x= n +1 xl n;(x n n +1 L n 1;(x: For ech N nd n N, we cll L n; (x intervl Legendre polynomil. Exmple 1. In Tble 1 we present the intervl Legendre polynomils for n =0; 1; ; 3. Their grphs re plotted in Figs. 1 nd for = 50.

4 18 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Tble 1 Intervl Legendre polynomils of degree 0, 1,, 3 [ L 0; (x= 1 1 ; 1+ 1 ] [ L 1; (x= 1 1 ; 1+ 1 ] x L ; (x= 3 [ 1 1 ; 1+ 1 ] x 1 [ 1 1 ; 1+ 1 ] L 3; (x= 5 [ 1 1 ; 1+ 1 ] x 3 3 [ 1 1 ; 1+ 1 ] x Fig.1.Intervl Legendre polynomils of degree zero nd degree one Fig..Intervl Legendre polynomils of degree two nd degree three. In the next theorem we estblish reltion between the intervl Legendre polynomils nd the well nown Legendre polynomils n (x;n N.

5 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Theorem 1. The intervl Legendre polynomil L n; (x is equl to the intervl polynomil obtined from the Legendre polynomil n (x considering their coecients multiplied by [1 1 ; 1+ 1 ]. Proof. Follows from the denitions of L n; (x nd n (x. Theorem. The intervl Legendre polynomils L n; (x;n N, stisfy 1. If n is even then the rel polynomils q + ;r + ;q nd r for L n; re given by n= ( r + (x=q + (x= j 1+ ( 1j ( 1 j x j ; n= ( r (x=q (x= j 1+ ( 1j+1 ( 1 j x j ; j = (n +j! n (j!( n + j!( n j!;. If n is odd then the rel polynomils q + ;r + ;q nd r for L n; re given by r (x=q + (x= q (x=r + (x= n 1 n 1 ( j 1+ ( 1j ( 1 n 1 j x j+1 ; (n +1+j! j = n (j 1!( n+1 + j!( n+1 j! ; 3. For n m; L n; ; L m; 0; +. ( j 1+ ( 1j+1 ( 1 n 1 j x j+1 ; Proof. Using the denitions of L n; ; q + ;r + ;q nd r we esily get 1 nd. Finlly, the behviour of L n; ; L n; when +, tht is 3, is consequence of the orthogonlity of the Legendre polynomils n in [ 1; 1]. Attending to the lst result, we remr tht the set {L n; ;n N 0 } stises L n ; L m 0 when +.Attending to this fct we sy tht the set of the intervl Legendre polynomils is symptoticlly orthogonl.

6 0 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Minimum squre pproximtions 4.1. Continuous dt Let us consider now the spce of intervl Legendre polynomils L n; equl to n L n; = j L j; (x; j R : of degree less or Denition 6. Let F(x;x [ 1; 1], be given intervl function nd proj Ln; F(x= F; L j; L j; ; L j; L j;(x; where :; : is dened by (3.We cll proj Ln; F the symptoticlly orthogonl projection of degree n of F into L n;. Exmple. Let Y (x=e x ;x [ 1; 1].This function cn be seen s rel intervl function.in the following we compute proj L; Y.We hve proj L; Y (x= We compute ech term of proj L; Y (x: Y; L 0; L 0; ; L 0; L 0;(x= Y; L 0; L 0; ; L 0; L 0;(x+ Y; L 1; L 1; ; L 1; L 1;(x+ Y; L ; L ; ; L ; L ;(x: e 1[1 1 e ; 1+ 1 ] [ (1 1 ; (1 + 1 ] L 0;(x = e 1 e [ ( 1 ( ] +1 ; ; +1 1 Y; L 1; L 1; ; L 1; L e 1;(x= [(1 e 1 +1; (e ] 3 [(1 1 ; (1 + 1 ] [ ( 1 = 3 e ( +1 L 1; (x ((1 e 1 +1 ( +1 x; ( 1 ((e 1 1 ] +1 x : For the computtion of ( Y; L ; = L ; ; L ; L ; (x we strt by noting tht Y; L ; = 1 [ e (4 e +e 7; ] (e 4 + e 7 :

