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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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Buletinul Institutului Politehnic din Iaşi
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published by the “Gheorghe Asachi” Technical University of Iaşi
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REMARKS ON THE REMAINDER OF CERTAIN FORMULAS FOR APPROXIMATING A DIFFERENCE DIVIDED BY DERIVATIVES
- Let us consider a functionffdefined and having a derivativenn-th continuous on the finite and closed interval[a,b],(a < b)[a, b],(a<b).
Leta=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+1distinct or non-distinct points belonging to the interval[a,b][a, b]such as the extremitieshashas,bbare among these points. It follows thatn( > 0)n(>0)is a natural number. Let us denote bya=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}=bthe distinct points among the pointsx_(alpha)x_{\alpha}and bek_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}their respective multiplicity orders. Therefore, we have2 <= p <= n+1,k_(1),k_(2),dots,k_(p)2 \leqq p \leqq n+1, k_{1}, k_{2}, \ldots, k_{p}are natural numbers,k_(1)+k_(2)+dots++k_(p)=n+1k_{1}+k_{2}+\ldots+ +k_{p}=n+1Andmax(k_(1),k_(2),dots,k_(p)) <= n\max \left(k_{1}, k_{2}, \ldots, k_{p}\right) \leqq n.
In accordance with the point systemx_(alpha),x_(alpha)^(')x_{\alpha}, x_{\alpha}^{\prime}, We have
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),
(k_(1)+k_(2)+cdots+k_(alpha-1)k_{1}+k_{2}+\cdots+k_{\alpha-1}is replaced by 0 ifalpha=1\alpha=1
There exists a continuous functionG(x)G(x), completely determined, such that one has
There exists a continuous functionG(x)G(x), completely determined, such that one has
{:(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*}
where we have designated by[x_(1),x_(2),dots,x_(n+1);f]\left[x_{1}, x_{2}, \ldots, x_{n+1}; f\right]the divided difference of the functionffon the nodesx_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1).
Formula (1) is a special case of a formula from R. v. Mises [4]. Whenn=max(k_(1),k_(2),dots,k_(p))n=\max \left(k_{1}, k_{2}, \ldots, k_{p}\right)which happens if and only ifp=2p=2Andk_(1)=1k_{1}=1Ork_(2)=1k_{2}=1, the property results from the formula (1 <= k <= n1 \leqq k \leqq n)
[a,a,dots,a,b,b,dots,b;f]=[a, a, \ldots, a, b, b, \ldots, b ; f]=
{:(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*}
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).
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 nodesx_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1), are distinct, therefore thata=x_(1) < x_(2) < dots < x_(n) < x_(n+1)=ba=x_{1}<x_{2}<\ldots<x_{n}<x_{n+1}=bLet's ask.l(x)==prod_(alpha=1)^([)(x-x_(alpha))l(x)= =\prod_{\alpha=1}^{[ }\left(x-x_{\alpha}\right)and consider the fundamental interpolation polynomialsl_(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), related to the nodesx_(alpha),(alpha=1,2,dots,n)x_{\alpha},(\alpha=1,2, \ldots, n)Finally, let us designate byL(x_(1),x_(2),dots,x_(n);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)the Lagrange interpolation polynomial of the functionffon these nodes. G. Kowalewski obtains [3] the remainderf(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]of the Lagrange interpolation formula in the following form
{:(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*}
If we now askL(x)=l(x)(x-x_(n+1))L(x)=l(x)\left(x-x_{n+1}\right)we obtain
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) .
By askingx=x_(n-1-1)=bx=x_{n-1-1}=bIn (3), we deduce
{:(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*}
- Formula (4) shows us that if the nodes are distinct the functionG(x)G(x)of formula (1) is continuous and even, ifn > 2n>2, has a continuous derivative of ordern-2n-2on[a,b][a, b]and reduces to a polynomial of degreen-1n-1on each of the partial intervals[x_(alpha),x_(alpha+1)],(alpha=1,2,dots,n)\left[x_{\alpha}, x_{\alpha+1}\right],(\alpha=1,2, \ldots, n)We formerly called such a function an elementary function
of ordern-1n-1and we have shown its importance in the theory of higher-order convex functions [6]. Today they are also called "spline" functions of degreen-1n-1IJ Schoenberg used them [9], [10] in interesting research on approximate quadrature.
