Stepsize Control for Cubic Spline Interpolation

Transcription

1 Nonme mnuscript No. (will be inserted by te editor) Stepsize Control for Cubic Spline Interpoltion Dvod Kojste Slkuye Received: dte / Accepted: dte Abstrct Numericl lgoritms wic include stepsize re ffected by globl error, wic consists of bot trunction error nd round-off error. In tese lgoritms s te stepsize decreses, te trunction error lso decreses, but te round-off error my increse. Te problem is now to find te stepsize wic minimizes te globl error. In tis pper, by using te CESTAC metod nd te CADNA librry procedure is proposed to control te stepsize for determining interpolting cubic spline functions. Some numericl experiments re given to sow te effectiveness of te proposed procedure. Keywords Cubic spline stepsize CESTAC metod DSA CADNA librry. Mtemtics Subject Clssifiction (2000) 65D07 65G50. 1 Introduction Finite precision computtions my ffect te stbility of lgoritms nd te ccurcy of computed solutions. In fct, te numericl result provided by n lgoritm is ffected by globl error, wic consists of bot trunction error nd round-off error. Computtion of te solution of mny problems in scientific computing involves stepsize. As decreses, te trunction error lso decreses, but te round-off error in te metod my increse. Here, te problem is to find te optiml stepsize opt wic minimizes te globl error. In generl, it is difficult to estimte te optiml stepsize in n lgoritm. In [4], Cesneux nd Jézéquel sowed tt by using te CESTAC (Contrôle et Estimtion Stocstique des Arrondis de Clcul) metod [20, 19] nd te CADNA (Control of Accurcy nd Debugging for Numericl Appliction) librry [2, 3, 17], one cn estimte te optiml stepsize for te numericl computtion of integrls using te trpezoidl nd Simpson s rules. Ten, Abbsbndy nd Argi [1] developed tis metod to generl closed Newton-Cotes integrtion rules. Te development of te metod to multiple integrls cn be found in [12]. In [14], Slkuye et l. proposed procedure wit stepsize control for solving n one-dimensionl initil vlue problems. In tis pper, procedure is presented to control te stepsize for te cubic spline interpoltions wit eqully spced knots. Tis pper is orgnized s follows. In section 2, some new results on te cubic splines re given. Section 3 is devoted to min results for definition of te procedure. A brief review of te CESTAC D. K. Slkuye Fculty of Mtemticl Sciences, University of Guiln, P.O. Box 1914, Rst, Irn Tel.: Fx: E-mil: kojste@guiln.c.ir, slkuye@gmil.com

2 2 Dvod Kojste Slkuye metod, te DSA nd te CADNA librry is given in Section 4. In section 5, some numericl experiments re given. Section 6 is devoted to some concluding remrks. 2 Some results on te cubic splines Let f be rel-vlued function defined on [, b] nd T n = { = x 0 < x 1 < x 2 <... < x n 1 < x n = b} be prtition of [, b] wit eqully spced knots nd stepsize. Let S n be te clmped interpolting cubic spline function on te prtition T n stisfying S n (x i ) = f(x i ) = f i, i = 0, 1,..., n, nd S n(x 0 ) = f (x 0 ), S n(x n ) = f (x n ). (1) It cn be seen tt (see [15,9]) were S n (x) = α j + β j (x x j ) + γ j (x x j ) 2 + δ j (x x j ) 3, x [x j, x j+1 ], (2) α j = f j, in wic M j s stisfy suc tt β j = f j+1 f j + 2M j + M j+1, γ j = M j 6 2, δ j = M j+1 M j M 0 d 0 M 1 d 1 M 2 = d 2, (3).. M n d n, d j = 6 2 (f j+1 2f j + f j 1 ), j = 1, 2,..., n 1, d 0 = 6 (f 1 f 0 f (x 0 )), d n = 6 (f (x n ) f n f n 1 ). It is well-known tt, if f C 2 [, b], ten S n(x) tends to f (x) s tends to zero for ll x [, b], nd E n := (f (x) n(x)) 2 dx = f (x) 2 dx n(x) 2 dx 0, (4) were equlity olds if nd only if f = S n (see [8,13,15]). Terefore, te evlution of E n would be very useful, becuse it my elp us to stte stopping criterion for coosing suitble n. Hence, we re going to investigte ow tis difference depends on te stepsize. We begin wit te following teorem. Teorem 1 Let S n be te clmped cubic spline interpolting f. Ten Proof From (2), one cn see tt n(x) 2 dx = n 1 3 (M n(x) = M j x j+1 x j=1 n 1 Mj 2 + Mn 2 + j=0 M j M j+1 ). x x j + M j+1, x [x j, x j+1 ]. By tis reltion it is esy to see tt xj+1 x j n(x) 2 dx = 3 (M 2 j + M j M j+1 + M 2 j+1).

