Remarks on the Remainder in Certain Formulas Approximating a Divided Difference by Derivatives

Tiberiu Popoviciu
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Tiberiu Popoviciu
Institutul de Calcul

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T. Popoviciu, Remarques sur le reste de certaines formules d’approximation d’une différence divisée par des dérivées, Bul. Inst. Politehn. Iaşi (N.S.), 13(17) (1967) fasc. 3-4, pp. 103-109 (in French)

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1967 b -Popoviciu- Bul. Inst. Politehn. Iasi – Remarques sur le reste de certaines formules d’approximation d’une difference divisee par des derivees

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Buletinul Institutului Politehnic din Iaşi

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published by the “Gheorghe Asachi” Technical University of Iaşi

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1967 b -Popoviciu- Bul. Inst. Politehn. Iasi - Remarks on the rest of certain d_approx formulas
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REMARKS ON THE REMAINDER OF CERTAIN FORMULAS FOR APPROXIMATING A DIFFERENCE DIVIDED BY DERIVATIVES

BYTIBERIU POPOVICIUmember of the Academy of the Socialist Republic of Romania

  1. Let us consider a function f f fffdefined and having a derivative n n nnn-th continuous on the finite and closed interval [ has , b ] , ( has < b ) [ has , b ] , ( has < b ) [a,b],(a < b)[a, b],(a<b)[has,b],(has<b).
Let has = x 1 ≦ x 2 ≦ … ≦ x n ≦ x n − 1 = b , n + 1 has = x 1 ≦ x 2 ≦ … ≦ x n ≦ x n − 1 = b , n + 1 a=x_(1) <= x_(2) <= dots <= x_(n) <= x_(n-1)=b,n+1a=x_{1} \leqq x_{2} \leqq \ldots \leqq x_{n} \leqq x_{n-1}=b, n+1has=x1≦x2≦…≦xn≦xn−1=b,n+1distinct or non-distinct points belonging to the interval [ has , b ] [ has , b ] [a,b][a, b][has,b]such as the extremities has has hashashas, b b bbbare among these points. It follows that n ( > 0 ) n ( > 0 ) n( > 0)n(>0)n(>0)is a natural number. Let us denote by has = x 1 = x 1 ′ < x 2 ′ < … < x p ′ = x n + 1 = b has = x 1 = x 1 ′ < x 2 ′ < … < x p ′ = x n + 1 = b a=x_(1)=x_(1)^(') < x_(2)^(') < dots < x_(p)^(')=x_(n+1)=ba=x_{1}=x_{1}^{\prime}<x_{2}^{\prime}<\ldots<x_{p}^{\prime}=x_{n+1}=bhas=x1=x1′<x2′<…<xp′=xn+1=bthe distinct points among the points x α x α x_(alpha)x_{\alpha}xαand be k 1 , k 2 , … , k p k 1 , k 2 , … , k p k_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}k1,k2,…,kptheir respective multiplicity orders. Therefore, we have 2 ≦ p ≦ n + 1 , k 1 , k 2 , … , k p 2 ≦ p ≦ n + 1 , k 1 , k 2 , … , k p 2 <= p <= n+1,k_(1),k_(2),dots,k_(p)2 \leqq p \leqq n+1, k_{1}, k_{2}, \ldots, k_{p}2≦p≦n+1,k1,k2,…,kpare natural numbers, k 1 + k 2 + … + + k p = n + 1 k 1 + k 2 + … + + k p = n + 1 k_(1)+k_(2)+dots++k_(p)=n+1k_{1}+k_{2}+\ldots+ +k_{p}=n+1k1+k2+…++kp=n+1And max ( k 1 , k 2 , … , k p ) ≦ n max k 1 , k 2 , … , k p ≦ n max(k_(1),k_(2),dots,k_(p)) <= n\max \left(k_{1}, k_{2}, \ldots, k_{p}\right) \leqq nmax(k1,k2,…,kp)≦n.
In accordance with the point system x α , x α ′ x α , x α ′ x_(alpha),x_(alpha)^(')x_{\alpha}, x_{\alpha}^{\prime}xα,xα′, We have
x k 1 + k 2 + … + k α − 1 + β = x α ′ , ( β = 1 , 2 , … , k α ; α = 1 , 2 , … , p ) , x k 1 + k 2 + … + k α − 1 + β = x α ′ , β = 1 , 2 , … , k α ; α = 1 , 2 , … , p , x_(k_(1)+k_(2)+dots+k_(alpha-1)+beta)=x_(alpha)^('),(beta=1,2,dots,k_(alpha);alpha=1,2,dots,p),x_{k_{1}+k_{2}+\ldots+k_{\alpha-1}+\beta}=x_{\alpha}^{\prime},\left(\beta=1,2, \ldots, k_{\alpha} ; \alpha=1,2, \ldots, p\right),xk1+k2+…+kα−1+β=xα′,(β=1,2,…,kα;α=1,2,…,p),
( k 1 + k 2 + ⋯ + k α − 1 k 1 + k 2 + ⋯ + k α − 1 k_(1)+k_(2)+cdots+k_(alpha-1)k_{1}+k_{2}+\cdots+k_{\alpha-1}k1+k2+⋯+kα−1is replaced by 0 if α = 1 α = 1 alpha=1\alpha=1α=1
There exists a continuous function G ( x ) G ( x ) G(x)G(x)G(x), completely determined, such that one has
(1) [ x 1 , x 2 , … , x n + 1 ; f ] = ∫ has b G ( x ) f ( n ) ( x ) d x , (1) x 1 , x 2 , … , x n + 1 ; f = ∫ has b   G ( x ) f ( n ) ( x ) d x , {:(1)[x_(1),x_(2),dots,x_(n+1);f]=int_(a)^(b)G(x)f^((n))(x)dx",":}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1}; f\right]=\int_{a}^{b} G(x) f^{(n)}(x) \mathrm{d} x, \tag{1} \end{equation*}(1)[x1,x2,…,xn+1;f]=∫hasbG(x)f(n)(x)dx,
where we have designated by [ x 1 , x 2 , … , x n + 1 ; f ] x 1 , x 2 , … , x n + 1 ; f [x_(1),x_(2),dots,x_(n+1);f]\left[x_{1}, x_{2}, \ldots, x_{n+1}; f\right][x1,x2,…,xn+1;f]the divided difference of the function f f fffon the nodes x α , ( α = 1 , 2 , … , n + 1 ) x α , ( α = 1 , 2 , … , n + 1 ) x_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1)xα,(α=1,2,…,n+1).
