CONTRIBUTTII LA INTEGRAREA NUMERICĂ A FUNCTYILOR DE MAI MULTE VARIABILE

Comunicare prezentată la ședinţa din 24 septembrie 1956 a Filialei Cluj a Academiei R.P.R.

§ 1. Considerațiuni generale

1.

Să notăm eu $D$ un domeniu din spațiul euclidian $s$ - dimensional $E_{s}$ , cu $dv_{M}$ un element de volum din $D$ , cu $f(M)$ și $K(M)$ două funcții de punctul $M=M\left(t^{1},t^{2},\ldots,t^{S}\right)$ , integrabile în domeniul considerat.

Prin formulă de integrare numerică sau formulă de cubatură se înțelege o formulă de forma

\iint\ldots\int_{D}K(M)f(M)dv_{M}=\sum_{i=1}^{N}c_{i}f\left(M_{i}\right)+\rho

(1)

unde numerele $c_{i}$ - care se numesc coeficienții formulei - depind numai de nodurile $M_{i}$ ale formulei de cubatură, care sînt puncte din $D;\rho$ este restul acestei formule.

Suma finită din membrul al doilea

J(f)=\sum_{i=1}^{N}c_{i}f\left(M_{i}\right)

(2)

reprezintă o evaluare aproximativă a funcționalei

I(f)=\iint\ldots\int_{D}K(M)f(M)dv_{M}

(3)

unde funcțiunea $K(M)$ se presupune că e aleasă odată pentru totdeauna pentru funcționala $I(f)$ și că păstrează un semn constant în $D$ .

Aşadar, o formulă de cubatură este o formulă care permite să se dea o evaluare aproximativă a unei integrale definite dintr-o funcție $f(M)$ , multiplicată cu o functie pondere $K(M)$ , printr-o anumită combinație liniară a valorilor funcției $f(M)$ pe un număr finit de puncte distincte. Avem și formule
de cubatură în care intervin și valorile derivatelor parțiale ale lui $f(M)$ în anumite puncte.

Restul $\rho$ al formulei (1) reprezintă valoarea, dată de, această formulă, a unei funcționale aditive și omogene, pe care, pentru a indica funcția $f(M)$ , o vom nota de asemenea cu $\rho(f)$ .
2. Vom spune că formula (1) are gradul parțial de exactitate ( $n_{1},n_{2},\ldots n_{s}$ ). dacă :
a) $\rho(P)=0$ pentru orice polinom $P(M)$ de grad $\left(n_{1},n_{2},\ldots,n_{s}\right)$ .
b) $\rho(P)\neq 0$ pentru cel puttin un polinom $P(M)$ de grad $\left(n_{1}+1\right.$ , $\left.n_{2}+1,\ldots,n_{s}+1\right)$ .
3. Problemele de bază care se pun acum sînt următoarele:
a) A construi formulele de cubatură, adică a determina coeficienții $c_{i}$ și nodurile $M_{i}$ .
b) A da și studia expresia restului acestor formule pentru a putea evalua eroarea care se comite cînd pentru $I(f)$ se ia valoarea aproximativă $J(f)$ .
Se știe că o metodă generală de construire a formulelor de cubatură constă în a înlocui funcția $f(M)$ prin expresia sa dată de o formulă de interpolare.

Dacă presupunem că $f(M)$ este dezvoltabilă în seria uniform convergentă în $D$

f(M)=\sum_{i_{1},\ldots,i_{s}=0}^{\infty}a_{i_{1}\ldots i_{s}}\left(t^{1}\right)^{i_{1}}\ldots\left(t^{s}\right)^{i_{s}}

(4)

şi punem

I_{i_{1}\ldots i_{s}}=\iint\ldots\int_{D}K\left(t^{1},\ldots,t^{s}\right)\left(t^{1}\right)^{i_{1}}\ldots\left(t^{s}\right)^{i_{s}}dv_{M}

restul va fi reprezentat de seria

\rho=\sum_{i_{1},\ldots,i_{s}=0}^{\infty}a_{i_{1}\ldots i_{s}}\left[I_{i_{1}\ldots i_{s}}-\sum_{j=1}^{N}p_{j}\left(t_{j}^{1}\right)^{i_{1}}\ldots\left(t_{j}^{s}\right)^{i_{s}}\right],

(5)

cantitătile intre parantezele drepte fiind toate indepedente de $f(M)$ .
Dacă $f(M)$ e un polinom de gradul ( $n_{1},n_{2},\ldots,n_{s}$ ), avem

I(f)=J(f)

oricare ar fi coeficienții $a_{i_{1}}\ldots i_{s}$ pentru care $i_{k}\leqq n_{k}(k=1,s)$ .
Aceasta ne conduce la următoarele ecuatii

\sum_{j=1}^{N}p_{j}\left(t^{1}\right)^{i_{1}}\ldots\left(t^{s}\right)^{i_{s}}=I_{i_{1}\ldots i_{s}}

4.

E, firesc să căutăm să mărim precizia formulei de cubatură despre care vorbim, căutînd să determinăm cele $sN$ necunoscute ( $t_{i}^{1},t_{i}^{2},\ldots,t_{i}^{s}$ ), $i=1,N$ astfel ca să se anuleze, oricare ar fi functia $f(M)$ , şi cei $sN$ termeni următori din seria (5), deducînd în acest $\bmod sN$ ecuații noi de forma

\sum_{j=1}^{N}p_{j}\left(t_{j}^{1}\right)^{i_{1}}\ldots\left(t_{j}^{s}\right)^{i_{s}}=I_{i_{1}\ldots i_{s}}

În felul acesta se obține în total un sistem de $(s+1)N$ ecuații cu tot atîtea necunoscute

c_{i};t_{i}^{1},t_{i}^{2},\ldots,t_{i}^{s}\quad(i=\overline{1,N})

Pentru ca problema să fie posibilă va trebui ca sistemul liniar care se obține să fie compatibil și să ne conducă la $N$ puncte $M_{i}$ reale aparținînd domeniului $D$ și fără să fie situate pe o hipersuprafată de ordinul $\left(n_{1},n_{2},\ldots,n_{s}\right)$ pentru a nu se anula anumiți determinanti care vor interveni la numitorii expresiilor coeficientiilor $c_{\mathrm{i}}$ .
5. Intr-o lucrare precedentă [3], relativ 1a sistemul de $N=\left(n_{1}+1\right)\ldots\ldots\left(n_{s}+1\right)$ noduri

M_{i_{1}\ldots i_{s}}=M_{i_{1}\ldots i_{s}}\left(t_{i_{1}}^{1},t_{i_{1}i_{2}}^{2},\ldots,t_{i_{1}\ldots i_{s}}^{s}\right)

(6)

am dat formula de interpolare

f(M)=L_{n_{1}n_{2}\ldots n_{s}}(M)+R_{s}(M)