3 Stepsize Control for Cubic Spline Interpoltion 3 Terefore n(x) 2 dx = n 1 xj+1 i=0 x j n(x) 2 dx = n 1 3 (M n Mj 2 + Mn 2 + M j M j+1 ). j=1 j=0 Tis completes te proof. Teorem 2 Let f be sufficiently smoot function nd S n be te clmped cubic spline interpolting f. Ten f (x) 2 dx = S n(x) 2 dx + b 720 (f (4) (c)) O( 6 ), (5) for some c in [, b]. Proof It is esy to see tt (see [15, pge 140]) (f (x) n(x)) 2 dx = (f(x) S n (x))f (4) (x)dx. (6) For te ske of simplicity, let g(x) = (f(x) S(x))f (4) (x). By te Euler-Mclurin formul [15, pge 156] we ve xi+1 x i g(x)dx = 2 (g(x i) + g(x i+1 )) 2 12 (g (x i+1 ) g (x i )) (g(3) (x i+1 ) g(3) (x + i )) (g(5) (x i+1 ) g(5) (x + i )) + O(8 ). (7) Obviously, g(x i ) = g(x i+1 ) = 0. We now recll tt (see [16, pge 121] nd [9, pge 487]) f (x j ) = S n(x j ) f (5) (x j ) + O( 6 ), j = i, i + 1, (8) f (x j ) = n(x j ) f (4) (x j ) + O( 4 ), j = i, i + 1, (9) f (3) (x i ) = S n (3) (x + i ) 2 f (4) (x i ) 2 12 f (5) (x i ) + O( 4 ), (10) f (3) (x i+1 ) = S n (3) (x i+1 ) + 2 f (4) (x i+1 ) 2 12 f (5) (x i+1 ) + O( 4 ). (11) Differentiting g nd using Eq. (8) we get g (x i+1 ) g (x i ) = (f (5) (x i+1 )f (4) (x i+1 ) f (5) (x i )f (4) (x i )) 4 + O( 6 ). (12) On te oter nd, by tree times differentiting of g nd using Eqs. (8), (9) nd (10), we obtin g (3) (x + i ) = (f (3) S (3) n )(x + i )f (4) (x i ) + 3(f n)(x + i )f (5) (x i ) +3(f S n)(x + i )f (6) (x i ) + (f S n )(x + i )f (7) (x i ) = 2 (f (4) (x i )) f (5) (x i )f (4) (x i ) + O( 4 ). (13) In te sme mnner, from Eqs. (8), (9) nd (11), we deduce g (3) (x i+1 ) = 2 (f (4) (x i+1 )) f (5) (x i+1 )f (4) (x i+1 ) + O( 4 ). (14) Terefore, from Eqs. (13) nd (14), we see tt