Formula (1) is a special case of a formula from R. v. Mises [4]. When n = max ( k 1 , k 2 , … , k p ) n = max k 1 , k 2 , … , k p n=max(k_(1),k_(2),dots,k_(p))n=\max \left(k_{1}, k_{2}, \ldots, k_{p}\right)n=max(k1,k2,…,kp)which happens if and only if p = 2 p = 2 p=2p=2p=2And k 1 = 1 k 1 = 1 k_(1)=1k_{1}=1k1=1Or k 2 = 1 k 2 = 1 k_(2)=1k_{2}=1k2=1, the property results from the formula ( 1 ≦ k ≦ n 1 ≦ k ≦ n 1 <= k <= n1 \leqq k \leqq n1≦k≦n)
[ has , has , … , has , b , b , … , b ; f ] = [ has , has , … , has , b , b , … , b ; f ] = [a,a,dots,a,b,b,dots,b;f]=[a, a, \ldots, a, b, b, \ldots, b ; f]=[has,has,…,has,b,b,…,b;f]=
(2) = 1 ( k − 1 ) ! ( n − k ) ! ( b − a ) n ∫ a b ( x − a ) n − k ( b − x ) k − 1 f ( n ) ( x ) d x (2) = 1 ( k − 1 ) ! ( n − k ) ! ( b − a ) n ∫ a b   ( x − a ) n − k ( b − x ) k − 1 f ( n ) ( x ) d x {:(2)=(1)/((k-1)!(n-k)!(b-a)^(n))int_(a)^(b)(x-a)^(n-k)(b-x)^(k-1)f^((n))(x)dx:}\begin{equation*} =\frac{1}{(k-1)!(n-k)!(b-a)^{n}} \int_{a}^{b}(x-a)^{n-k}(b-x)^{k-1} f^{(n)}(x) \mathrm{d} x \tag{2} \end{equation*}(2)=1(k−1)!(n−k)!(b−has)n∫hasb(x−has)n−k(b−x)k−1f(n)(x)dx
which is easy to obtain, by calculating the integral of the right-hand side by repeated integrations by parts.
2. Formula (1) can also be deduced from another by G. Kowalewski [3] relating to the remainder of the Lagrange interpolation formula (the Lagranze-Hernite formula in general).
To simplify, let's assume that the nodes x α , ( α = 1 , 2 , … , n + 1 ) x α , ( α = 1 , 2 , … , n + 1 ) x_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1)xα,(α=1,2,…,n+1), are distinct, therefore that a = x 1 < x 2 < … < x n < x n + 1 = b a = x 1 < x 2 < … < x n < x n + 1 = b a=x_(1) < x_(2) < dots < x_(n) < x_(n+1)=ba=x_{1}<x_{2}<\ldots<x_{n}<x_{n+1}=bhas=x1<x2<…<xn<xn+1=bLet's ask. l ( x ) == ∏ α = 1 [ ( x − x α ) l ( x ) == ∏ α = 1 [   x − x α l(x)==prod_(alpha=1)^([)(x-x_(alpha))l(x)= =\prod_{\alpha=1}^{[ }\left(x-x_{\alpha}\right)L(x)==∏α=1[(x−xα)and consider the fundamental interpolation polynomials l α ( x ) = l ( x ) ( x − x α ) l ′ ( x α ) , ( α = 1 , 2 , … , n ) l α ( x ) = l ( x ) x − x α l ′ x α , ( α = 1 , 2 , … , n ) l_(alpha)(x)=(l(x))/((x-x_(alpha))l^(')(x_(alpha))),(alpha=1,2,dots,n)l_{\alpha}(x)=\frac{l(x)}{\left(x-x_{\alpha}\right) l^{\prime}\left(x_{\alpha}\right)},(\alpha=1,2, \ldots, n)Lα(x)=L(x)(x−xα)L′(xα),(α=1,2,…,n), related to the nodes x α , ( α = 1 , 2 , … , n ) x α , ( α = 1 , 2 , … , n ) x_(alpha),(alpha=1,2,dots,n)x_{\alpha},(\alpha=1,2, \ldots, n)xα,(α=1,2,…,n)Finally, let us designate by L ( x 1 , x 2 , … , x n ; f ∣ x ) L x 1 , x 2 , … , x n ; f ∣ x L(x_(1),x_(2),dots,x_(n);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)L(x1,x2,…,xn;f|x)the Lagrange interpolation polynomial of the function f f fffon these nodes. G. Kowalewski obtains [3] the remainder f ( x ) − L ( x 1 , x 2 , … , x n ; f ∣ x ) = l ( x ) [ x 1 , x 2 , … , x n , x ; f ] f ( x ) − L x 1 , x 2 , … , x n ; f ∣ x = l ( x ) x 1 , x 2 , … , x n , x ; f f(x)-L(x_(1),x_(2),dots,x_(n);f∣x)=l(x)[x_(1),x_(2),dots,x_(n),x;f]f(x)-L\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)=l(x)\left[x_{1}, x_{2}, \ldots, x_{n}, x ; f\right]f(x)−L(x1,x2,…,xn;f|x)=L(x)[x1,x2,…,xn,x;f]of the Lagrange interpolation formula in the following form
