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On the Link Between Finite Difference and Derivative of Polynomials Kolosov Petro

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On the Link Between Finite Difference and Derivative of Polynomials Kolosov Petro On the link between finite difference and derivative of polynomials Kolosov Petro To cite this version: Kolosov Petro. On the link between finite difference and derivative of polynomials. 2017. hal- 01350976v4 HAL Id: hal-01350976 https://hal.archives-ouvertes.fr/hal-01350976v4 Preprint submitted on 6 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS KOLOSOV PETRO Abstract. The main aim of this paper to establish the relations between for- ward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polyno- mials. Keywords. finite difference, divided difference, high order finite difference, derivative, ode, pde, partial derivative, partial difference, power, power func- tion, polynomial, monomial, power series, high order derivative 2010 Math. Subject Class. 46G05, 30G25, 39-XX e-mail: kolosov [email protected] ORCID: http://orcid.org/0000-0002-6544-8880 Social media links Twitter - Kolosov Petro Youtube - Kolosov Petro Mendeley - Petro Kolosov Academia.edu - Petro Kolosov LinkedIn - Kolosov Petro Google Plus - Kolosov Petro Facebook - Kolosov Petro Vimeo.com - Kolosov Petro VK.com - Math Facts Publications on other resources HAL.fr articles - Most recent updated ArXiV.org articles Archive.org articles ViXrA.org articles Figshare.com articles Datahub.io articles Contents 1. Introduction 2 2. Definitions for xg distribution 3 3. Difference and derivative of power function 4 4. Difference of polynomials 6 5. Relation with Partial derivatives 8 6. Relations between finite differences 10 7. The error of approximation 11 8. Summary 11 9. Conclusion 12 1 2 KOLOSOV PETRO References 12 10. Appendix 1. Difference table up to tenth power 13 1. Introduction Let introduce the basic definition of finite difference. Finite difference is dif- ference between function values with constant increment. There are three types of finite differences: forward, backward and central. Generally, the first order forward difference could be noted as: ∆hf(x) = f(x + h) − f(x) backward, respec- 1 1 tively, is rhf(x) = f(x) − f(x − h) and central δhf(x) = f(x + 2 h) − f(x − 2 h), where h = const, (see [1], [2], [3]). When the increment is enough small, but constant, we can say that finite difference divided by increment tends to deriv- ative, but not equals. The error of this approximation could be counted next: 0 ∆hf(x) h − f (x) = O(h) ! 0, where h - increment, such that, h ! 0. By means of induction as well right for backward difference. More exact approximation we have 0 δhf(x) 2 using central difference, that is: h − f (x) = O(h ), note that function should be twice differentiable. The finite difference is the discrete analog of the derivative (see [4]), the main distinction is constant increment of the function's argument, while difference to be taken. Backward and forward differences are opposite each other. More generally, high order finite differences (forward, backward and central, respectively) could be denoted as (see [7]): n X n (1.1) ∆k f(x) = ∆k−1f(x + h) − ∆k−1f(x) = (−1)k · f(x + (n − k)h) h k k=0 n X n n (1.2) δnf(x) = (−1)k · f x + − k h h k 2 k=0 n X n (1.3) rk f(x) = rk−1f(x) − rk−1f(x − h) = (−1)k · f(x − kh) h k k=0 Let describe the main properties of finite difference operator, they are next (see [5]) (1) Linearity rules ∆(f(x) + g(x)) = ∆f(x) + ∆g(x) δ(f(x) + g(x)) = δf(x) + δg(x) r(f(x) + g(x)) = rf(x) + rg(x) (2) ∆(C · f(x)) = C · ∆f(x); r(C · f(x)) = C · rf(x); δ(C · f(x)) = C · δf(x) (3) Constant rule ∆C = rC = δC = 0 Strictly speaking, divided difference (see [6]) with constant increment is discrete analog of derivative, when finite difference is discrete analog of function's differ- ential. They are close related to each other. To show this, let define the divided difference. Definition 1.4. Divided difference of fixed increment definition (forward, centaral, backward respectively) + f(xj) − f(xi) f [xi; xj] := ; j > i; ∆x ≥ 1 xj − xi ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 3 − f(xi) − f(xj) f [xi; xj] := ; j < i; rx ≥ 1 xi − xj f(x ) − f(x ) f c[x ] := i+m i−m i 2m Hereby, divided diffrence could be represented from the finite difference, let be j = i ± const, backward as −, respectively + as forward and c as centered ∆f(x ) rf(x ) f ±[x ; x ] ≡ j ≡ i i j ∆x rx δf(x ) δf(x ) f [x ] ≡ i±m ≡ i±m c i 2m δx The n-order ∆nf(x ) rnf(x ) f ±[x ; x ]n ≡ j ≡ i i j ∆xn rxn δnf(x ) δnf(x ) f [x ]n ≡ i±m ≡ i±m c i (2m)n δxn Each properties, which holds for finite differences holds for divided differences as well. 2. Definitions for xg distribution Let be