(7)

unde

L_{n_{1}\ldots n_{s}}(M)=\sum_{i_{1}=1}^{n_{1}+1}\ldots\sum_{i_{s}=1}^{n_{s}+1}l_{i_{1}}^{1}\left(t^{1}\right)\ldots l_{i_{1}\ldots i_{s}}^{s}\left(t^{s}\right)f\left(M_{i_{1}\ldots i_{s}}\right)

(8)

\begin{gathered}l_{i_{1}\ldots i_{k}}^{k}\left(t^{k}\right)=\frac{u_{i_{1}\ldots i_{k-1}}^{k}\left(t^{k}\right)}{\left(t^{k}-t_{i_{1}\ldots i_{k}}^{k}\right)\dot{u}_{i_{1}\ldots i_{k-1}}^{k}\left(t_{i_{1}\ldots i_{k}}^{k}\right)}\\ u_{i_{1}\ldots i_{k-1}}^{k}\left(t^{k}\right)=\prod_{i_{k}=1}^{n_{k}+1}\left(t^{k}-t_{i_{1}\ldots i_{k}}^{k}\right)\end{gathered}

e polinomul de interpolare de gradul ( $n_{1},n_{2},\ldots,n_{s}$ ) care coincide cu $f(M)$ pe nodurile (6):

Restul formulei (7) are expresia

R_{s}(M)=\sum_{p=1}^{s}\sum_{i_{1}=1}^{n_{1}+1}\ldots\sum_{i_{p-1}=1}^{n_{p-1}+1}l_{i_{1}}^{1}\left(t^{1}\right)\ldots l_{i_{1}\ldots i_{p-1}}^{p-1}\left(t^{p-1}\right)u_{i_{1}\ldots i_{p-1}}^{p}S_{i_{1}\ldots i_{p-1}}^{p},

unde

S_{i_{1}\ldots i_{p-1}}^{p}=\left[t^{p},t_{i_{1}\ldots i_{p-1}1}^{p},\ldots,t_{i_{1}\ldots i_{p-1}}^{p},n_{p+1};f\left(t_{i_{1}}^{1},\ldots,t_{i_{1}\ldots i_{p-1}}^{p-1},t^{p},\ldots,t^{s}\right)\right]

Aici în membrul drept avem diferența divizată pe nodurile $t^{p},t_{i_{1}i_{2}\ldots i_{p-1}}^{p},\ldots\ldots,t_{i_{1}i_{2}\ldots i_{p-1}}n_{p}+1$ aplicată variabilei $t^{p}$ a funcției

f\left(t_{i_{1}}^{1},t_{i_{1}i_{2}}^{2},\ldots,t_{i_{1}\ldots i_{p-1}}^{p-1},t^{p},t^{p+1},\ldots,t^{s}\right).

6.

Să presupunem că punctele (6) aparțin domeniului $D$ . Dacă se utilizează formula (7) se obține formula de cubatură

\iint\ldots\int_{D}K(M)f(M)dv_{M}=\sum_{i_{1}=1}^{n_{1}+1}\ldots\sum_{i_{\mathrm{s}}=1}^{n_{s}+1}A_{i_{1}\ldots i_{s}}t\left(M_{i_{1}\ldots i_{s}}\right)+\rho_{s}

(9)

unde coeficienții sînt dați de formula

A_{i_{1}\ldots i_{s}}=\iint\ldots\int_{D}K\left(t^{1},\ldots,t^{s}\right)l_{i_{1}}^{1}\left(t^{1}\right)\ldots l_{i_{1}\ldots i_{s}}^{s}\left(t^{s}\right)dt^{1}\ldots dt^{s}

(10)

iar restul e dat de

\rho_{s}=\iint\cdots\int_{D}K(M)R_{s}(M)dM

(11)

7.

Exemplu. Considerînd în cazu1 $s=2$ nodurile

	$\displaystyle M_{1}(-a,-b),M_{2}\left(-a,-\frac{b-c}{2}\right),M_{3}(-a,c)$
	$\displaystyle M_{4}\left(0,-\frac{b+c}{2}\right),M_{6}(0,0),M_{6}\left(0,\frac{b+c}{2}\right)$		(12)
	$\displaystyle M_{7}(a,-c),M_{8}\left(a,\frac{b-c}{2}\right),M_{9}(a,b)$

formula (7) ne conduce la formula de interpolare

		$\displaystyle f(x,y)=\frac{1}{a^{2}(b+c)^{2}}x(x-a)\left(y+\frac{b-c}{2}\right)(y+b)f(-a,c)-$
		$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}x(x-a)(y+b)(y-c)f\left(-a,\frac{c-b}{2}\right)+$
		$\displaystyle+\frac{1}{a^{2}(b+c)^{2}}x(x-a)(y-c)\left(y+\frac{b-c}{2}\right)f(-a,-b)-$
		$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}\left(x^{2}-a^{2}\right)y\left(y+\frac{b+c}{2}\right)f\left(0,\frac{b+c}{2}\right)+$

	$\displaystyle+\frac{4}{a^{2}(b+c)^{2}}\left(x^{2}-a^{2}\right)\left[y^{2}-\frac{(b+c)^{2}}{4}\right]f(0,0)-$
	$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}\left(x^{2}-a^{2}\right)\left(y-\frac{b+c}{2}\right)yf\left(0,-\frac{b+c}{2}\right)+$		(13)
	$\displaystyle+\frac{1}{a^{2}(b+c)^{2}}x(x+a)(y+c)\left(y-\frac{b-c}{2}\right)f(a,b)-$
	$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}x(x+a)(y-b)(y+c)f\left(a,\frac{b-c}{2}\right)+$
	$\displaystyle+\frac{1}{a^{2}(b+c)^{2}}x(x+a)(y-b)\left(y-\frac{b-c}{2}\right)f(a,-c)+r(x,y)$

unde restul are expresia

	$\displaystyle r(x,y)=x\left(x^{2}-a^{2}\right)[-a,0,a,x;f(x,y)]+$
	$\displaystyle+\frac{1}{2a^{2}}x(x-a)(y+b)(y-c)\left(y+\frac{b-c}{2}\right)\cdot\left[c,\frac{c-b}{2},-b,y;f(a,y)\right]-$
	$\displaystyle-\frac{1}{a^{2}}\left(x^{2}-a^{2}\right)y\left(y^{2}-\frac{(b+c)^{2}}{4}\right)\left[\frac{b+c}{2},0,-\frac{b+c}{2},y;f(0,y)\right]+$
	$\displaystyle+\frac{1}{2a^{2}}x(x+a)(y-b)\left(y-\frac{b-c}{2}\right)(y+c)\left[b,\frac{b-c}{2},-c,y;f(-a,y)\right]$		(14)