(3) l ( x ) [ x 1 , x 2 , … , x n , x ; f ] = ∑ α = 1 n l α ( x ) ∫ x α x ( x α − u ) n − 1 ( n − 1 ) ! f ( n ) ( u ) d u (3) l ( x ) x 1 , x 2 , … , x n , x ; f = ∑ α = 1 n   l α ( x ) ∫ x α x   x α − u n − 1 ( n − 1 ) ! f ( n ) ( u ) d u {:(3)l(x)[x_(1),x_(2),dots,x_(n),x;f]=sum_(alpha=1)^(n)l_(alpha)(x)int_(x_(alpha))^(x)((x_(alpha)-u)^(n-1))/((n-1)!)f^((n))(u)du:}\begin{equation*} l(x)\left[x_{1}, x_{2}, \ldots, x_{n}, x ; f\right]=\sum_{\alpha=1}^{n} l_{\alpha}(x) \int_{x_{\alpha}}^{x} \frac{\left(x_{\alpha}-u\right)^{n-1}}{(n-1)!} f^{(n)}(u) \mathrm{d} u \tag{3} \end{equation*}(3)L(x)[x1,x2,…,xn,x;f]=∑α=1nLα(x)∫xαx(xα−u)n−1(n−1)!f(n)(u)du
If we now ask L ( x ) = l ( x ) ( x − x n + 1 ) L ( x ) = l ( x ) x − x n + 1 L(x)=l(x)(x-x_(n+1))L(x)=l(x)\left(x-x_{n+1}\right)L(x)=L(x)(x−xn+1)we obtain
l ( x n + 1 ) = L ′ ( x n + 1 ) , l α ( x n + 1 ) = − L ′ ( x n + 1 ) L ′ ( x α ) , ( 0 = 1 , 2 , … , n ) . l x n + 1 = L ′ x n + 1 , l α x n + 1 = − L ′ x n + 1 L ′ x α , ( 0 = 1 , 2 , … , n ) . l(x_(n+1))=L^(')(x_(n+1)),quadl_(alpha)(x_(n+1))=-(L^(')(x_(n+1)))/(L^(')(x_(alpha))),quad(0=1,2,dots,n).l\left(x_{n+1}\right)=L^{\prime}\left(x_{n+1}\right), \quad l_{\alpha}\left(x_{n+1}\right)=-\frac{L^{\prime}\left(x_{n+1}\right)}{L^{\prime}\left(x_{\alpha}\right)}, \quad(0=1,2, \ldots, n) .L(xn+1)=L′(xn+1),Lα(xn+1)=−L′(xn+1)L′(xα),(0=1,2,…,n).
By asking x = x n − 1 − 1 = b x = x n − 1 − 1 = b x=x_(n-1-1)=bx=x_{n-1-1}=bx=xn−1−1=bIn (3), we deduce
(4) [ x 1 , x 2 , … , x n + 1 ; f ] = − ∑ α = 1 n 1 L ′ ( x α ) ∫ x α b ( x α − u ) n − 1 ( n − 1 ) ! f ( n ) ( u ) d u . (4) x 1 , x 2 , … , x n + 1 ; f = − ∑ α = 1 n   1 L ′ x α ∫ x α b   x α − u n − 1 ( n − 1 ) ! f ( n ) ( u ) d u . {:(4)[x_(1),x_(2),dots,x_(n+1);f]=-sum_(alpha=1)^(n)(1)/(L^(')(x_(alpha)))int_(x_(alpha))^(b)((x_(alpha)-u)^(n-1))/((n-1)!)f^((n))(u)du.:}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=-\sum_{\alpha=1}^{n} \frac{1}{L^{\prime}\left(x_{\alpha}\right)} \int_{x_{\alpha}}^{b} \frac{\left(x_{\alpha}-u\right)^{n-1}}{(n-1)!} f^{(n)}(u) \mathrm{d} u . \tag{4} \end{equation*}(4)[x1,x2,…,xn+1;f]=−∑α=1n1L′(xα)∫xαb(xα−u)n−1(n−1)!f(n)(u)du.
  1. Formula (4) shows us that if the nodes are distinct the function G ( x ) G ( x ) G(x)G(x)G(x)of formula (1) is continuous and even, if n > 2 n > 2 n > 2n>2n>2, has a continuous derivative of order n − 2 n − 2 n-2n-2n−2on [ a , b ] [ a , b ] [a,b][a, b][has,b]and reduces to a polynomial of degree n − 1 n − 1 n-1n-1n−1on each of the partial intervals [ x α , x α + 1 ] , ( α = 1 , 2 , … , n ) x α , x α + 1 , ( α = 1 , 2 , … , n ) [x_(alpha),x_(alpha+1)],(alpha=1,2,dots,n)\left[x_{\alpha}, x_{\alpha+1}\right],(\alpha=1,2, \ldots, n)[xα,xα+1],(α=1,2,…,n)We formerly called such a function an elementary function
    of order n − 1 n − 1 n-1n-1n−1and we have shown its importance in the theory of higher-order convex functions [6]. Today they are also called "spline" functions of degree n − 1 n − 1 n-1n-1n−1IJ Schoenberg used them [9], [10] in interesting research on approximate quadrature.