7 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( It is not simple ts to prove tht where nd L ; ; L ; =[ ;b ]; ( 9 = 10 b + b + ( 9 b = b 10 b ( 9 +(b 10 (5 i + b 5 s+ b ( 3 s 3 i 1 (b i + ; (4 5 3 ( b ; (5 =1 1 ; b= 1 ; +1 i = 3( 1 ; 1 s = 3( +1 : Then, ttending tht for 0; (4 e +e 7 0, we obtin Y; L L ; ; L ; L ;(x= 1 [ ( ( 1 e (4 e +e 1 7 f (x; (e 4 + e 7 b ] f + (x where L ; (x=[f (x;f + (x]. In order to give n illustrtion of the behviour of proj L; Y (x we only give n estimtion for this intervl function.we too [ 5 ; 5 ] L ;; L ; nd we dene F (x by F (x=[f ; (x;f ;+ (x] proj L; Y (x; x [ 1; 1] F ; (x= 1 ( ( e 1 1 ( 1+3 (1 e+1 x e( ( 4e (4 e +e 7 f (x; x [ 1; 1]; F ;+ (x= +1 ( ( e 1 1 ( +1+3 (e 1+1 x e( ( 4e (e 4 + e 7 f + (x; x ]0; 1]: In Figs. 3 nd 4 we plot the grphs of Y nd F for dierent vlues of. In the following we study the behviour of d(f; proj Ln; F for certin metric d(:; : dened for intervl functions. Let us consider C[; b].let F; G C[; b], F(x=[f (x;f + (x]; G(x=[g (x;g + (x] for x [; b].in order to dene metric on the lst set we sy tht F =G if f (x=g (x;g + (x=f + (x for x [; b]. Let us dene now metric in C[; b].

8 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Fig.3.The intervl function F Fig.4.The intervl function F 50. Theorem 3. For F; G C[; b] we dene d L (F; g by d L (F; G = mx{ f g L [;b]; f + g + L [;b]}: The lst denition induces metric in C[; b]. Proof. In fct, it is esy to prove tht 1. d L (F; G = 0 if nd only if F = G in C[; b],. d L (F; G 6 d L (F; H+d L (H;G for F; G; H C[; b]. Exmple 3. Let us consider the following intervl function Exp (x=[1 ; 1+]e x ; x [ 1; 1]; (0; 1: The minimum squre pproximtion of rst degree proj L1; Exp (x is given by proj L1; Exp (x=[a; B]+[C; D]x A = (e 1(1 ( 1 e( +1 ;

9 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Fig.5.The squre of the distnce of F to proj L1; Exp 0:5 s function of. B = (e 1(1 + ( +1 ; e( 1 ( ( (1 + ( +1 e(1 + (1 ( +1+e(1 + C =3( 1 ; D=3( 1 : ( +1 e( 1 Let us loo to the behviour of d L (F; proj L1; Exp.For 10 we hve ( ( (1 + (e 4 1 B(e 1 + C( e d L (F; proj L1; Exp =(1+ + D e e + B(B + D C+ D + C : 3 In Fig. 5 we plot d L (F; proj L1; Exp 0:5. Let us consider now proj L; Exp (x.we hve proj L; Exp (x = proj L1; Exp (x+ Exp ; L ; L ; ; L ; L ;(x [ Exp ; L ; L ; ; L ; L Exp; ;(x= ; Exp ] ;+ L ; (x; b where nd b re dened by (4 nd (5, nd ( ( 3 Exp ; =(1 1 1 e 5 1 ( 1+ 1 e 1 e e ( ( 3 Exp ;+ =( e 5 1 ( 1 1 e 1 e e Then d L (F; proj L; Exp = 1+ e ; : ( (1 + (e 4 1 ( (B F(e 1+C + E(e 5 e

10 4 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Fig.6.The squre of the distnce of F to proj L; Exp 0:5 s function of Fig.7.The behviour of d L (F; proj L1; Exp 0:5 nd d L (F; proj L; Exp 0:5. ( ( E + B 3 F ( + (B + F +E ( E +(D C ( E 5 F 3 4 F (1 + +B + C + D ; 3 E = 3 Exp ;+ ( +1; F = 1 Exp ;+ ( 1: In Fig. 6 we plot d L (F; proj L; Exp 0:5. Finlly in Fig. 7 we plot d L (F; proj L1; Exp 0:5 nd d L (F; proj L; Exp 0: Discrete dt Let us consider discrete set of dt {(x i ;Y i ;i=1;:::;n}, where x i R nd Y i =[y (1 i ;y ( i ]is compct rel intervl.in wht follows we construct n intervl function which pproximtes the discrete set given.we strt by dening nother set of discrete dt {(ˆx i ; ŷ i ;i=1;:::;n}