Luînd ca domeniu de integrare paralelogramul $D$ de vîrfuri

M_{1}(-a,-b),M_{3}(-a,c),M_{7}(a,-c),M_{9}(a,b)

(15)

și făcînd $K(x,y)=1$ , formula de cubatură (9) devine

\displaystyle\begin{array}[]{l}\iint_{D}f(x,y)dxdy=\frac{a}{90(b+c)}\left\{\left(22bc-b^{2}-c^{2}\right)[f(a,b)+f(-a,c)+\right.\\ +f(-a,-b)+f(a,-c)]+8\left(7b^{2}+26bc+7c^{2}\right)f(0,0)+\\ +16\left(2b^{2}+bc+2c^{2}\right)\left[f\left(0,\frac{b+c}{2}\right)+f\left(-a,\frac{c-b}{2}\right)+\right.\\ \left.\left.+f\left(0,-\frac{b+c}{2}\right)+f\left(a,\frac{b-c}{2}\right)\right]\right\}+\rho^{\prime}\end{array}

\mathrm{p}^{\prime}=\iint_{D}r(x,y)dxdy

(17)

Formula de cubatură (16) are gradul parțial de exactitate ( 2,2 ) și gradul global de exactitate egal cu 3.

Dacă facem $d=c$ obținem următoarea formulă de cubatură de grad parțial de exactitate $(3,3)$

	$\displaystyle\iint_{D}f(x,y)dxdy=$
	$\displaystyle=\frac{ab}{9}\{f(a,b)+f(-a,b)+f(-a,-b)+f(a,-b)+4[f(0,b)+$
	$\displaystyle+f(-a,0)+f(0,-b)+f(a,0)]+16f(0,0)\}+\rho$		(18)

care este tocmai formula clasică a lui Cavalieri-Simpson extinsă la două variabile. Domeniul de integrare $D$ este în acest caz dreptunghiul definit de inegalitățile

-a\leqq x\leqq a,\quad-b\leqq y\leqq b

(19)

Restul formulei de cubatură a lui Cavalieri-Simpson e dat de formula

\rho=\iint_{D}R(x,y)dxdy

(20)

unde

	$\displaystyle R(x,y)=x\left(x^{2}-a^{2}\right)[-a,0,a,x;f(x,y)]+$
	$\displaystyle+y\left(y^{2}-b^{2}\right)[-b,0,b,y;f(x,y)]-$		(21)
	$\displaystyle-x\left(x^{2}-a^{2}\right)y\left(y^{2}-b^{2}\right)\left[\begin{array}[]{l}-a,0,\quad a,x\\ -b,0,-b,y\end{array};f(x,y)\right].$

8.

In continuare vom căuta să dăm o evaluare a restului (20).

Se observă că putem scrie

	$\displaystyle\rho=\int_{-a}^{a}\int_{-b}^{b}R(x,y)dxdy=\int_{-a}^{a}\left\{x(x^{2}-a^{2})\left[-a,0,a,x;\int_{-}^{b}f(x,y)dy\mid-\right.\right.$
	$\displaystyle-x\left(x^{2}-a^{2}\right)\left[-a,0,a,x\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b;f(x,y)]dy\mid\right\}dx+(22)$		(22)
	$\displaystyle+\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{\varepsilon}\right)[-b,0,b,y;f(x,y)]dxdy$

Utilizînd formula elementară

\int_{-A}^{A}F(u)du=\int_{0}^{A}[F(t)+F(-t)]dt

(23)

vom putea scrie succesiv

\rho=\int_{0}^{a}\left\{t\left(t^{2}-a^{2}\right)\left[-a,0,a,t,;\int_{-b}^{b}f(t,y)dy\right]-\right.

\begin{gathered}\left.-t\left(l^{2}-a^{2}\right)\mid-a,0,a,t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\cdot[b,0,-b,y;f(t,y)]dy\right]-\\ -t\left(t^{2}-a^{2}\right)\left|a,0,-a,t;\int_{-b}^{b}f(t,y)dy\right|+\\ \left.+t\left(t^{2}-a^{2}\right)\left[a,0,-a,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]\right\}dt+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;/(x,y)]dtdy=\int_{0}^{a}t\left(t^{2}-a^{2}\right)([-a,0,a,t;\\ \left.\left.\int_{-b}^{b}f(t,y)dy\right]-\left[a,0,-a,t;\int_{-b}^{b}f(t,y)dy\right]\right)-t\left(t^{2}-a^{2}\right)([-a,0,a,t;\\ \left.\int_{-b}^{b}y\left(y^{2};-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]-\mid a,0,-a,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\\ \cdot[b,0,-b,y;/(t,y)]dy])\}+\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f(x,y)]dxdy\end{gathered}

Dar în baza formulei de recurență a diferențelor divizate avem

\begin{gathered}\left.\mid-a,0,a,t;\int_{-b}^{b}f(t,y)dy\right]-\left[a,0,-a,-t;\int_{-b}^{b}f(t,y)dy\right]=\\ =2t\left[-a,0,a,t,-t;\int_{-b}^{b}l(t,y)dy\right]\\ {\left[-a,0,a,t;\int_{-b}^{a}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]-}\\ \left.-\mid a,0,-a,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]=\\ =2t\left[-a,0,a,t,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]\end{gathered}

Cu acestea vom putea scrie în continuare

\begin{gathered}\rho=2\int_{0}^{a}t^{2}\left(t^{2}-a^{2}\right)\left\{\left|-a,0,a,t,-t;\int_{-b}^{b}f(t,y)dy\right|-\right.\\ -\left\{-a,0,a,t,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\mid\right\}dt+\\ +\int_{-a}^{\prime\prime}\int_{-b}^{b}y\left(y^{2}-b^{-}\right)[-b,0,b,y,;f(x,y)]dxdy\end{gathered}

6 - Studii și cercetări

Deoarece $t^{2}\left(t^{2}-a^{2}\right)$ păstrează un semn constant în intervalul de integrare, putem aplica formula mediei și găsim

\begin{gathered}\rho=2\left\{\left[-a,0,a,\xi^{\prime},-\xi^{\prime};\int_{-b}^{b}f\left(\xi^{\prime},y\right)dy\right]-\left[-a,0,a,\xi^{\prime},-\xi^{\prime}\right.\right.\\ \left.\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[b,0,-b,y;f\left(\xi^{\prime},y\right)\right]dy\right\}\int_{0}^{a}t^{2}\left(t^{2}-a^{2}\right)dt+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f]dxdy=-\frac{2.2a^{5}}{4!15}\left(\int_{-b}^{b}\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}dy-\right.\\ \left.-\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[b,0,-b,y;\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}\right]dy\right)+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f]dxdy=-\frac{a^{5}}{90}\int_{-b}^{b}\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}dy+\\ +\frac{a^{5}}{90}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[b,0,-b,y;\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}\right]dy+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f]dxdy=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\\ +\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[-b,0,b,y;\int_{-a}^{a}f(x,y)dx+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}\right]dy\end{gathered}