When the nodes x α , ( α = 1 , 2 , … , n + 1 ) x α , ( α = 1 , 2 , … , n + 1 ) x_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1)xα,(α=1,2,…,n+1)They are not distinct but are grouped into distinct nodes. x α ′ x α ′ x_(alpha)^(')x_{\alpha}^{\prime}xα′order k α k α k_(alpha)k_{\alpha}kαof respective multiplicity, the previous properties are only partially verified. Let's extend the function G ( x ) G ( x ) G(x)G(x)G(x)by the value 0 outside the interval [ a , b ] [ a , b ] [a,b][a, b][has,b]The function G ( x ) G ( x ) G(x)G(x)G(x)thus extended is defined on ( − ∞ , + ∞ − ∞ , + ∞ -oo,+oo-\infty,+\infty−∞,+∞), is continuous on the open interval ( a , b a , b a,ba, bhas,band reduces to a polynomial of degree n − 1 n − 1 n-1n-1n−1on each of the intervals ( − ∞ , x 1 ′ ) , ( x p ′ , + ∞ ) , ( x α ′ , x α + 1 ′ ) , ( α = 1 − ∞ , x 1 ′ , x p ′ , + ∞ , x α ′ , x α + 1 ′ , ( α = 1 (-oo,x_(1)^(')),(x_(p)^('),+oo),(x_(alpha)^('),x_(alpha+1)^(')),(alpha=1\left(-\infty, x_{1}^{\prime}\right),\left(x_{p}^{\prime},+\infty\right),\left(x_{\alpha}^{\prime}, x_{\alpha+1}^{\prime}\right),(\alpha=1(−∞,x1′),(xp′,+∞),(xα′,xα+1′),(α=1, 2 , … , p − 1 ) 2 , … , p − 1 ) 2,dots,p-1)2, \ldots, p-1)2,…,p−1)On the node x α ′ , ( α = 1 , 2 , … , p ) x α ′ , ( α = 1 , 2 , … , p ) x_(alpha)^('),(alpha=1,2,dots,p)x_{\alpha}^{\prime},(\alpha=1,2, \ldots, p)xα′,(α=1,2,…,p)the extended function G ˙ ( x ) G ˙ ( x ) G^(˙)(x)\dot{G}(x)G˙(x)is continuous if n ≧ 1 + k α n ≧ 1 + k α n >= 1+k_(alpha)n \geqq 1+k_{\alpha}n≧1+kαand has a continuous derivative of order n − 1 − k k n − 1 − k k n-1-k_(k)n-1-k_{k}n−1−kkif n > 1 + k α n > 1 + k α n > 1+k_(alpha)n>1+k_{\alpha}n>1+kα4.
The function G ( x ) G ( x ) G(x)G(x)G(x)of formula (1) is non-negative and has a positive integral over [ a , b ] [ a , b ] [a,b][a, b][has,b]Indeed, if we ask f = x n f = x n f=x^(n)f=x^{n}f=xnwe deduce
∫ a b G ( x ) d x = 1 n ! ∫ a b   G ( x ) d x = 1 n ! int_(a)^(b)G(x)dx=(1)/(n!)\int_{a}^{b} G(x) \mathrm{d} x=\frac{1}{n!}∫hasbG(x)dx=1n!
The non-negativity of G ( x ) G ( x ) G(x)G(x)G(x)on [ a , b ] [ a , b ] [a,b][a, b][has,b]results from the Cauchy mean formula
(5) [ x 1 , x 2 , … , x n + 1 ; f ] = − 1 n ! f ( n ) ( ξ ) , ξ ∈ ( a , b ) . (5) x 1 , x 2 , … , x n + 1 ; f = − 1 n ! f ( n ) ( ξ ) , ξ ∈ ( a , b ) . {:(5)[x_(1),x_(2),dots,x_(n+1);f]=-(1)/(n!)f^((n))(xi)","quad xi in(a","b).:}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=-\frac{1}{n!} f^{(n)}(\xi), \quad \xi \in(a, b) . \tag{5} \end{equation*}(5)[x1,x2,…,xn+1;f]=−1n!f(n)(ξ),ξ∈(has,b).
Indeed, if the function continues G ( x ) G ( x ) G(x)G(x)G(x)is not non-negative; there exists a subinterval [ α , β ] [ α , β ] [alpha,beta][\alpha, \beta][α,β]of non-zero length [ a , b ] [ a , b ] [a,b][a, b][has,b]on which G ( x ) < 0 G ( x ) < 0 G(x) < 0G(x)<0G(x)<0So then f ( x ) f ( x ) f(x)f(x)f(x)a (continuous) function, defined on [ a , b ] [ a , b ] [a,b][a, b][has,b]whose derivative n n nnn-th is
(6) f ( n ) ( x ) = { 0 pour x ∈ [ a , α ] ∪ [ β , b ] , ( x − α ) ( β − x ) ≠ 0 pour x ∈ ( α , β ) . (6) f ( n ) ( x ) = 0  pour  x ∈ [ a , α ] ∪ [ β , b ] , ( x − α ) ( β − x ) ≠ 0  pour  x ∈ ( α , β ) . {:(6)f^((n))(x)={[0," pour "x in[a","alpha]uu[beta","b]","],[(x-alpha)(beta-x)!=0," pour "x in(alpha","beta).]:}:}f^{(n)}(x)=\left\{\begin{array}{cl} 0 & \text { pour } x \in[a, \alpha] \cup[\beta, b], \tag{6}\\ (x-\alpha)(\beta-x) \neq 0 & \text { pour } x \in(\alpha, \beta) . \end{array}\right.(6)f(n)(x)={0 For x∈[has,α]∪[β,b],(x−α)(β−x)≠0 For x∈(α,β).
On the one hand, from (5) it follows that [ x 1 , x 2 , … , x n + 1 ; f ] ⩾ 0 x 1 , x 2 , … , x n + 1 ; f ⩾ 0 [x_(1),x_(2),dots,x_(n+1);f] >= 0\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right] \geqslant 0[x1,x2,…,xn+1;f]⩾0(the function f f fffis non-concave of order n − 1 n − 1 n-1n-1n−1). On the other hand, from (6) it follows that
∫ a b G ( x ) f ( n ) ( x ) d x = ∫ a β G ( x ) f ( n ) ( x ) d x < 0 ∫ a b   G ( x ) f ( n ) ( x ) d x = ∫ a β   G ( x ) f ( n ) ( x ) d x < 0 int_(a)^(b)G(x)f^((n))(x)dx=int_(a)^(beta)G(x)f^((n))(x)dx < 0\int_{a}^{b} G(x) f^{(n)}(x) \mathrm{d} x=\int_{a}^{\beta} G(x) f^{(n)}(x) \mathrm{d} x<0∫hasbG(x)f(n)(x)dx=∫hasβG(x)f(n)(x)dx<0
The properties of the function G ( x ) G ( x ) G(x)G(x)G(x)of formula (1) have also been studied by DV Ionescu [2].