variable xg : xg = g · C; C = xg+1 − xg = const;C 2 R>0 ! xg 2 R>0; g 2 Z. To define the finite difference of function of such argument, we take C = h and rewrite forward, backward and central differences of some analytically defined function f(xi) next way: ∆f(xi+1) = f(xi+1) − f(xi); rf(xi−1) = f(xi) − f(xi−1); δf(xi) = f x 1 − f x 1 . The n-th differences of such a function i+ 2 i− 2 could be written as n X n (2.1) ∆nf(x ) = ∆n−1f(x ) − ∆n−1f(x ) = (−1)k · f(x ) i+1 i+1 i k i+n−k k=0 n n X n k (2.2) δ f(xi) = (−1) · f xi+ n −k k 2 k=0 n X n (2.3) rnf(x ) = rn−1f(x ) − rn−1f(x ) = (−1)k · f(x ) i−1 i i−1 k i−n+k k=0 Let be differences ∆f(xi+1); δf(xi); rf(xi−1), such that i 2 Z and differences is taken starting from point i, which divides the space Z into Z = Z− [ Z+ symmetri- cally (note that +=− symbols mean the left and right sides of start point i = 0, i.e backward and forward direction), this way we have (i + 1) 2 Z+; (i − 1) 2 Z−; i 6= 0 2 (Z+; Z−). Let derive some properties of that distribution: (1) max(Z−) = min(Z+) = i (2) Forward difference is taken starting from min(Z+), while backward from max(Z−) + − P P (3) card( ) = card( ), i.e + 1 = − 1 Z Z k2Z k2Z (4) Maximal order of forward difference in which it is not equal to zero is max(Z+) 4 KOLOSOV PETRO (5) Maximal order of backward difference in which it is not equal to zero is min(Z−) (6) Maximal order of central difference in which it is not equal to zero is max(Z+) (7) Forward and backward difference equal each other by absolute value, while to be taken from i = 0 Limitation 2.4. Note that most expression generated as case of i = 0, so the initial start point of each difference and inducted expressions are 0. Definitions 2.5. Generalized definitions complete this section + (1) Z := N1 - positive integers (2) Z− := {−1; −2;:::; min(Z−)g - negative integers n (3) ff; f(x); f(xi)g := x - power function, value of power function in point i of difference table (4) i = 0 - initial point of every differentiating process, δf(x0) exist only for operator of centered difference (as per limitation 2.4) P (5) xi := i·∆x ≡ rx ≡ (δx)=2 = ∆x - value of function's argument in point i of difference table (6)∆ x ≡ rx ≡ (δx)=2 - function's argument differentials, constant values 2 R>0 (7)∆ f(xi+1); rf(xi−1) - forward and backward finite differences in points i + 1 and i − 1 of difference table (8) δf(xi) - centered finite difference in point i of difference table (9)∆ 0f ≡ δ0f ≡ r0f ≡ f 3. Difference and derivative of power function Since the n-order polynomial defined as summation of argument to power mul- tiplied by coefficient, with higher power n, let describe a few properties of finite (divided) difference of power function. Lemma 3.1. For each power function with natural number as exponent holds the equality between forward, backward and central divided differences, and derivative with order respectively to exponent and equals to exponent under factorial sign mul- tiplied by argument differential to power. 0 Proof. Let be function f(x) = xn; n 2 N. The derivative of power function, f (x) = nxn−1, so k-th derivative f (k)(x) = n · (n − 1) ··· (n − k + 1) · xn−k, n > k. Using (m) (n) limit notation, we have: limm!n− f (x) = f (x) = n!: Let rewrite expressions (2.1, 2.2, 2.3) according to definition xi = i · ∆x, note that ∆x ≡ rx ≡ δx=2. By means of power function multiplication property (i · ∆x)n = in · ∆xn, we can rewrite the n-th finite difference equations (2.1, 2.2, 2.3) as follows m X m m (3.2) ∆m(xn ) = (−1)k · i + m − k · ∆xm; m < n 2 i+1 k N k=0 Using limit notation on divided by ∆x to power (3.2), we obtain m n m ∆ (x ) X m m (3.3) lim i+1 = lim (−1)k · i + m − k m!n− ∆xm m!n− k k=0 ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 5 n X n n−0 = (−1)k · i + n − k = n! k k=0 Similarly, going from (2.3), backward n-th difference equals: m n m r (x ) X m m (3.4) lim i−1 = lim (−1)k · i − m + k m!n− rxm m!n− k k=0 n X n n−0 = (−1)k · i − n + k = n! k k=0 And n-th central (2.2), respectively m m m δ f(xi) X m m (3.5) lim = lim (−1)k · i + − k m!n− δxm m!n− k 2 k=0 n X n n n−0 = (−1)k · i + − k = n! k 2 k=0 As we can see the next conformities hold ∆nf ∆nf (3.6) lim ≡ lim ≡ n! ∆x!0 ∆xn ∆x!C ∆xn δnf δnf (3.7) lim ≡ lim ≡ n! δx!0 δxn δx!C δxn rnf rnf (3.8) lim ≡ lim ≡ n! rx!0 rxn rx!C rxn n n n ∆ f δ f r f + (3.9) lim ≡ lim ≡ lim 8(C 2 R ) ∆x!C ∆xn δx!C δxn rx!C rxn In partial case when C = 0 dnf δnf rnf (3.10) ≡ lim ≡ lim dxn δx!0 δxn rx!0 rxn As well holds df rf (3.11) (x0) = lim (x0) dx rx!0 rx n n n n d f ∆ f δ f r f + (3.12) ≡ lim ≡ lim ≡ lim ; 8(C 2 R ) dxn ∆x!C ∆xn δx!C δxn rx!C rxn where f = xn.
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  • Generalized J-Factorial Functions, Polynomials, and Applications
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