Aplicînd iarăşi formula (23), primim succesiv

		$\displaystyle\rho=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\int_{0}^{b}\left(\tau(\tau^{2}-b^{2})\left[-b,0,b,\tau;\int_{-a}^{a}f(x,\tau)dx+\right.\right.$
		$\displaystyle\left.\left.+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]-\tau\left(\tau^{2}-b^{2}\right)\left[b,0,-b,-\tau;\int_{-a}^{a}f(x,\tau)dx+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]\right)d\tau=$
		$\displaystyle=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\int_{0}^{b}\tau\left(\tau^{2}-b^{2}\right)\left(\left[-b,0,b,\tau;\int_{-a}^{a}f(x,\tau)dx+\right.\right.$
		$\displaystyle\left.\left.+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]-\left[-b,0,b,-\tau;\int_{-a}^{a}f(x,\tau)dx+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]\right)d\tau=$
		$\displaystyle=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+2\int_{0}^{b}\tau^{2}\left(\tau^{2}-b^{2}\right)\left[-b,0,b,\tau,-\tau;\int_{-a}^{a}f(x,\tau)dz+\right.$
		$\displaystyle\left.+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]d\tau=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+2\left[-b,0,b,\eta^{\prime},-\eta^{\prime};\int_{-a}^{a}f\left(x,\eta^{\prime}\right)dx+\right.$

	$\displaystyle+$	$\displaystyle\left.\frac{a^{5}}{90}\frac{\partial^{4}f\left(\xi,\eta^{\prime}\right)}{\partial\xi^{4}}\right]\int_{-b}^{b}\tau^{2}\left(\tau^{2}-b^{2}\right)d\tau=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}-$
		$\displaystyle-\frac{4b^{5}}{15}\cdot\frac{1}{24}\int_{-a}^{a}\frac{\partial^{4}f(x,\eta)}{\partial\eta^{4}}dx-\frac{4a^{5}b^{5}}{15\cdot 24\cdot 90}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}=$
		$\displaystyle=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}-\frac{ab^{5}}{45}\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\partial\eta^{4}}-\frac{a^{5}b^{5}}{90^{2}}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}$

Dacă introducem o natatie simbolică dată de J. F. Steffensen [6] și mai nou utilizată mult de Ş. E. Mikel a d z e [1]

\frac{\partial^{p}f\left(\xi,\eta_{1}\right)}{\partial\xi^{p}}=D_{\xi}^{p},\frac{\partial^{q}f\left(\xi_{1},\eta\right)}{\partial\eta^{q}}=D_{\eta}^{q},\frac{\partial^{p+q}f(\xi,\eta)}{\partial\xi^{p}\eta^{q}}=D_{\xi}^{p}D_{\xi}^{q}

(24)

restul (20) se va scrie în definitiv astfel

\rho=-\frac{ab}{45}\left[a^{4}D_{\xi}^{4}+b^{4}D_{\eta}^{4}+\frac{a^{4}b^{4}}{180}D_{\xi}^{4}D_{\eta}^{4}\right]

(25)

Observattii. $1^{0}$ . Restul dat de Ș. E. Mike1adze la pag. 491 a lucrării [1] trebuie rectificat întrucît în loc de factorul $\frac{1}{180}$ care multiplică derivata $D_{\xi}^{4}D_{\eta}^{4}$ , în formula sa figurează $\frac{1}{45}$ ; această inexactitate provine din formula mai generală care o precede pe aceasta.
$2^{0}$ . Formula (18), cu restul (25) a fost dată în cazul patratului cu centrul în origine şi cu latura unu de către J. F. Steffensen [6], însă fără să se arate că $\xi$ și $\eta$ care intervin mai sus sînt aceiaşi.

§ 2. Unele formule practice de cubatură pentru integralele $s$ -uple

9.

Pentru formulele pe care le vom da va fi util să introducem un operator $S_{m}$ definit astfel

	$\displaystyle S_{i}\varphi(M)=S_{i}\varphi\left(t^{1},\ldots,\mathrm{t}^{i-1},t^{i},t^{i+1},\ldots,t^{s}\right)=$		(26)
	$\displaystyle=\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}+h_{i}t^{i},t^{i+1},\ldots,t^{s}\right)+\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}-h_{i}t^{i},t^{i+1},\right.$

Oricare sînt numerele naturale $i,k,(i<k;i,k\leqq s)$ , se stabileşte imediat că avem

	$\displaystyle S_{i}\left(S_{k}\varphi\right)=S_{k}\left(S_{i}\varphi\right)=$
	$\displaystyle\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}+h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}+h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)+$
	$\displaystyle+\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}-h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}+h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)+$		(27)
	$\displaystyle+\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}+h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}-h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)+$
	$\displaystyle+\rho\left(t^{1},\ldots,t^{i-1},t_{0}^{i}-h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}-h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)$

10.

Ne vor si utile dovă lence inportante.

Lema 1. Relativ la iunctia $/(M)$ si la nodurile $M_{i_{1}\ldots i_{s}}\left(t_{i_{1}}^{1},l_{i_{2}}^{2},\ldots,l_{i_{s}}^{s}\right)$ a căror coordonată de ordinal $\hbar$ par urge valorile

	$\displaystyle t_{0}^{k}-p_{k}h_{k},\ldots,t_{0}^{k}-h_{k},t_{0}^{k},l_{0}^{k}+h_{k},\ldots,l_{0}^{k}+p_{k}h_{k}$		(28)
	$\displaystyle(k=1,2,\ldots,s)$

avem formula de interpolare

	$\displaystyle\frac{1}{2^{s}}S_{1}S_{2}\ldots S_{s}f(M)=\frac{(-1)^{N}}{P}Uf\left(M_{0}\right)+$
	$\displaystyle+\sum_{i=1}^{s}\frac{U_{i}}{P_{i}}\sum_{j_{i}=1}^{p_{i}}(-1)^{N-j_{i}}\frac{v_{j_{i}}^{i}\left(t^{i}\right)}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!}S_{j_{i}}^{0,i_{i}}f\left(M_{0}\right)+$
	$\displaystyle+\sum_{i=1}^{s-1}\sum_{\begin{subarray}{c}k=2\\ (i<k)\end{subarray}}^{s}\frac{U_{i,k}}{P_{i,k}}\sum_{j_{i}=1}^{p_{i}}\sum_{j_{k}=1}^{p_{k}}(-1)^{N-j_{i}-j_{k}}\frac{v^{i}{}_{ji}\left(t^{i}\right)}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!\left(p_{k}-j_{k}\right)!\left(p_{k}+j_{k}\right)!}$		(29)
	$\displaystyle\text{ - }S_{j_{i}}^{0,i}S_{j_{k}}^{0,k}f(Mo)$
	$\displaystyle+\sum_{j_{1}=1}^{p_{1}}\cdots\sum_{j_{s}=1}^{p_{s}}(-1)^{N-j_{1}-\cdots-j_{s}}\frac{v^{1}{}_{j_{1}}\left(l^{1}\right)}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!}\cdots\frac{v_{j_{s}}^{s}\left(l^{s}\right)}{\left(p_{s}-j_{s}\right)!\left(p_{s}+j_{s}\right)!}.$
	$\displaystyle\text{ - }S_{j_{1}}^{0,1}\ldots S_{j_{s}}^{0,s}f\left(M_{0}\right)+R_{12\ldots s}(f;M)\text{, }$