5. Cauchy's formula (59) suggests the approximation formula
(7) [ x 1 , x 2 , … , x n + 1 ; f ] ≈ 1 n ! f ( n ) ( x 0 ) , (7) x 1 , x 2 , … , x n + 1 ; f ≈ 1 n ! f ( n ) x 0 , {:(7)[x_(1),x_(2),dots,x_(n+1);f]~~(1)/(n!)f^((n))(x_(0))",":}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right] \approx \frac{1}{n!} f^{(n)}\left(x_{0}\right), \tag{7} \end{equation*}(7)[x1,x2,…,xn+1;f]≈1n!f(n)(x0),
Or x 0 x 0 x_(0)x_{0}x0is a given point. The degree of accuracy of this formula is ≧ n ≧ n >= n\geqq n≧nand this degree of accuracy is > n > n > n>n>nif and only if in (7) we take x 0 = 1 n + 1 ∑ α = 1 n + 1 x α x 0 = 1 n + 1 ∑ α = 1 n + 1   x α x_(0)=(1)/(n+1)sum_(alpha=1)^(n+1)x_(alpha)x_{0}=\frac{1}{n+1} \sum_{\alpha=1}^{n+1} x_{\alpha}x0=1n+1∑α=1n+1xαSo the degree of accuracy is n + 1 n + 1 n+1n+1n+1and we have the approximation formula
(8) [ x 1 , x 2 , … , x n + 1 ; f ] = 1 n ! f ( n ) ( x 1 + x 2 + ⋯ + x n + 1 n − 1 ) + R (8) x 1 , x 2 , … , x n + 1 ; f = 1 n ! f ( n ) x 1 + x 2 + ⋯ + x n + 1 n − 1 + R {:(8)[x_(1),x_(2),dots,x_(n+1);f]=(1)/(n!)f^((n))((x_(1)+x_(2)+cdots+x_(n+1))/(n-1))+R:}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\frac{1}{n!} f^{(n)}\left(\frac{x_{1}+x_{2}+\cdots+x_{n+1}}{n-1}\right)+R \tag{8} \end{equation*}(8)[x1,x2,…,xn+1;f]=1n!f(n)(x1+x2+⋯+xn+1n−1)+R
This formula is a Gaussian formula [7] and therefore has a remainder of simple form [8]. A simple calculation gives us the remainder R R RRRin the form
R = 1 ( n + 1 ) ⋅ ( n + 2 ) ! [ ( n + 1 ) ∑ α = 1 n + 1 x α 2 − ( ∑ α = 1 n + 1 x α ) 2 ] ⋅ [ ξ 1 , ξ 2 , ξ 3 ; f ( n ) ] R = 1 ( n + 1 ) ⋅ ( n + 2 ) ! ( n + 1 ) ∑ α = 1 n + 1   x α 2 − ∑ α = 1 n + 1   x α 2 ⋅ ξ 1 , ξ 2 , ξ 3 ; f ( n ) R=(1)/((n+1)*(n+2)!)[(n+1)sum_(alpha=1)^(n+1)x_(alpha)^(2)-(sum_(alpha=1)^(n+1)x_(alpha))^(2)]*[xi_(1),xi_(2),xi_(3);f^((n))]R=\frac{1}{(n+1) \cdot(n+2)!}\left[(n+1) \sum_{\alpha=1}^{n+1} x_{\alpha}^{2}-\left(\sum_{\alpha=1}^{n+1} x_{\alpha}\right)^{2}\right] \cdot\left[\xi_{1}, \xi_{2}, \xi_{3} ; f^{(n)}\right]R=1(n+1)⋅(n+2)![(n+1)∑α=1n+1xα2−(∑α=1n+1xα)2]⋅[ξ1,ξ2,ξ3;f(n)]
Or ξ 1 , ξ 2 , ξ 3 ξ 1 , ξ 2 , ξ 3 xi_(1),xi_(2),xi_(3)\xi_{1}, \xi_{2}, \xi_{3}ξ1,ξ2,ξ3are three distinct points of the interval ( a , b a , b a,ba, bhas,b(generally dependent on the function) f f fff).
If the function f f fffhas a derivative of order n + 2 n + 2 n+2n+2n+2on ( a , b ) ( a , b ) (a,b)(a, b)(has,b), we also have
(9) R = 1 2 ( n + 1 ) ⋅ ( n + 2 ) ! [ ( n + 1 ) ∑ α = 1 n + 1 x α 2 − ( ∑ α = 1 n + 1 x α ) 2 ] f ( n + 2 ) ( ξ ) , ξ ξ ( a , b ) R = 1 2 ( n + 1 ) ⋅ ( n + 2 ) ! ( n + 1 ) ∑ α = 1 n + 1   x α 2 − ∑ α = 1 n + 1   x α 2 f ( n + 2 ) ( ξ ) , ξ ξ ( a , b ) quad R=(1)/(2(n+1)*(n+2)!)[(n+1)sum_(alpha=1)^(n+1)x_(alpha)^(2)-(sum_(alpha=1)^(n+1)x_(alpha))^(2)]f^((n+2))(xi),quad xi xi(a,b)\quad R=\frac{1}{2(n+1) \cdot(n+2)!}\left[(n+1) \sum_{\alpha=1}^{n+1} x_{\alpha}^{2}-\left(\sum_{\alpha=1}^{n+1} x_{\alpha}\right)^{2}\right] f^{(n+2)}(\xi), \quad \xi \xi(a, b)R=12(n+1)⋅(n+2)![(n+1)∑α=1n+1xα2−(∑α=1n+1xα)2]f(n+2)(ξ),ξξ(has,b).