11 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( ˆx 1 =ˆx = x 1 ; ˆx 3 =ˆx 4 = x ;:::; ˆx n 1 =ˆx n = x n nd ŷ 1 = y (1 1 ; ŷ = y ( 1 ; ŷ 3 = y (1 ; ŷ 4 = y ( ;:::;ŷ n 1 = y n (1 ; ŷ n = y n (. For the lst set we dene ˆ (x by m ˆ (x= j j (x; where j (x isthej Legendre polynomil nd j ;;:::;m, re the solutions of the minimiztion problem n min 0;:::; m R j=1 n (ŷ j (x j = (ŷ j ˆ (x j (6 j=1 (x = m j j (x; j R.Tht is ˆ (x is the minimum squre pproximtion for the set {(ˆx i ; ŷ i ;i=1;:::;n} when Legendre polynomils re considered. Denition 7. Let ˆL (x be dened by m ˆL (x= j L j; (x; where j ;;:::;m, stisfy (6.We cll ˆL (x the intervl Legendre minimum squre pproximtion for the discrete set {(x i ;Y i ;i=1;:::;n}, where x i R nd Y i =[y (1 i ;y ( i ]. In wht follows we consider the performnce of the dened intervl Legendre minimum squre pproximtion on the computtion of n intervl function which pproximtes set of points. Exmple 4. The theoreticl models for certin strs enble to obtin the loction of those strs using the so clled Hertzsprung Russel (HR digrm.the loction is obtined dening two curves which re the solutions of certin dierentil equtions rising on the model.the Hipprcos mission hs provided very ccurte dt for hundred dis strs of spectrl types F to K.In [] those observtions were nlysed by mens of the stellr models computed using the most recent input physics nd the positions of the objects versus stndrd theoreticl isochrones, corresponding to their chemicl composition nd ge, were exmined.some discrepncy between the theoreticl position nd the position observed ws detected. Tble ws ten from [] for certin strs.the position of the str is (log T e ;M bol nd Mbol is the bsolute error of the observed vlue M bol, tht is, for the vlue T e the corresponding vlue M bol belongs to the intervl [M bol bol ;M bol + bol ].We use the minimum squre Legendre polynomils on the computtion of n estimte to the HR digrm. The minimum Intervl Legendre polynomil of rst degree coecients presenting n exct deciml digit is [ ˆL (x=83:6 1 1 ] [ ; 1+1 1:0 1 1 ] ; 1+1 x: Its grph is plotted in Fig. 8 = 00.

12 6 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( Tble T e M bol Mbol T e M bol Mbol T e M bol Mbol Fig.8.The minimum squre intervl Legendre polynomil of degree one = Fig.9.The minimum squre intervl Legendre polynomil of degree two =10 4. The minimum intervl Legendre polynomil of degree two coecients presenting n exct deciml digit is [ ˆL (x= 887:3 1 1 ] [ ; :6 1 1 ] [ ; 1+1 x 70:3 1 1 ] ; 1+1 x ; nd its grph is plotted in Fig. 9 for =10 4.

13 F. Ptrcio et l. / Journl of Computtionl nd Applied Mthemtics 154 ( We remr tht the behviour of the minimum squre intervl Legendre polynomil of degree three is similr to the behviour of the minimum squre intervl Legendre polynomil of degree two A piecewise version of the intervl Legendre polynomils cn be dened nd more ecient curves contining ll the observed vlues cn be constructed. References [1] J.A.Ferreir, F.Ptrcio, F.Oliveir, On the computtion of the zeros of intervl polynomils, J.Comput.Appl. Mth.136 ( [] Y.Lebreton, M.-N.Perrin, R.Cyrel, A.Bglin, J.Fernndes, The Hipprcos HR digrm of nerby strs in the metllicity rnge: 1 [Fe=H] 0:3 new constrint on the theory of stellr interiors nd model tmospheres, Astron.Astrophys.350 ( [3] R.E. Moore, Intervl Anlysis, Prentice-Hll Inc, Englewood Clis, [4] R.E. Moore, Methods nd Applictions of the Intervl Anlysis, SIAM, Phildelphi, 1979.