unde

	$\displaystyle\left\{\begin{array}[]{l}U=u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)\ldots u_{s}\left(t^{s}\right)\\ U_{i}=u_{1}\left(t^{1}\right)\ldots u_{i-1}\left(t^{i-1}\right)u_{i+1}\left(t^{i+1}\right)\ldots u_{s}\left(t^{s}\right)\\ U_{ik}=u_{1}\left(t^{1}\right)\ldots u_{i-1}\left(t^{i-1}\right)u_{i+1}\left(t^{i-1}\right)\ldots u_{k-1}\left(t^{k-1}\right)u_{k+1}\left(t^{k+1}\right)\ldots u_{s}\left(t^{s}\right)\\ U_{12}\cdots s=1;\end{array}\right.$		(30)
	$\displaystyle\left\{\begin{array}[]{l}u_{i}\left(t^{i}\right)=\left[\left(t^{i}\right)^{2}-1^{2}\right]\left[\left(t^{i}\right)^{2}-2^{2}\right]\ldots\left[\left(t^{i}\right)^{2}-p_{i}{}^{2}\right]\\ v_{ji}\left(t^{i}\right)=\left(t^{i}\right)^{2}\left[\left(t^{i}\right)^{2}-1^{2}\right]\ldots\left[\left(t^{i}\right)^{2}-\left(j_{i}-1^{2}\right)\right]\left[\left(t^{i}\right)^{2}-\left(i_{i}+1\right)^{2}\ldots\left[\left(t^{j}\right)^{2}-p_{i}{}^{2}\right];\right.\end{array}\right.$
	$\displaystyle\left\{\begin{array}[]{l}\mathrm{P}=\left(p_{1}!\right)^{2}\left(p_{2}!\right)^{2}\ldots\left(p_{s}!\right)^{2}\\ \mathrm{P}_{i}=\left(p_{1}!\right)^{2}\ldots\left(p_{i-1}!\right)^{2}\left(p_{i+1}!\right)^{2}\ldots\left(p_{s}!\right)^{2}\\ \mathrm{P}_{ik}=\left(p_{1}!\right)^{2}\ldots\left(p_{i-1}!\right)^{2}\left(p_{i+1}!\right)^{2}\ldots\left(p_{k-1}!\right)^{2}\left(p_{k+1}!\right)^{2}\ldots\left(p_{s}!\right)^{2}\\ \mathrm{P}_{12}\ldots s=1;\end{array}\right.$

	$\displaystyle N=p_{1}+p_{2}+\ldots+p_{s};$		(33)
	$\displaystyle S_{j_{i}}^{0,i}f\left(M_{0}\right)=f\left(t_{0}^{1},\ldots,t_{o}^{i-1},t_{o}^{i}+j_{i}h_{i},t_{o}^{i+1},\ldots,t_{o}^{s}\right)+f\left(t_{o}^{1},\ldots,t_{o}^{i-1},t_{o}^{i}-j_{i}h_{i},t_{o}^{i+1},\ldots,t_{o}^{s}\right).$

Restul are următoarea expresie

R_{12}\ldots s(f;M)=\frac{1}{2s}\rho_{12}\ldots s(f;M)

	$\displaystyle\rho_{12}\cdots s(j;M)=2\sum_{i=1}^{s}h_{i}^{2p_{i}+2}\left(t^{i}\right)^{2}u_{i}\left(t^{i}\right)\left[t_{o}^{i},l_{o}^{i}\pm h_{i},\ldots,t_{o}^{i}\pm p_{i}h_{i},l_{o}^{i}\pm h_{i}l^{i};F_{i}\right]-$
	$\displaystyle-4\sum_{\begin{subarray}{c}i=1\\ (i<k)\end{subarray}}^{s-1}\sum_{k=2}^{s}h_{i}^{2p_{i}+2}h_{k}^{2p_{k}+2}\left(t^{i}\right)^{2}\left(t^{k}\right)^{2}u_{i}\left(t^{i}\right)u_{k}\left(l^{k}\right).$		(35)

		$\displaystyle+\quad\left[\begin{array}[]{l}t_{o}^{i},t_{o}^{i}\pm h_{i},\ldots,t_{o}^{i}\pm p_{i},h_{i},t_{o}^{i}\pm h_{i}t^{i}\\ t_{o}^{k},t_{o}^{k}\pm h_{k},\ldots,t_{o}^{k}\pm p_{k}h_{k},t_{o}^{k}\pm h_{k}t^{k}\end{array}\right]F_{ik}$
		$\displaystyle+(1)^{s-1}h_{1}^{2p_{1}+2}\ldots h_{s}^{2p_{s}+2}\left(t^{1}\right)^{2}\ldots\left(t^{s}\right)^{2}u_{1}\left(t^{1}\right)\ldots u_{s}\left(t^{s}\right)\ldots$
		$\displaystyle\quad\left[\begin{array}[]{l}t_{o}^{1},t_{o}^{1}\pm h_{1},\ldots,t_{o}^{1}\pm p_{1}h_{1},t_{o}^{1}\pm h_{1}t^{1}\\ \cdots\\ t_{o}^{s},t_{o}^{s}\pm h_{s},\ldots,t_{o}^{s}\pm p_{s}h_{s},t_{o}^{s}\pm h_{n}t^{s}\end{array}\right]$
		unde
		$\displaystyle\begin{array}[]{l}F_{i}=S_{1}S_{2}\ldots S_{i-1}S_{i+1}\ldots S_{s}f(M)\\ F_{ik}=S_{1}\ldots S_{i-1}S_{i+1}\ldots S_{k-1}S_{k+1}\ldots S_{s}f(M)\\ F_{12}\ldots s=f(M)\end{array}$

Pentru demonstratetie se consideră formula de interpolare a lui Lagrange pentru s valabile, cu restul sub forma dată de J. F. Steffensen [6]. Interpolarea se face peo retea hiperparalelipipedică cu coordonatele nodurilor echidistante. Că această formulă se poate aduce la forma (29) se poate demonstra prin inductie completă ¹ ).

