Journal of Mathematical Extension Vol. 7, No. 4, (2013), 63-69 A Concise Formula for Two Dimensional Divided Differences with Multiple Knots R. Roozegar∗ Yazd University A. A. Jafari Yazd University Abstract. The well-known formulas for divided differences in the case of one and two dimensional with multiple knots have played an important role in applied mathematics, particularly in numerical analysis, polynomial and lagrange interpolations. In this article we obtain a new formula for two dimensional divided differences in the case of multiple knots. This formula has a simpler form than the known formula given in literature. It is friendly using for computations and analysis. AMS Subject Classification: 41A63; 65D05 Keywords and Phrases: Divided difference, multiple knots, one and two dimensional divided differences 1. Introduction The divided difference has appeared to be a powerful tool in certain areas of applied mathematics, as numerical analysis, calculus of finite differences, interpolation theory and probability theory. Two good sources are the texts by Gelfond [2] and Milne-Thomson [4]. Let f : R → C be a differentiable function, where C is the complex number. Assume that s0 , s1 , ..., sm are distinct real numbers subject to s0 < s1 < · · · < sm ; such points are known as knots. The divided differences of f are recursively given by the following formula,[3] and [7]: [f ; s0 ] = f (s0 ), ∗ Received: June 2013; Accepted: December 2013 Corresponding author 63 64 R. ROOZEGAR AND A. A. JAFARI [f ; s1 , . . . , sm ] − [f ; s0 , s1 , . . . , sm−1 ] if sm 6= s0 . sm − s0 From the recursive formulas, it is clear that if s = s0 , then [f ; s0 , s1 , . . . , sm ] = [f ; s, s0 ] = f (s) − f (s0 ) . s − s0 If repetitions are permitted in the arguments and the function f is smooth enough, then lim [f ; s, s0 ] s→s0 = lim s→s0 f (s) − f (s0 ) s − s0 0 = f (s0 ). This gives the definition of first-order divided difference with repeated points, 0 [f ; s0 , s0 ] = f (s0 ). In general, let s0 6 s1 6 · · · 6 sm ; the divided differences with repeated points obey the following recursive formula: ( [f ;s ,··· ,s ]−[f ;s ,s ,··· ,s 1 m 0 1 m−1 ] , if sm 6= s0, sm −s0 (1) [f ; s0 , s1 , · · · , sm ] = (m) f (s0 ) , if sm = s0. m! When s0 , s1 , · · · , sm are all distinct, it follows inductively from (1) that [f ; s0 , s1 , · · · , sm ] = m X f (sk ) , v (1) (sk ) k=0 (2) Qm d where v(s) = j=0 (s − sj ) and v (1) (s) = ds v(s), [5]. We recall that (2) is the m-th order divided differences of f on the knots s0 , s1 , ..., sm . Let us introduce the following notations. Let N = {1, 2, . . .}, N0 = N ∪ {0}, m, n ∈ N0 , and let k0 , k1 , . . . , km , l0 , l1 , . . . , ln ∈ N so that k0 + k1 + · · · + km = M +1 and l0 +l1 +· · ·+ln = N +1. Also let α = max{k0 −1, k1 −1, . . . , km −1} and β = max{l0 − 1, l1 − 1, . . . , ln − 1}; I, J ⊆ R be intervals, I × J be a bidimensional interval and Dα,β (I × J) be the set of all real valued bivariate ∂f i+j functions f with the property that exists ∂s i ∂tj on I ×J, where i ∈ {0, 1, . . . , α} and j ∈ {0, 1, . . . , β}. In the following, the one dimensional divided differences with multiple knots is (k ) (k ) (k ) denoted by [f ; s0 0 , s1 1 , . . . , smm ] where s0 , s1 , . . . , sm are distinct and (kj ) sj = sj , sj , . . . , sj . | {z } kj times AACONCISE FORMULA FORFOR TWOTWO DIMENSIONAL ... CONCISE FORMULA DIMENSIONAL ... 