4 4 Dvod Kojste Slkuye g (3) (x i+1 ) g(3) (x + i ) = 2 ((f (4) (x i )) 2 + (f (4) (x i+1 )) 2 ) (f (5) (x i+1 )f (4) (x i+1 ) f (5) (x i )f (4) (x i )) + O( 4 ) = (f (4) (ξ i )) (f (5) (x i+1 )f (4) (x i+1 ) f (5) (x i )f (4) (x i )) + O( 4 ), for some ξ i [x i, x i+1 ]. It is mentioned tt te ltter equlity s been obtined by te intermedite vlue teorem. Substituting (12) nd (15) in Eq.(11), yields xi+1 x i g(x)dx = (f (4) (ξ i )) (f (5) (x i+1 )f (4) (x i+1 ) f (5) (x i )f (4) (x i )) (g(5) (x i+1 ) g(5) (x + i )) + O(8 ). = (f (4) (ξ i )) (f (5) (x i+1 )f (4) (x i+1 ) f (5) (x i )f (4) (x i )) g(6) (η i ) + O( 8 ), for some η i (x i, x i+1 ). Here, it is noted tt te ltter equlity is obtined by te men vlue teorem. Tis togeter wit (6) results in (f (x) n(x)) 2 dx = n 1 xi+1 i=0 x i (f(x) S n (x))f (4) (x)dx = 1 n (f (4) (ξ i )) (f (5) (b)f (4) (b) f (5) ()f (4) ()) 6 + O( 6 ) i=0 = b 720 (f (4) (c)) O( 6 ), for some c in [, b] nd te proof is completed. Tis teorem sows tt S n(x) 2 dx pproximtes f (x) 2 dx nd te pproximtion error is of order 4. In te next section, we present teorem wic elps us to propose procedure to control te stepsize for determining interpolting cubic spline functions. 3 Definition of te procedure We first recll te following definition [4, 10]. Definition 1 Let r nd s be two rel numbers. Te number of exct significnt digits tt re common to r nd s cn be defined in (, + ) by 1. for r s, 2. C r,r = +. C r,s = log 10 r + s 2(r s).

5 Stepsize Control for Cubic Spline Interpoltion 5 Teorem 3 (Jézéquel [10]). Let I() be n pproximtion of order p of n exct vlue, i.e., I I() = C p + O( q ) wit 1 p < q nd C R. If I m is n pproximtion computed wit te stepsize 0 /2 m, ten C Im,I m+1 = C Im,I + log 10 ( 2 p 1 ) + O(2m(p q) ). Teorem 4 Let f be sufficiently smoot function nd S m be te clmped cubic spline interpolting f wit stepsize (b )/2 m. Moreover, let Ten Proof Let I = In tis cse, by Teorem 2 we ve 2 p f (x) 2 dx, nd I m = m(x) 2 dx. C Im,I m+1 = C Im,I + log 10 ( ) + O( 1 ). (15) 4m I() = n(x) 2 dx, = b n. I I() = C 4 + O( 6 ), were C = b 720 (f (4) (c)) 2. Now, letting p = 4, q = 6 nd 0 = b, from Teorem 3 te desired reltion is obtined. Tis teorem sows tt, if te convergence zone is reced, i.e., if te term O( 1 4 ) becomes m negligible, ten te significnt digits common to two successive pproximtions I m nd I m+1 re lso in common wit I, up to one. Note tt te term log 10 ( ) corresponds t most to one bit. Tese teoreticl results ve been obtined by tking into ccount only te trunction error on two successive pproximtions I m nd I m+1. However computed results re lso ffected by roundoff error propgtion. Next, we describe ow round-off errors cn be estimted wit probbilistic pproc in order to determine te exct significnt digits of ny computed result. 4 Te CESTAC metod, te DSA nd te CADNA librry Te Discrete Stocstic Aritmetic (DSA) [18] is probbilistic pproc for nlyzing te round-off error propgtion wic is bsed on te syncronous implementtion of te CESTAC metod nd te stocstic order reltions. In te CESTAC metod te floting-point ritmetic is replced by rndom ritmetic [5,7,18,20]. In fct, computer progrm is run N times in prllel nd in tis wy new ritmetic clled stocstic ritmetic is defined (for more detils see [6]). Te min ide is to use rndom rounding to produce severl smples of ec result of ny ritmetic opertion nd te number of common digits in tese smples estimtes te number of exctly significnt digits in te floting-point result. From tis point of view, Vignes [21] introduced te concept of computtionl zero wic is denoted A computtionl zero is computed result wic s no significnt digit or wic is te mtemticl zero. Te CADNA softwre [2,3,17] is librry wic implements utomticlly te DSA in ny code written in Fortrn. Using te CADNA librry, ec stndrd FP types ve teir corresponding stocstic types. Every intrinsic function nd opertor re overloded for tose types. Wen stocstic vrible is printed, only its significnt digits re displyed to point out its ccurcy. If number s no significnt digit (i.e., computtionl zero), te is displyed. In te next section, we describe ow te use of te CADNA librry llows us to control te stepsize for te clmped interpolting cubic spline functions.