Lema 2. - Oricare ar fi funcția $f(M)$ integrabilă în hiperparalelipipedul

t_{0}^{i}-mh_{i}\leqq t^{i}\leqq t_{0}^{i}+m_{i}h_{i},(i=\overline{1,s})

(37)

avem formula

\int l_{0}^{1}+m_{1}h_{1}\\ \ldots\\ l_{0}^{1}-m_{1}h_{1}\int_{l_{0}^{s}-m_{s}h_{s}}^{l_{0}^{s}+m_{s}h_{s}}f(M)dM=h_{1}\ldots h_{s}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}S_{1}\ldots S_{s}f(M)dM.

Demonstrația se face de asemenea prin inducție completă asupra lui $s$ .

⁰⁰footnotetext: ¹⁾ Vezi lucrarea [4].

11. Bazați pe aceste leme vom putea enunța următoarea
teoremå: Relativ la funcția $f(M)$ şi la nodurile pe care le foloseste formula (29) avem formula de cubatură

$\displaystyle\iint\cdots$	$\displaystyle\int_{D}f(M)dM=\int_{t_{0}^{1}-m_{1}h_{1}}^{t_{0}^{1}+m_{1}h_{1}}\int_{t_{0}^{s}-m_{s}h_{s}}^{t_{0}^{s}+m_{s}h_{s}}f(M)dM=$
$\displaystyle=$	$\displaystyle h_{1}\ldots h_{s}\left[A_{0}^{0}f\left(M_{0}\right)+\sum_{i=1}^{s}\sum_{j_{i}=1}^{p_{i}}A_{j_{i}}^{i}S_{j_{i}}^{0,i}f\left(M_{0}\right)+\right.$	(35)
	$\displaystyle+\sum_{i=1}^{s-1}\sum_{\begin{subarray}{c}k=2\\ (i<k)\end{subarray}}^{s}\sum_{j_{k}=1}^{p_{i}}\sum_{j_{k}=1}^{p_{k}}A_{j_{i}j_{k}}^{i,k}S_{j_{i}}^{0,i}S_{j_{k}}^{0,k}f\left(M_{0}\right)+$

	$\displaystyle\frac{1}{2^{s}}A_{0}^{0}=(-1)^{N}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\frac{U}{P}dM$
	$\displaystyle\frac{1}{2^{s}}A_{j_{i}}^{i}=\frac{(-1)^{N-j_{i}}}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\frac{U_{i}}{P_{i}}v_{j_{i}}^{i}dM$
	$\displaystyle\frac{1}{2^{s}}A_{j_{i}j_{k}}^{i,k}=\frac{(-1)^{N-j_{i}-j_{k}}}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!\left(p_{k}-j_{k}\right)!\left(p_{k}+j_{k}\right)!}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\frac{U_{ik}}{P_{ik}}v_{j_{i}}^{i}v_{j_{k}}^{k}dM$		(40)
	$\displaystyle\frac{1}{2^{s}}A_{j_{1}j_{2}\cdots j_{s}}^{1,2,\ldots,s}=\frac{(-1)^{N-j_{1}-\cdots-j_{s}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\ldots!\left(p_{s}-j_{s}\right)!\left(p_{s}+j_{s}\right)!}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}v_{j_{1}}^{1}\ldots v_{j_{s}}^{s}dM$

iar

r_{s}=h_{1}\ldots h_{s}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\rho_{12}\cdots_{s}(f;M)dM

(41)

Relativ la rest vom demonstra următoarea
teoremă: Dacă în domeniul $D$ funcția $f(M)$ e continuă împreună $cu$ derivatele sale partiale de ordinul $\left(2p_{1}+2,2p_{2}+2,\ldots,2p_{s}+2\right)$ , atunci pentru restul (41) se obține evaluarea

\frac{1}{2^{s}}r_{s}=\sum_{i=1}^{s}\frac{N_{i}h_{1}\ldots h_{i-1}h_{i}^{2p_{i}+3}h_{i+1}\ldots h_{s}}{\left(2p_{i}+2\right)!}B_{i}D_{\xi_{i}}^{2p_{i}+2}-

	$\displaystyle-\sum_{\begin{subarray}{c}i=1\\ (i<k)\end{subarray}}^{s-1}\sum_{k=1}^{s}\frac{N_{ik}h_{1}\ldots h_{i-1}h_{i}^{2p_{i}+3}h_{i+1}\ldots h_{k-1}h_{k}^{2p_{k}+3}h_{k+1}\cdots h_{s}}{\left(2p_{i}+2\right)!\left(2p_{k}+2\right)!}B_{i}B_{k}D_{\mathrm{\xi}_{i}}^{2p_{i}+2}D_{\mathrm{\xi}_{k}}^{2p_{k}+3}$
	$\displaystyle\quad+\ldots\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot h_{s}^{2p_{s}+3}$		(42)
	$\displaystyle+(-1)^{s-1}\frac{h_{1}^{2p_{1}+3}}{\left(2p_{1}+2\right)!\ldots\left(2p_{s}+2\right)!}B_{1}B_{2}\ldots B_{s}D_{\mathrm{\xi}_{1}}^{2p_{1}+2}\cdots D_{\mathrm{\xi}_{s}}^{2p_{s}+2}$

unde

	$\displaystyle N_{i}=m_{1}\ldots m_{i-1}m_{i+1}\ldots m_{s}$
	$\displaystyle N_{ik}=m_{1}\ldots m_{i-1}m_{i+1}\ldots m_{k-1}m_{k+1}\ldots m_{s}$		(43)
	$\displaystyle\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$
	$\displaystyle N_{12}\ldots s=1$

și

B_{i}=\int_{0}^{m_{i}}\left(t^{i}\right)^{2}u_{i}\left(t^{i}\right)dt^{i}

(44)

Demonstrația o vom da în cazul $s=3$ ; în cazul lui $s$ oarecare se va proceda exact la fe1, t,inînd numai seama de formula (35) și de lema 2.