When the nodesx_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1)They are not distinct but are grouped into distinct nodes.x_(alpha)^(')x_{\alpha}^{\prime}orderk_(alpha)k_{\alpha}of respective multiplicity, the previous properties are only partially verified. Let's extend the functionG(x)G(x)by the value 0 outside the interval[a,b][a, b]The functionG(x)G(x)thus extended is defined on (-oo,+oo-\infty,+\infty), is continuous on the open interval (a,ba, band reduces to a polynomial of degreen-1n-1on each of the intervals(-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=1,2,dots,p-1)2, \ldots, p-1)On the nodex_(alpha)^('),(alpha=1,2,dots,p)x_{\alpha}^{\prime},(\alpha=1,2, \ldots, p)the extended functionG^(˙)(x)\dot{G}(x)is continuous ifn >= 1+k_(alpha)n \geqq 1+k_{\alpha}and has a continuous derivative of ordern-1-k_(k)n-1-k_{k}ifn > 1+k_(alpha)n>1+k_{\alpha}4.
The functionG(x)G(x)of formula (1) is non-negative and has a positive integral over[a,b][a, b]Indeed, if we askf=x^(n)f=x^{n}we deduce
The functionG(x)G(x)of formula (1) is non-negative and has a positive integral over[a,b][a, b]Indeed, if we askf=x^(n)f=x^{n}we deduce
int_(a)^(b)G(x)dx=(1)/(n!)\int_{a}^{b} G(x) \mathrm{d} x=\frac{1}{n!}
The non-negativity ofG(x)G(x)on[a,b][a, b]results from the Cauchy mean formula
{:(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*}
Indeed, if the function continuesG(x)G(x)is not non-negative; there exists a subinterval[alpha,beta][\alpha, \beta]of non-zero length[a,b][a, b]on whichG(x) < 0G(x)<0So thenf(x)f(x)a (continuous) function, defined on[a,b][a, b]whose derivativenn-th is
{:(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.
On the one hand, from (5) it follows that[x_(1),x_(2),dots,x_(n+1);f] >= 0\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right] \geqslant 0(the functionffis non-concave of ordern-1n-1). On the other hand, from (6) it follows that
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
The properties of the functionG(x)G(x)of formula (1) have also been studied by DV Ionescu [2].
5. Cauchy's formula (59) suggests the approximation formula
5. Cauchy's formula (59) suggests the approximation formula
{:(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*}
Orx_(0)x_{0}is a given point. The degree of accuracy of this formula is>= n\geqq nand this degree of accuracy is> n>nif and only if in (7) we takex_(0)=(1)/(n+1)sum_(alpha=1)^(n+1)x_(alpha)x_{0}=\frac{1}{n+1} \sum_{\alpha=1}^{n+1} x_{\alpha}So the degree of accuracy isn+1n+1and we have the approximation formula
{:(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*}
This formula is a Gaussian formula [7] and therefore has a remainder of simple form [8]. A simple calculation gives us the remainderRRin the form
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]
Orxi_(1),xi_(2),xi_(3)\xi_{1}, \xi_{2}, \xi_{3}are three distinct points of the interval (a,ba, b(generally dependent on the function)ff).
If the functionffhas a derivative of ordern+2n+2on(a,b)(a, b), we also have
(9)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).
(9)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).