6 6 Dvod Kojste Slkuye Tble 1 Results for Exmple 1. m I m I m I m E E E E E E E E E E E E E E E E E E E E E E E E E E E Numericl exmples According to te previous section nd Teorem 4, by using te CESTAC metod nd te CADNA librry, we propose to use I m I m 1 s stopping criterion. Te use of tis stopping criterion llows us to estimte te optiml stepsize s soon s te difference between I m nd I m 1 is equl to te computtionl zero. In tis cse, by te teoreticl results presented in Section 3, te common significnt digits between I m nd I m 1 re lso te common significnt digits between I m nd I, up to one digit. By te proposed metod, te best pproximtion of I provided by te computer is cosen. In tis cse, S m is te best pproximtion of f provided by te computer. It is necessry to mention tt in ll te presented exmples I m s ve been computed by Teorem 1. Let us now present te exmples nd te results wic we obtined by te Fortrn code of te clmped interpolting cubic spline functions combined wit te CADNA librry, version BETA [17]. All te numericl experiments were computed in double precision. Te dt st function of te CADNA librry s been used to tke into ccount te ssignment error of some dt of te exmples wic re not integer numbers nd cnnot be exctly coded on computer. Exmple 1 Consider te Runge s function f(x) = x 2 on [ 3, 3]. Numericl results re given in Tble 1. As we observe I 15 I 14 Terefore te optiml stepsize is estimted s = 6/2 15 = E 004. To sow te effectiveness of te proposed metod te vlues of S 15 (x) nd f(x) for midpoint of ten subintervls (denoted by x j ) of [ 3, 3], for te obtined optiml stepsize, re given in Tble 2. In column 1 of tis tble te index of te subintervl is given. Here we mention tt te vlues of f re computed by MATLAB code wit 20 digits of ccurcy nd re presented by 15 digits wit rounding. As we see, ll of te digits of S 15 ( x j ), up to one, coincide wit tt of te f( x j ) for ll j. Exmple 2 In tis exmple, we consider f(x) = 3 + x + 5 sin x on [0, 2]. Numericl results re given in Tble 3. Here, we ve I 10 I 9 Terefore te optiml stepsize is estimted s = 2/2 10 = E 003. Similr to Exmple 1, te vlues of S 10 (x) nd f(x) for midpoint of ten subintervls of [0, 2] for te obtined optiml stepsize re given in Tble 4. Oter ssumptions re s before. As we see, ll of te digits of S 10 ( x j ), up to one, coincide wit tt of te f( x j ). Exmple 3 Tis exmple is devoted to te function f(x) = tn x on te intervl [0, 1]. Numericl results re given in Tble 5. In tis exmple I 11 I 10 Terefore te optiml stepsize is