Formula (8) was examined in the particular case p = 2 p = 2 p=2p=2p=2And k 1 = 1 k 1 = 1 k_(1)=1k_{1}=1k1=1Or k 2 = 1 k 2 = 1 k_(2)=1k_{2}=1k2=1, by Laura Gotusso [1] who obtained, in this case, the remainder with some imprecision. Laura Gotusso's correct formula is obtained by setting x 1 = x 2 = ⋯ = x n = x , x n + 1 = x + h x 1 = x 2 = ⋯ = x n = x , x n + 1 = x + h x_(1)=x_(2)=cdots=x_(n)=x,x_(n+1)=x+hx_{1}=x_{2}=\cdots=x_{n}=x, x_{n+1}=x+hx1=x2=⋯=xn=x,xn+1=x+hOr x 1 = x + h , x 2 = x 3 = ⋯ == x n + 1 = x x 1 = x + h , x 2 = x 3 = ⋯ == x n + 1 = x x_(1)=x+h,x_(2)=x_(3)=cdots==x_(n+1)=xx_{1}=x+h, x_{2}=x_{3}=\cdots= =x_{n+1}=xx1=x+h,x2=x3=⋯==xn+1=xin (8) and (9). This gives us the approximation formula (with remainder)
f ( x + h ) = ∑ α = 0 n − 1 h α α ! f ( α ) ( x ) + h n n ! f ( n ) ( x + h n + 1 ) + n h n + 2 2 ( n + 1 ) ⋅ ( n + 2 ) ! f ( n + 2 ) ( ξ ) f ( x + h ) = ∑ α = 0 n − 1   h α α ! f ( α ) ( x ) + h n n ! f ( n ) x + h n + 1 + n h n + 2 2 ( n + 1 ) ⋅ ( n + 2 ) ! f ( n + 2 ) ( ξ ) f(x+h)=sum_(alpha=0)^(n-1)(h^(alpha))/(alpha!)f^((alpha))(x)+(h^(n))/(n!)f^((n))(x+(h)/(n+1))+(nh^(n+2))/(2(n+1)*(n+2)!)f^((n+2))(xi)f(x+h)=\sum_{\alpha=0}^{n-1} \frac{h^{\alpha}}{\alpha!} f^{(\alpha)}(x)+\frac{h^{n}}{n!} f^{(n)}\left(x+\frac{h}{n+1}\right)+\frac{n h^{n+2}}{2(n+1) \cdot(n+2)!} f^{(n+2)}(\xi)f(x+h)=∑α=0n−1hαα!f(α)(x)+hnn!f(n)(x+hn+1)+nhn+22(n+1)⋅(n+2)!f(n+2)(ξ), Or ξ ξ xi\xiξis inside the smallest interval containing the points x , x + h x , x + h x,x+hx, x+hx,x+h6.
The previous results can be obtained without using formula (1). Indeed, we have shown [5] that if the function f f fffis convex of order n − 1 n − 1 n-1n-1n−1we have inequality
[ x 1 , x 2 , … , x n + 1 ; f ] > 1 n ! f ( n ) ( x 1 + x 2 + ⋯ + x n + 1 n + 1 ) x 1 , x 2 , … , x n + 1 ; f > 1 n ! f ( n ) x 1 + x 2 + ⋯ + x n + 1 n + 1 [x_(1),x_(2),dots,x_(n+1);f] > (1)/(n!)f^((n))((x_(1)+x_(2)+cdots+x_(n+1))/(n+1))\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]>\frac{1}{n!} f^{(n)}\left(\frac{x_{1}+x_{2}+\cdots+x_{n+1}}{n+1}\right)[x1,x2,…,xn+1;f]>1n!f(n)(x1+x2+⋯+xn+1n+1)
(Or x α x α x_(alpha)x_{\alpha}xαare not all confused). The simplicity of the rest of formula (8) then follows [8].
7. Other approximation formulas for the divided difference (1) can be obtained by applying any quadrature formula to the integral of the right-hand side. I will limit myself to examining one more particular case.
Let's start with some preliminary calculations. The moments
c n = ∫ a b G ( x ) x n d x , ( n = 0 , 1 , … ) c n = ∫ a b   G ( x ) x n d x , ( n = 0 , 1 , … ) c_(n)=int_(a)^(b)G(x)x^(n)dx,quad(n=0,1,dots)c_{n}=\int_{a}^{b} G(x) x^{n} \mathrm{~d} x, \quad(n=0,1, \ldots)cn=∫hasbG(x)xn dx,(n=0,1,…)
can be calculated using well-known symmetric functions w r = ∑ α 1 + α 2 + … + α n + 1 = r x 1 α 1 x 2 α 2 … x n + 1 α n + 1 , ( r = 0 , 1 , … ; w 0 = 1 ) w r = ∑ α 1 + α 2 + … + α n + 1 = r   x 1 α 1 x 2 α 2 … x n + 1 α n + 1 , r = 0 , 1 , … ; w 0 = 1 w_(r)=sum_(alpha_(1)+alpha_(2)+dots+alpha_(n+1)=r)x_(1)^(alpha_(1))x_(2)^(alpha_(2))dotsx_(n+1)^(alpha_(n+1)),(r=0,1,dots;w_(0)=1)w_{r}=\sum_{\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n+1}=r} x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} \ldots x_{n+1}^{\alpha_{n+1}},\left(r=0,1, \ldots ; w_{0}=1\right)wr=∑α1+α2+…+αn+1=rx1α1x2α2…xn+1αn+1,(r=0,1,…;w0=1), the summation being extended to the non-negative integer solutions of the Diophantine equation a 1 + α 2 + ⋯ + α n + 1 = r a 1 + α 2 + ⋯ + α n + 1 = r a_(1)+alpha_(2)+cdots+alpha_(n+1)=ra_{1}+\alpha_{2}+\cdots+\alpha_{n+1}=rhas1+α2+⋯+αn+1=rWe can calculate the w r w r w_(r)w_{r}wrusing the recurrence formula
(10) w r − p 1 w r − 1 + p 2 w r − 2 − ⋯ + ( − 1 ) n + 1 p n + 1 w r − n − 1 = 0 , ( r = 1 , 2 , … ) , (10) w r − p 1 w r − 1 + p 2 w r − 2 − ⋯ + ( − 1 ) n + 1 p n + 1 w r − n − 1 = 0 , ( r = 1 , 2 , … ) , {:(10)w_(r)-p_(1)w_(r-1)+p_(2)w_(r-2)-cdots+(-1)^(n+1)p_(n+1)w_(r-n-1)=0","(r=1","2","dots)",":}\begin{equation*} w_{r}-p_{1} w_{r-1}+p_{2} w_{r-2}-\cdots+(-1)^{n+1} p_{n+1} w_{r-n-1}=0,(r=1,2, \ldots), \tag{10} \end{equation*}(10)wr−p1wr−1+p2wr−2−⋯+(−1)n+1pn+1wr−n−1=0,(r=1,2,…),
by asking w 0 = 1 , w − 1 = w − 2 = ⋯ = w − n = 0 , p 1 , p 2 , … , p n + 1 w 0 = 1 , w − 1 = w − 2 = ⋯ = w − n = 0 , p 1 , p 2 , … , p n + 1 w_(0)=1,w_(-1)=w_(-2)=cdots=w_(-n)=0,p_(1),p_(2),dots,p_(n+1)w_{0}=1, w_{-1}=w_{-2}=\cdots=w_{-n}=0, p_{1}, p_{2}, \ldots, p_{n+1}w0=1,w−1=w−2=⋯=w−n=0,p1,p2,…,pn+1being the fundamental symmetric functions of x 1 , x 2 , … , x n + 1 x 1 , x 2 , … , x n + 1 x_(1),x_(2),dots,x_(n+1)x_{1}, x_{2}, \ldots, x_{n+1}x1,x2,…,xn+1.