Formula (8) was examined in the particular casep=2p=2Andk_(1)=1k_{1}=1Ork_(2)=1k_{2}=1, by Laura Gotusso [1] who obtained, in this case, the remainder with some imprecision. Laura Gotusso's correct formula is obtained by settingx_(1)=x_(2)=cdots=x_(n)=x,x_(n+1)=x+hx_{1}=x_{2}=\cdots=x_{n}=x, x_{n+1}=x+hOrx_(1)=x+h,x_(2)=x_(3)=cdots==x_(n+1)=xx_{1}=x+h, x_{2}=x_{3}=\cdots= =x_{n+1}=xin (8) and (9). This gives us the approximation formula (with remainder)
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), Orxi\xiis inside the smallest interval containing the pointsx,x+hx, x+h6.
The previous results can be obtained without using formula (1). Indeed, we have shown [5] that if the functionffis convex of ordern-1n-1we have inequality
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), Orxi\xiis inside the smallest interval containing the pointsx,x+hx, x+h6.
The previous results can be obtained without using formula (1). Indeed, we have shown [5] that if the functionffis convex of ordern-1n-1we have inequality
[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)
(Orx_(alpha)x_{\alpha}are 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.
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)=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)
can be calculated using well-known symmetric functionsw_(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), the summation being extended to the non-negative integer solutions of the Diophantine equationa_(1)+alpha_(2)+cdots+alpha_(n+1)=ra_{1}+\alpha_{2}+\cdots+\alpha_{n+1}=rWe can calculate thew_(r)w_{r}using the recurrence formula
{:(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*}
by askingw_(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}being the fundamental symmetric functions ofx_(1),x_(2),dots,x_(n+1)x_{1}, x_{2}, \ldots, x_{n+1}.
If we askf=x^(n+r)f=x^{n+r}In (1), we obtain,
hence
hence
{:[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}
- Suppose the nodesx_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1), are symmetrically distributed with respect to the origin, therefore thata=-b,x_(alpha)+x_(n+2-alpha)=0a=-b, x_{\alpha}+x_{n+2-\alpha}=0,(alpha=1,2,dots,n+1)(\alpha=1,2, \ldots, n+1)In this case we havep_(alpha)=0p_{\alpha}=0Foralpha\alphaodd and from (10) it follows thatw_(r)=0w_{r}=0, therefore alsoc_(r)=0c_{r}=0, Forrrany odd number.
We have the approximation formula (m >= 1m \geqq 1)
{:(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*}
of which the rest
{:(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*}
is the degree of accuracy2m-12 m-1and is indeed of the simple form, therefore of the form
{:(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*}
THExi_(alpha)\xi_{\alpha}being2m+12 m+1distinct points of the interval (-b,b-b, b). In (12) we assume that the functionffhas a continuous derivative of order2m-12 m-1, but the rest is of the form (13) even ifffhas a continuous derivative of order2m-22 m-2only. The rest can therefore be in the form
(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]
xi_(1),xi_(2),xi_(3)\xi_{1}, \xi_{2}, \xi_{3}being three distinct points of the interval (-b,b-b, b).
We deduce that if the nodesx_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1), are symmetrically distributed with respect to the origin and ifx_(n+1)=-x_(1)=b( > 0)x_{n+1}=-x_{1}=b(>0)we have the approximation formula
We deduce that if the nodesx_(alpha),(alpha=1,2,dots,n+1)x_{\alpha},(\alpha=1,2, \ldots, n+1), are symmetrically distributed with respect to the origin and ifx_(n+1)=-x_(1)=b( > 0)x_{n+1}=-x_{1}=b(>0)we have the approximation formula
{:[[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}
Orffhas a continuous derivative of ordern+2m-2sur(-b,b)n+2 m-2 \operatorname{sur}(-b, b)Andxi_(1),xi_(2),xi_(3)\xi_{1}, \xi_{2}, \xi_{3}are 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
Babeş-Bolyai University
BIBLIOGRAPHY
- 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.
- Kowalewski G., Interpolation und genäherte Quadratur. Leipzig u. Berlin, 1932.
- v. Mises s R., Über allgemeine Quadraturformeln. J.f. queen u. angew. Math., 1936, 174, S. 56-67.