65 A CONCISE FORMULA FOR TWO DIMENSIONAL ... A CONCISE FORMULA FOR TWO DIMENSIONAL ... 65 65 65 We for] for distinct knotsknots only. only. We use use[f; [fs;;0ss,0s,,1ss, 1. .,, ... ,..s..m,, ss] m distinct We use [f knots only. 0which 1 are available m ] for distinct The formulas in the literature for onefor dimensional divided divided The formulas which are available in the the literature one dimensional dimensional We use [f ; s0 , which s1 , . . . ,are sm ]available for distinct knots only. The formulas in literature for complicated. one differences with multiple knots are computationally rather It is divided differences with multiple knots are are computationally rather complicated. It is is The formulas which are available in computationally the literature forrather one dimensional divided differences with multiple knots complicated. It the ratio ofoftwo determinants, namely, the ratio two determinants, namely, differences with multiple knots are computationally rather complicated. It is the ratio of two determinants, namely, the ratio of two determinants, namely, (k0 ) (k (k ) (k10)), . . (k (km ) , s1(k . , 1s)mm ) s(k (k (k ) )(k1 )(k ) (km )(k ) (W f)(s (W0 ff )(s )(s s(k ) mm )(3) 0 0 ) ,, s 1 1 ) ,, .. .. .. ,, s m [f(s, t);t); s0 s0(k ,00s)1, s(k, .11.)., ,.s.m ](ks m =) ]s = (W m [f (s, (3) . , s (k ) (k ) (km )) , 1 (k 1m) (k0 ) (k m ] = (W (k100)), 0. .,(k (k ) ), [f (s, t); s00(k0 ), s11(k1 ), . . . , sm (3) 1s) m s , . . . , s m ,(ss)(s . , ) m (km ) s V (s0 V f 0 1 s(k s(k mm ) ) , 0 0 ) ,, s 1 1 ) ,, .. .. .. ,, s [f (s, t); s0 , s1 , . . . , sm ]s = V (s1(k (3) m (k0 ) 1 (k1 ) (km )) 0 V (s , s , . . . , s ) m 0 1 where where (k0 ) (k(k)1 ) (k ) (km ) where (km ) (4) (W(W f)(s )(s ,(k s 0 ,s.(k. 1.1 ), ,s.m. . ,)s(k ) (4) where mm ) ) 0 10 ) ,, s 1 (W ff0)(s (4) (k0 ) 1 (k1 ), . . . , sm 0 (km ) (4) (W f )(s0 , s1 , . . . , sm ) 1 s 2 2 M −1 · · ·· · · sM−1 f(s0 ) f (s0 ) 0s0 s0 s2 0 1 s M −1 0 0 s0 · · · s0M −1M −2 f (s0 ) 1 s M−2 · · ·· · · (M −(M 1)s− f´(s0 ) f´´ 001 1s100 2s2s 0s2 0s01)sM (s0 )) 00 0 −2 0 1 2s · · · (M − 1)s f (s 0 0 M −2 ´ 0 · ·· 0·· · · ···1· · 2s · · ·· · · · · · −· 1)s ··· · · · (M f 0 0 ··· ·(s ·· ··0 ) M−k0 M −k (k00−1) 0··0·· ·· 0··0·· ·· 0 0 · · ··· ·· ··(M (M (k −1) − 1)(M − 2).− . . 2). . .(M f−k (sf0(k ) 00 −1) ·− ·.(M ·k0 +−1)s · · · (s0 ) 0 + 1)sM − 1)(M . . . k 0 0 − k0 + 1)s00M −k0· · · f (k0 0 · · ·· · · (M − 1)(M − ·2). 0) · ··00·· · · ···00· · −1) = ,· (s · · .. ..·..·.(M = · · · · · · (M − 1)(M − 2). .(M − k + 1)s f (s0 ) 0 0 0 = · · · · · · · · · · · · · · · 2 M−1 1·1· · sms· · · sms2 · · ·· · · M −1 sm sM f(sm ) f (s = · ·−1 · · ·m · )) m 1 sm m m s2m ··· sm M−2 ´ m 2 M −1 M −2 0 1 2s · · · (M − 1)s f (sm ) ff´´(s m m 1 s s s 0 1 2s · · · (M − 1)s (s ) M −2 m m m m m m (M 1)sm f´(s m) m· · ·· · · M −2 · ··00·· · · ···11· · 2s ··· − ··· · · · · · · · · · 2s (M − 1)s f (s m m ··· ··· · · k· m+ 1)sM−km Mf −k ··· ) (km −1) (k −1) 0·0· · 0·0· · 0 0 · · ··· ·· ·(M − 1)(M − 2).