7 Stepsize Control for Cubic Spline Interpoltion 7 Tble 2 Comprison between exct nd numericl results for Exmple 1 t some points. j x j = (x j + x j+1 )/2 f( x j ) S 15 ( x j ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-002 Tble 3 Results for Exmple 2. m I m I m I m E E E E E E E E E E E E E E E E E Tble 4 Comprison between exct nd numericl results for Exmple 2 t some points. j x j = (x j + x j+1 )/2 f( x j ) S 15 ( x j ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+001 estimted s = 1/2 11 = E 004. Te vlues of S 11 (x) nd f(x) for some points re given in Tble 6. Oter ssumptions re similr to previous exmples. As we see, ll of te digits of S 11 ( x j ), up to one, coincide wit tt of f( x j ). 6 Conclusion In tis pper, teorem s been stted to provide stopping criterion to control te stepsize in te clmped interpolting cubic spline functions. We observed tt te use of te CESTAC metod nd te CADNA librry llows us to estimte te optiml stepsize. Numericl exmples sow tt te proposed metod is effective. It is esy to sow tt te results of tis pper old for oter type of splines. Acknowledgements Te utor would like to tnk to te nonymous referee for is/er comments. Tis work is prtilly supported by University of Guiln.

8 8 Dvod Kojste Slkuye Tble 5 Results for Exmple 3. m I m I m I m E E E E E E E E E E E E E E E E E E E Tble 6 Comprison between exct nd numericl results for Exmple 3 t some points. j x j = (x j + x j+1 )/2 f( x j ) S 15 ( x j ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+001 References 1. S. Abbsbndy nd M. A. Friborzi Argi, A stocstic sceme for solving definite integrls, Appl. Numer. Mt., 55 (2005) J. M. Cesneux, L ritmétique stocstique et le logiciel CADNA, Hbilittion à diriger des recerces, Université Pierre et Mrie Curie, Pris, J. M. Cesneux, CADNA: An ADA tool for round-off errors nlysis nd for numericl debugging, In ADA in Aerospce, Brcelone, Spin, December J. M. Cesneux nd F. Jézéquel, Dynmicl Control of Computtions Using te Trpezodil nd Simpson s rules, J. Universl Comput. Sci., 4 (1998) J. M. Cesneux, Study of te computing ccurcy by using probbilistic pproc, in: C. Ullric (Ed.), Contribution to Computer Aritmetic nd Self-Vlidting Numericl Metods, IMACS, New Brunswick, NJ, J. M. Cesneux, Stocstic ritmetic properties, IMACS Comput. Appl. Mt. (1992) J. M. Cesneux nd J. Vignes, Les fondements de l ritmétique stocstique, C. R. Acd. Sci. Pris, Sér. I Mt., 315 (1992) C. de Boor, A prcticl guide to splines, Revised edition, Springer-Verlg F. B. Hildebrnd, Introduction to numericl nlysis, second edition, Dover Publictions, Inc., New York, F. Jézéquel, A dynmicl strtegy for pproximtion metods, C. R. Mécnique, 334 (2006) F. Jézéquel nd J. M. Cesneux, CADNA: librry for estimting round-off error propgtion, Computer Pysics Communictions, 178 (2008) F. Jézéquel, Contrôle dynmique de métodes d pproximtion, Hbilittion á diriger des recerces, Université Pierre et Mrie Curie, Pris, Februry 2005, Avilble t: ttp://www-np.lip6.fr/ jezequel/hdr.tml. 13. A. Qurteroni, R. Scco nd F. Sleri, Numericl mtemtics, Springer-Verlg, New York, D. K. Slkuye, F. Toutounin nd H. S. Yzdi, A procedure wit stepsize control for solving n one-dimensionl IVPs, Mtemtics nd Computers in Simultion, 79 (2008) J. Stoer nd R. Bulirsc, Introduction to Numericl Anlysis, Springer-Verlg, Tird Edition, S. S. Sstry, Introductory metods of numericl nlysis, Fourt edition, Prentice-Hll of Indi, Te CADNA librry, URL ddress: ttp:// 18. J. Vignes, Discrete stocstic ritmetic for vlidting results of numericl softwre, Numericl Algoritms, 37 (2004) J. Vignes, New metods for evluting te vlidity of te results of mtemticl computtions, Mt. Comp. Simul., 20 (1978) J. Vignes, A stocstic ritmetic for relible scientific computtion, Mt. Comp. Simul., 35 (1993) J. Vignes, Zéro mtémtique et zéro informtique, C. R. Acd. Sci. Pris Sér. I Mt., 303 (1986)