If we ask f = x n + r f = x n + r f=x^(n+r)f=x^{n+r}f=xn+rIn (1), we obtain,
hence
w r = [ x 1 , x 2 , … , x n + r ; x n − r ] = ( n + r ) ! r ! ∫ a b G ( x ) x r d x c r = r ! ( n + r ) ! w r , ( r = 0 , 1 , … ; 0 ! = 1 ) w r = x 1 , x 2 , … , x n + r ; x n − r = ( n + r ) ! r ! ∫ a b   G ( x ) x r d x c r = r ! ( n + r ) ! w r , ( r = 0 , 1 , … ; 0 ! = 1 ) {:[w_(r)=[x_(1),x_(2),dots,x_(n+r);x^(n-r)]=((n+r)!)/(r!)int_(a)^(b)G(x)x^(r)dx],[c_(r)=(r!)/((n+r)!)w_(r)","(r=0","1","dots;0!=1)]:}\begin{gathered} w_{r}=\left[x_{1}, x_{2}, \ldots, x_{n+r} ; x^{n-r}\right]=\frac{(n+r)!}{r!} \int_{a}^{b} G(x) x^{r} \mathrm{~d} x \\ c_{r}=\frac{r!}{(n+r)!} w_{r},(r=0,1, \ldots ; 0!=1) \end{gathered}wr=[x1,x2,…,xn+r;xn−r]=(n+r)!r!∫hasbG(x)xr dxcr=r!(n+r)!wr,(r=0,1,…;0!=1)
  1. Suppose the nodes x α , ( α = 1 , 2 , … , n + 1 ) x α , ( α = 1 , 2 , … , n + 1 ) x_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1)xα,(α=1,2,…,n+1), are symmetrically distributed with respect to the origin, therefore that a = − b , x α + x n + 2 − α = 0 a = − b , x α + x n + 2 − α = 0 a=-b,x_(alpha)+x_(n+2-alpha)=0a=-b, x_{\alpha}+x_{n+2-\alpha}=0has=−b,xα+xn+2−α=0, ( α = 1 , 2 , … , n + 1 ) ( α = 1 , 2 , … , n + 1 ) (alpha=1,2,dots,n+1)(\alpha=1,2, \ldots, n+1)(α=1,2,…,n+1)In this case we have p α = 0 p α = 0 p_(alpha)=0p_{\alpha}=0pα=0For α α alpha\alphaαodd and from (10) it follows that w r = 0 w r = 0 w_(r)=0w_{r}=0wr=0, therefore also c r = 0 c r = 0 c_(r)=0c_{r}=0cr=0, For r r rrrany odd number.
We have the approximation formula ( m ≧ 1 m ≧ 1 m >= 1m \geqq 1m≧1)
(11) ∫ − b b G ( x ) f ( x ) d x ≈ ∫ − b b G ( x ) [ ∑ α = 0 2 m − 1 x α α ! f ( α ) ( 0 ) ] d x = ∑ α = 0 m − 1 c 2 α ( 2 α ) ! f ( 2 α ) ( 0 ) (11) ∫ − b b   G ( x ) f ( x ) d x ≈ ∫ − b b   G ( x ) ∑ α = 0 2 m − 1   x α α ! f ( α ) ( 0 ) d x = ∑ α = 0 m − 1   c 2 α ( 2 α ) ! f ( 2 α ) ( 0 ) {:(11)int_(-b)^(b)G(x)f(x)dx~~int_(-b)^(b)G(x)[sum_(alpha=0)^(2m-1)(x^(alpha))/(alpha!)f^((alpha))(0)]dx=sum_(alpha=0)^(m-1)(c_(2alpha))/((2alpha)!)f^((2alpha))(0):}\begin{equation*} \int_{-b}^{b} G(x) f(x) \mathrm{d} x \approx \int_{-b}^{b} G(x)\left[\sum_{\alpha=0}^{2 m-1} \frac{x^{\alpha}}{\alpha!} f^{(\alpha)}(0)\right] \mathrm{d} x=\sum_{\alpha=0}^{m-1} \frac{c_{2 \alpha}}{(2 \alpha)!} f^{(2 \alpha)}(0) \tag{11} \end{equation*}(11)∫−bbG(x)f(x)dx≈∫−bbG(x)[∑α=02m−1xαα!f(α)(0)]dx=∑α=0m−1c2α(2α)!f(2α)(0)
of which the rest
(12) ∫ − b b G ( x ) x 2 m [ x , 0 , 0 , … , 0 ⏟ 2 m ; f ] d x (12) ∫ − b b   G ( x ) x 2 m [ x , 0 , 0 , … , 0 ⏟ 2 m ; f ] d x {:(12)int_(-b)^(b)G(x)x^(2m)[x","ubrace(0,0,dots,0ubrace)_(2m);f]dx:}\begin{equation*} \int_{-b}^{b} G(x) x^{2 m}[x, \underbrace{0,0, \ldots, 0}_{2 m} ; f] \mathrm{d} x \tag{12} \end{equation*}(12)∫−bbG(x)x2m[x,0,0,…,0⏟2m;f]dx
is the degree of accuracy 2 m − 1 2 m − 1 2m-12 m-12m−1and is indeed of the simple form, therefore of the form
(13) c 2 m [ ξ 1 , ξ 2 , … , ξ 2 m + 1 ; f ] , (13) c 2 m ξ 1 , ξ 2 , … , ξ 2 m + 1 ; f , {:(13)c_(2m)[xi_(1),xi_(2),dots,xi_(2m+1);f]",":}\begin{equation*} c_{2 m}\left[\xi_{1}, \xi_{2}, \ldots, \xi_{2 m+1} ; f\right], \tag{13} \end{equation*}(13)c2m[ξ1,ξ2,…,ξ2m+1;f],