In cazul lui $E_{3}$ restul (35) se scrie explicit astfel

	$\displaystyle\rho_{123}(f;M)=2h_{1}^{2p_{1}+2}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]$		(45)
	$\displaystyle+2h_{2}^{2p_{2}+2}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)\left[t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}-h_{2}t^{2};F_{2}\right]$
	$\displaystyle+2h_{3}^{2p_{3}+2}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)\left[t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3};F_{0}\right]$
	$\displaystyle-4h_{1}^{2p_{1}+2}h_{2}^{2p_{2}+2}\left(t^{1}t^{2}\right)^{2}u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)\left[\begin{array}[]{l}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1}\\ t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}\pm h_{2}t^{2};F_{12}\end{array}\right]$
	$\displaystyle-4h_{1}^{2p_{1}+2}h_{3}^{2p_{3}+2}\left(t^{1}t^{3}\right)^{2}u_{1}\left(t^{1}\right)u_{3}\left(t^{3}\right)\left[\begin{array}[]{l}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{13}\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3}\end{array}\right]$
	$\displaystyle-4h_{2}^{2p_{2}+2}h_{3}^{2p_{3}+2}\left(t^{2}t^{3}\right)^{2}u_{2}\left(t^{2}\right)u_{3}\left(t^{3}\right)\left[\begin{array}[]{l}t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}+h_{2}t^{2}\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3}\end{array}\right]$
	$\displaystyle+8h_{1}^{2p_{1}+2}h_{2}^{2p_{2}+2}h_{3}^{2p_{3}+2}\left(t^{1}t^{2}t^{3}\right)^{2}u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)u^{3}\left(t^{3}\right)\times$
	$\displaystyle\times\left[\begin{array}[]{l}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1}\\ t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}\pm h_{2}t^{2};F_{123}\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3}\end{array}\right]$

unde

		$\displaystyle F_{1}=S_{2}S_{3}f\left(t^{1},l^{2},t^{3}\right)=f\left(t^{1},l_{0}^{2}+h_{2}t^{2},t_{0}^{3}+h_{3}t^{3}\right)+f\left(t^{1},t_{0}^{2}-h_{2}t^{2},t_{0}^{3}+h_{3}t^{3}\right)+f\left(t^{1},t_{0}^{2}+\right.$
		$\displaystyle\left.+h_{2}t^{2},t_{0}^{3}-h_{3}t^{3}\right)+f\left(t^{1},t_{0}^{2}-h_{2}t^{2},t_{0}^{3}-h_{3}t^{3}\right)$
		$\displaystyle\quad F_{2}=S_{1}S_{3}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{3}=S_{1}S_{2}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{12}=S_{3}f\left(t^{1},t^{2},t^{3}\right)=f\left(t^{1},t^{2},t_{0}^{3}+h_{3}t^{3}\right)+f\left(t^{1},t^{2},t_{0}^{3}-h_{3}t^{3}\right)$
		$\displaystyle\quad F_{13}=S_{2}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{23}=S_{1}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{123}=f\left(t^{1},t^{2},t^{3}\right).$

Aplicînd teorema mediei integralelor triple și ținînd seama că $/(M)$ admite în ( $D$ ) derivate partiale de ordinul ( $2p_{1}+2,2p_{2}+2,2p_{3}+2$ ) continue, putem scrie

		$\displaystyle\quad r_{3}^{1}=-h_{1}^{2p_{1}+3}h_{2}h_{3}\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}+h_{1}\xi_{1},t_{0}^{1}+h_{1}\xi_{1};F^{\prime}\right]_{1}+\right.$
		$\displaystyle\left.+\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}-h_{1}\xi_{1},t_{0}^{1}-h_{1}\xi_{1};F_{1}^{\prime}\right]\right)\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{2}}dt^{3}\int_{-m_{1}}^{0}Q_{1}\left(t^{1}\right)dt^{1}$

\begin{gathered}F_{1}^{\prime}=f\left(\xi_{1},t_{0}^{2}+h_{2}\eta_{11},t_{0}^{3}+h_{3}\zeta_{1}\right)+f\left(\xi_{1},t_{0}^{2}-h_{2}\eta_{1},t_{0}^{3}+h_{3}\zeta_{1}\right)+f\left(\xi_{1},t_{0}^{2}+\right.\\ \left.+h_{2}\eta_{1},t_{0}^{3}-h_{3}\zeta\right)+f\left(\xi_{1},t_{0}^{2}-h_{2}\eta_{1},t_{0}^{3}-h_{3}\zeta_{1}\right)\end{gathered}

Să calculăm acum integrala

I_{3}\left(\rho_{123}\right)=h_{1}h_{2}h_{3}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}\rho\left(t^{1},t^{2},t^{3}\right)dt^{1}dt^{2}dt^{3}

(46)

Vom evalua mai întîi această integrală pentru primul termen al restului (45) ; primim

		$\displaystyle r_{3}^{1}=2h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]dt^{1}dt^{2}dt^{3}$
		$\displaystyle=2h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{3}}dt^{3}\int_{0}^{m_{1}}t^{1}\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]\right)dQ_{1}\left(t^{1}\right)$

unde

Q_{1}\left(t^{1}\right)=\int_{-m_{1}}^{l^{1}}t^{1}u_{1}\left(t^{1}\right)dt^{1}

este o funcție care conform studiului făcut de J. F. Steffensen [6] ² ) păstrează un semn constant în intervalul $\left[-m_{1},0\right]$ .

Integrînd prin părți găsim

\begin{array}[]{r}r_{3}^{1}=-2h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{3}}dt^{3}\int_{-m_{1}}^{0}Q_{1}\left(t^{1}\right)\frac{\partial}{\partial t^{1}}\left\{t^{1}\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm\right.\right.\\ \left.\pm h_{1},t^{1};F_{1}\right]dt^{1}\end{array}

	$\displaystyle{\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]}$	$\displaystyle=\frac{1}{2h_{1}t^{1}}\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}+\right.\right.$
	$\displaystyle\left.+h_{1}t^{1};F_{1}\right]$	$\displaystyle\left.+\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}-h_{1}t^{1};F_{1}\right]\right)$

astfel că vom putea scrie în continuare

	$\displaystyle r_{3}^{1}=$	$\displaystyle-h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{3}}dt^{3}\int_{-m_{1}}^{0}Q_{1}\left(t^{1}\right)\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}+\right.\right.$
		$\displaystyle\left.\left.+h_{1}t^{1},t_{0}^{1}+h_{1}t^{1};F_{1}\right]+\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}-h_{1}t^{1}t_{0}^{1}-h_{1}t^{1};F_{1}\right]\right)dt^{1}$

⁰⁰footnotetext: ² ) pag. 155. Vezi de asemenea şi S. E. Mike1adze [1], pag, 312. ⁰⁰footnotetext: ³⁾ Vezi [4].