− . . 2). . .(M − (s ) m m··· m · .(M · · m − kmm+ 1)sM (M − 1)(M . . . f (sm )) · −k (k −1) m m f (km (s 0 0 0 · · · (M − 1)(M − 2). . . . .(M − km + 1)sm m −km m −1) f (sm ) 0 0 0 · · · (M − 1)(M − 2). . . . .(M − km + 1)sM m and and (k0 ) (k(k)1 ) (k ) (km ) and m) V (sV0 (s ,(ks 0 ) , (k · · 1· ), sm ) (km (5) ) (5) and m )) V (s00(k100 ),, ss11(k11 ),, ·· ·· ·· ,, ss(k (5) m (km ) V (s0 , s1 , · · · , sm ) (5) M sM 1 1 s0s0 s20 s220 · · · · · · 0 sM 0 s · · · s 1 s M−1 0 0 1 1 s 2s0 s02 · · · M −1 Ms0 M s0M 10 2s 2s00 ·· ·· ·· 00 0 −1 1 M s 0 · · · · · · −1 · · · M s· 0M ·· 0·· ·· ·· 1·· ·· 2s0· · · ·· ·· ·· · ·0· ·· M−k0 +1M −k 0 +1 0 1).−. . 1). . .(M k0· + ·− · 2)s M− (M . .. ..−.(M .(M k00 + + 2)s 2)sM ·00· · 0 ·00· · 0 00 · · · ·· ·· ··M(M 0 +1 −k 0 M (M − 1). . − k ,0 +1 0 −k · ···0· · · · ·· 0· · , == · ·· − · k0 + 2)s0M 0 · · · ·· ·· ·· M (M − 1). .· .· .· .(M , 0 = · · · · · · · · M 1 sm s2 2 · · · , M sm s· M = ·1· · s· m · · mss2m ·· ·· ·· ·· m 1 s s M−1 m 0 m m M −1 2 Msm M sM −1 2s 001 1s11m 2sm smm· · · ·· ·· ·· m M s 2s m m M −1 · · · M s· · · · ···0· · · · ·· 1· · 2sm· · · · · · m· ··· ··· · · · · · M−km +1 M −k +1 − 1).−. .1). . .(M −.(M km· + 2)s 0·0· · 0 ·0· · 0 0 · · · · ·M(M m m · · · M (M . . . − k + 2)s m + 2)sM m −km +1 0 0 0 · · · M (M − 1). . . . .(M − km M −km +1 0 0 0 · · · M (M − 1). . . . .(M − km + 2)sm m is determinant of Vandermonde, [5]. [5]. is the thegeneralized generalized determinant of Vandermonde, Vandermonde, is the generalized determinant of [5]. Similarly, the order divided differences of theoffunction f(s, .) :f (s, .) : Similarly, theN-th N -th -th order divided differences the is the generalized determinant of Vandermonde, [5].function the N order differences .) : l the function ln 1t n is defined JSimilarly, → R, s s∈∈ I,I,with respect todivided the knots tl00 , tl11tll,00.,. t.llof ,of by ff (s, n J → R, with respect to the knots , . . . , t is defined defined by: Similarly, the N -th order divided differences the function (s, by .) J → R, s ∈ I, with respect to the knots t00l0 , t11l11 , . . . , tnlnlnn is J → R, s∈ I, with respect to the knots t , t , . . . , t is defined (W f)0(t(l10 ) , t(l(l1 )0,).n. .(l, 1t)(ln ) ) (lnby ) n (l0 )(l (l ) 1 (l ) (W0ff )) (t (t1(l n nn) ) (l (ln=) 0 0 ) ,, tt(l 1 1 ) ,, .. .. .., ,, tt(l f(s, t);t); t0 t(l, 00t)1) , t(l , .11.)).,,.t.n. , t(l (W n (s, (l0 ) 1 (l (ln )) , 0 ff (s, 1n)) (l (l ) n ) = (l0 ) 1 n 0 1 (l (l t); t (l ), t (l ), . . . , t t n ) t = (t0,0). . .(l,,11tt)1)n, . .,).. ,. t.