THE ξ α ξ α xi_(alpha)\xi_{\alpha}ξαbeing 2 m + 1 2 m + 1 2m+12 m+12m+1distinct points of the interval ( − b , b − b , b -b,b-b, b−b,b). In (12) we assume that the function f f fffhas a continuous derivative of order 2 m − 1 2 m − 1 2m-12 m-12m−1, but the rest is of the form (13) even if f f fffhas a continuous derivative of order 2 m − 2 2 m − 2 2m-22 m-22m−2only. The rest can therefore be in the form
2 c 2 m ( 2 m ) ! [ ξ 1 , ξ 2 , ξ 3 ; f ( 2 m − 2 ) ] 2 c 2 m ( 2 m ) ! ξ 1 , ξ 2 , ξ 3 ; f ( 2 m − 2 ) (2c_(2m))/((2m)!)[xi_(1),xi_(2),xi_(3);f^((2m-2))]\frac{2 c_{2 m}}{(2 m)!}\left[\xi_{1}, \xi_{2}, \xi_{3} ; f^{(2 m-2)}\right]2c2m(2m)![ξ1,ξ2,ξ3;f(2m−2)]
ξ 1 , ξ 2 , ξ 3 ξ 1 , ξ 2 , ξ 3 xi_(1),xi_(2),xi_(3)\xi_{1}, \xi_{2}, \xi_{3}ξ1,ξ2,ξ3being three distinct points of the interval ( − b , b − b , b -b,b-b, b−b,b).
We deduce that if the nodes x α , ( α = 1 , 2 , … , n + 1 ) x α , ( α = 1 , 2 , … , n + 1 ) x_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1)xα,(α=1,2,…,n+1), are symmetrically distributed with respect to the origin and if x n + 1 = − x 1 = b ( > 0 ) x n + 1 = − x 1 = b ( > 0 ) x_(n+1)=-x_(1)=b( > 0)x_{n+1}=-x_{1}=b(>0)xn+1=−x1=b(>0)we have the approximation formula
[ x 1 , x 2 , … , x n + 1 ; f ] = ∑ α = 0 m − 1 w 2 α ( n + 2 α ) ! f ( n + 2 α ) ( 0 ) + + 2 w 2 m ( n + 2 m ) ! [ ξ 1 , ξ 2 , ξ 3 ; f ( n + 2 m − 2 ) ] x 1 , x 2 , … , x n + 1 ; f = ∑ α = 0 m − 1   w 2 α ( n + 2 α ) ! f ( n + 2 α ) ( 0 ) + + 2 w 2 m ( n + 2 m ) ! ξ 1 , ξ 2 , ξ 3 ; f ( n + 2 m − 2 ) {:[[x_(1),x_(2),dots,x_(n+1);f]=sum_(alpha=0)^(m-1)(w_(2alpha))/((n+2alpha)!)f^((n+2alpha))(0)+],[quad+(2w_(2m))/((n+2m)!)[xi_(1),xi_(2),xi_(3);f^((n+2m-2))]]:}\begin{aligned} & {\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\sum_{\alpha=0}^{m-1} \frac{w_{2 \alpha}}{(n+2 \alpha)!} f^{(n+2 \alpha)}(0)+} \\ & \quad+\frac{2 w_{2 m}}{(n+2 m)!}\left[\xi_{1}, \xi_{2}, \xi_{3} ; f^{(n+2 m-2)}\right] \end{aligned}[x1,x2,…,xn+1;f]=∑α=0m−1w2α(n+2α)!f(n+2α)(0)++2w2m(n+2m)![ξ1,ξ2,ξ3;f(n+2m−2)]

Or f f fffhas a continuous derivative of order n + 2 m − 2 sur ( − b , b ) n + 2 m − 2 sur ( − b , b ) n+2m-2sur(-b,b)n+2 m-2 \operatorname{sur}(-b, b)n+2m−2on(−b,b)And ξ 1 , ξ 2 , ξ 3 ξ 1 , ξ 2 , ξ 3 xi_(1),xi_(2),xi_(3)\xi_{1}, \xi_{2}, \xi_{3}ξ1,ξ2,ξ3are three distinct points in this interval.
Formula (11) is still of the Gaussian type, according to the definition of formulas of this type [7].
Regue on 13 II 1967
Babeş-Bolyai University

BIBLIOGRAPHY

  1. Gotusso L., Una valutasione approssimata del fin complementare della formula di Taylor. Atti del Seminario Mat. e Fizico dell’Univ. di Modena, 1964, XIII, pp. 221-229. 2. Ionescu DV, Cuadraturi Numerice. Ed. Tehn. Buc., 1957.
  2. Kowalewski G., Interpolation und genäherte Quadratur. Leipzig u. Berlin, 1932.
  3. v. Mises s R., Über allgemeine Quadraturformeln. J.f. queen u. angew. Math., 1936, 174, S. 56-67.
1967

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