(l , ntn) ) , t)1(l V (t(WV,f(t nn) ) t = 0V 1 , . . . , tn 0 0 ) ,, tt(l , f (s, t); t00 0 , t11 1 , . . . , tn(l (t n (l )) (l ) (l ) 1 0 1 0 t V (t0 , t1 , . . . , tn n ) , , , 66 R. ROOZEGAR AND A. A. JAFARI (l ) (l ) (l ) (l ) (l ) (l ) where (W f ) (t0 0 , t1 1 , . . . , tn n ) and V (t0 0 , t1 1 , . . . , tn n ) are defined as above. Soltani and Roozegar [6] give a new and compact formula for one dimensional divided differences with multiple knots. They use this formula to find the distribution of randomly weighted averages. This formula comes as a theorem in next section. In this article we present a concise and compact formula for two dimensional divided differences with multiple knots. The advantages of our formula are in its formulation and computation. It can be readily used in calculus of divided differences, Hermite interpolation polynomial and bivariate Lagrange interpolation polynomials. As an illustration, we derive two dimensional divided 1 . differences for the fw (s, t) = (w−s)(w−t) 2. Main Results In this section we give our formula for two dimensional divided differences with multiple knots, Theorem 2.2. First we present a compact formula for one dimensional divided differences, [6]. Theorem 2.1. Let f ∈ Dα (I), where Dα (I) is the set of all real functions f , α-times differentiable on I. Then (k0 ) (k1 ) [f ; s0 , s1 We let Qm (k0 ) m) , . . . , s(k ] m i=0, i6=j (si = (W f )(s0 (k ) (k1 ) , s1 (k ) (k ) , . . . , smm ) (k ) V (s0 0 , s1 1 , . . . , smm ) m X (−1)M +kj −1 dkj −1 f (sj ) = ( Qm ). k −1 ki j (kj − 1)! ds i=0, i6=j (si − sj ) j=0 j − sj )ki = 1 when m = 0. Now we notice to the following theorem which states the main theorem. The main difference between the following formula and the result obtained by Pop and Barbosu [5] is in its formulation, computation and application. No closed form expression for two dimensional divided differences in the case of multiple knots expressed in [5]. That formula is based on some long and complicated determinants and is used numerically in application. For instance, the formula in [5] is not applicable to find the distribution of two dimensional randomly weighted averages in [6], but the following result is a user-friendly formula for applicability and analysis. Theorem 2.2. Let f ∈ Dα,β (I × J). Then the following equalities i i hh (k ) (k ) (l ) (l ) n) m) ; t0 0 , t1 1 , . . . , t(l f (s, t); s0 0 , s1 1 , . . . , s(k m n s t 67 A CONCISE FORMULA FOR TWO DIMENSIONAL ... hh = (l ) (l ) n) f (s, t); t0 0 , t1 1 , . . . , t(l n i (k0 ) t m X n X (−1)M +N +ki +lj −2 = ; s0 (k1 ) , s1 m) , . . . , s(k m i s ∂ ki +lj −2 (ki − 1)!(lj − 1)! ∂ski −1 ∂tlj −1 j i i=0 j=0 f (si , tj ) Qn Qm kq lr r=0,r6=j (tr − tj ) q=0,q6=i (sq − si ) ! . Qm Qn hold. We let q=0,q6=i (sq − si )kq = 1 when m = 0 and r=0,r6=j (tr − tj )lr = 1 when n = 0. Proof. For m = n = 0, k0 = M + 1 and l0 = N + 1, it follows from the Definition 2.1. that 1 ∂M f (s0 , t), f (s, t); s0 , s0 , . . . , s0 = | {z } M ! ∂sM 0 (M +1) times s and 1 ∂N f (s, t0 ). f (s, t); t0 , t0 , . . . , t0 = | {z } N ! ∂tN 0 (N +1) times t Therefore, hh (k0 ) f (s, t); s0 i (l ) s ; t0 0 1 ∂M (l0 ) f (s , t); t 0 0 M ! ∂sM 0 t M 1 ∂ (l0 ) f (s0 , t); t0 M ! ∂sM 0 t i = t = = 1 ∂ M +N f (s0 , t0 ). N M !N ! ∂sM 0 ∂t0 For general m, n, by Theorem 2.1. we have hh (k0 ) f (s, t); s0 (k1 ) , s1 m) , . . . , s(k m i (l ) s (l ) n) ; t0 0 , t1 1 , . . . , t(l n i t 68 R. ROOZEGAR AND A. A. JAFARI " # m X f (si , t) (−1)M +ki −1 ∂ ki −1 (l0 ) (l1 ) (ln ) ( Qm = ); t0 , t1 , . . . , tn kq (ki − 1)! ∂ski i −1 q=0,q6=i (sq − si ) i=0 #t " m X (−1)M +ki −1 ∂ ki −1 f (si , t) (l ) (l ) ); t0 0 , t1 1 , . . . , tn(ln ) = ( Qm ki −1 kq (k − 1)! (s − s ) ∂s i q i q=0,q6=i i i=0 t ki −1 f (si ,tj ) ∂ m n (Q kq ) X (s −s ) (−1)M +ki −1 X (−1)N +lj −1 ∂ lj −1 ∂ski i −1 m q i q=0,q6=i = ) , ( Qn lj −1 lr (k − 1)! (l − 1)! (t − t ) i j r j ∂t r=0,r6 = j i=0 j=0 j which gives the result. Definition 2.3. The (m, n)-th order divided difference of the function f , where f ∈ Dα,β (I × J), with respect to the distinct knots (si , tj ) ∈ I × J, i ∈ {0, 1, . . . , m} and j ∈ {0, 1, . . . , n} is defined by # " (k ) (k ) (k ) s0 0 , s1 1 , . . . , smm f; (l ) (l ) (l ) t0 0 , t 1 1 , . . . , t n n = m X n X (−1)M +N +lj +ki −2 i=0 j=0 ∂ ki +lj −2 (ki − 1)!(lj − 1)! ∂ski −1 ∂tlj −1 i j f (si , tj ) Qn ). kq lr (s − s q i) q=0,q6=i r=0,r6=j (tr − tj ) ( Qm The following corollary has given by [1]. Corollary 2.4. If f (x, y) = g(x)h(y), (∀)(x, y) ∈ [a, b]×[c, d] where g : [a, b] → R, h : [c, d] → R, the following equality x , x , . . . , xm f; 0 1 = [g; x0 , x1 , . . . , xm ][h; y0 , y1 , . . . , yn ], y0 , y 1 , . . . , yn holds. Now, we present an example using the Theorem 2.2, Corollary 2.1 and the work of Soltani and Roozegar [6]. 1 and for given real numbers s0 , s1 , . . . , sm , Example 2.5. Let fw (s, t) = (w−s)(w−t) t0 , t1 , . . . , tn , w and integers ki > 1, i = 0, 1, . . . , m; li > 1, i = 0, 1, . . . , n then # " m n (k ) (k ) (k ) Y Y 1 1 s0 0 , s1 1 , . . . , smm = f; . (ln ) (l0 ) (l1 ) kj (w − s ) (w − ti )li t0 , t 1 , . . . , tn j j=0 i=0 A CONCISE FORMULA FOR TWO DIMENSIONAL ... 69 Acknowledgement The authors express their sincere thanks to the referees for valuable comments and to the Editor-in-Chief and Manager for his time and kind efforts. The authors are also thankful to the Yazd University for supporting this research. References [1] D. Barbosu, Two dimensional divided differences revisited, Creative Math. & Inf., 17 (2008), 1-7. [2] A. O. Gelfond, Calculus of Finite Differences, Hindustan publishing corporation, Delhi-7, India, 1971. [3] S. Karlin, C. A. Micchelli, and Y. Rinott, Multivariate splines: A probabilistic perspective. J. Multivariate Anal., 20(1) (1986), 69-90. [4] L. M. Milne-Thomson, The Calculus of Finite Differences, Mcmillan, New York, 1965. [5] O. T. Pop and D. Barbosu, Two dimensional divided differences with multiple knots, An. Şt. Univ. Ovidius Constanţa, 17 (2009), 181-190. [6] A. R. Soltani and R. Roozegar, On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach. Statist. Probab. Lett., 82(5) (2012), 1012-1020. [7] A. Xu and C. Wang, Some extensions of fa‘a di brunos formula with divided differences. Comput. Math. Appl., 59(6) (2010), 2047-2052. Rasool Roozegar Department of Statistics Assistant Professor of Statistics Yazd University P.O. Box 89195-741 Yazd, Iran E-mail: [email protected] Ali Akbar Jafari Department of Statistics Assistant Professor of Statistics Yazd University P.O. Box 89195-741 Yazd, Iran E-mail: [email protected]
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