Cu acestea

	$\displaystyle r_{3}^{4}=-$	$\displaystyle\frac{8m_{3}h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}}{\left(2p_{2}+2\right)!}\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)dt^{2}\int_{-m_{1}}^{0}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)$
		$\displaystyle\cdot\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};D_{\xi_{2}}^{2p_{2}+2}\right]dt^{1}$

Și acum, folosind iar evaluarea dată la primul termen al restului, se găseşte că

r_{3}^{4}=\frac{-8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}m_{3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}\int_{0}^{m_{1}}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)dt^{1}\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)dt^{2}

In mod analog se obțin

		$\displaystyle r_{3}^{5}=\frac{-8h_{1}^{2p_{1}+3}h_{2}h_{3}^{2p_{3}+3}m_{2}}{\left(2p_{1}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{1}}\left(t^{1}\right)^{2}u_{2}\left(t^{1}\right)dt^{1}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}$
		$\displaystyle r_{3}^{6}=\frac{-8h_{1}h_{2}^{2p_{3}+3}h_{3}^{2p_{3}+3}m_{1}}{\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{2}}^{3p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{1}\left(t^{2}\right)dt^{2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}$

A1 șaptelea și ultimul termen al restului este

	$\displaystyle\gamma_{3}^{7}=$	$\displaystyle 8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{2}}\left(t^{1}t^{2}t^{3}\right)^{2}u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)u_{3}\left(t^{3}\right)$
		$\displaystyle\cdot\left[\begin{array}[]{ll}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},&t_{0}^{1}\pm h_{1}t^{1}\\ t_{0}^{2},&t_{0}^{2}\pm h_{2},\ldots,\\ t_{0}^{2}\pm p_{2}h_{2},&t_{0}^{2}\pm h_{2}t^{2};/\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},&t_{0}^{3}\pm h_{3}t^{3}\end{array}\right].dt^{1}dt^{2}dt^{3}$

In baza celor precedente avem

\begin{gathered}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)\left[t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3};f\right]dt^{3}=\\ =\frac{1}{\left(2p_{3}+2\right)!}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}\end{gathered}

și

\begin{gathered}\frac{1}{\left(2p_{3}+2\right)!}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}\int_{0}^{m_{3}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)\left[t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}\pm\right.\\ \left.\quad\pm h_{2}t^{2};D_{\xi_{3}}^{2p_{3}+2}\right]dt^{2}=\frac{1}{\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}\times\\ \times\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)dt^{2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}\end{gathered}

Astfel că

\begin{gathered}r_{3}^{7}=\frac{8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{1}}\left(t^{1}\right)^{2}u_{1}dt^{1}\times\\ \times\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}dt^{2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}dt^{3}\end{gathered}

Aşadar în $E_{3}$ restul formulei de cubatură (39) poate fi exprimat prin formula

$\displaystyle\gamma_{3}$	$\displaystyle=\gamma_{3}^{1}+\ldots+\gamma_{3}^{7}=\frac{8m_{2}m_{3}h_{1}^{2p_{1}+3}h_{2}h_{3}}{\left(2p_{1}+2\right)!}B_{1}D_{\xi_{1}}^{2p_{1}+2}+$
$\displaystyle+$	$\displaystyle\frac{8m_{1}m_{3}h_{1}h_{2}^{2p_{2}+3}h_{3}}{\left(2p_{2}+2\right)!}B_{2}D_{\xi_{2}}^{2p_{2}+2}+\frac{8m_{1}m_{2}h_{1}h_{2}h_{3}^{2p_{3}+3}}{\left(2p_{3}+2\right)!}B_{3}D_{\xi_{3}}^{2p_{3}+2}-$
	$\displaystyle-\frac{8m_{3}h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!}B_{1}B_{2}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}-$
	$\displaystyle-\frac{8m_{2}h_{1}^{2p_{1}+3}h_{2}h_{3}^{2p_{3}+3}}{\left(2p_{1}+2\right)!\left(2p_{3}+2\right)!}B_{1}B_{3}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{3}}^{2p_{3}+2}-$	(47)
	$\displaystyle-\frac{8m_{1}h_{1}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}}{\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}B_{2}B_{3}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}+$
$\displaystyle+$	$\displaystyle\frac{8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}B_{1}B_{2}B_{3}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}$

unde

B_{i}=\int_{0}^{m_{i}}\left(t^{i}\right)^{2}u_{i}\left(t^{i}\right)dt^{i}

12.

O categorie importantă de formule de cubatură se obține din (39) dacă se ia $m_{i}=p_{i},i=\overline{1,s}$ . Asemenea formule le vom numi, împreună cu I. F. Steffensen [6] și S. E. Mikeladze [1], formule de tip închis.

In cazul cînd limitele integralei a $i-a(i=\overline{1,s})$ sînt în afara intervalului ( $t_{0}^{i}-p_{i}h_{i},t_{0}^{i}+p_{i}h_{i}$ ), cu alte cuvinte $m_{i}>p_{i}(i=\overline{1,s})$ , obținem aşa numitele formule de tip deschis.

Și, în sfîrşit, dacă $m_{i}<p_{i}$ , obținem formulele de cubatură cu noduri aşezate în afara domeniului de integrare.

§ 3. Cazuri particulare importante ale formulelor de cubatură precedente

13.

Vom considera acum anumite cazuri particulare, care ni se par mai interesante, ale formulei (39).

In cazul $s=1$ majoritatea formulelor care se obtin au fost date de către I. F. Steffensen [6], S. E. Mikeladze [1], W. E. Milne [2], etc.

Schimbînd puțin notațiile, formula (39) în cazul $s=1$ devine

	$\displaystyle\int_{x_{0}-mh}^{x_{0}+mh}f(x)dx=h\left[A_{0}^{0}f\left(x_{0}\right)+\sum_{j=1}^{p}A_{j}^{1}S_{j}^{0}f\left(x_{0}\right)\right]+r_{1}=$		(48)
	$\displaystyle=h\left[A_{0}^{0}f\left(x_{0}\right)+\sum_{j=1}^{p}A_{j}^{1}\left(f\left(x_{0}+jh\right)+f\left(x_{0}-jh\right)\right)\right]+r_{1}$

unde

	$\displaystyle\frac{1}{2}A_{0}^{0}=\frac{(-1)^{p}}{(p!)^{2}}\int_{0}^{m}\left(x^{2}-1\right)\ldots\left(x^{2}-p^{2}\right)dx$
	$\displaystyle\frac{1}{2}A_{j}^{0}=\frac{(-1)^{p-j}}{(p-j)!(p+j)!}\int_{0}^{m}x^{2}\left(x^{2}-1\right)\ldots\left(x^{2}-i-1^{2}\right)$		(52)

şi

\frac{1}{2}r_{1}=\frac{h^{2p+3}}{(2p+2)!}D_{\xi}^{2p+2}\int_{0}^{m}x^{2}\left(x^{2}-1\right)\ldots\left(x^{2}-p^{2}\right)dx

14.

Dacă se ia în (48) $p=0$ se obtine formula de cuadratură cu un nod

\int_{x_{0}-mh}^{x_{0}+mh}f(x)dx=2mhf\left(x_{0}\right)+\frac{m^{3}h^{3}}{3}f^{\prime\prime}(\xi)

(49)

iar dacă facem $p=1$ se găsește formula cu 3 noduri

	$\displaystyle\int_{x_{0}-mh}^{x_{0}+mh}f(x)dx=\frac{mh}{3}\left[2\left(3-m^{2}\right)f\left(x_{0}\right)+m^{2}f\left(x_{0}+h\right)+f\left(x_{0}-h\right)\right]$		(50)
	$\displaystyle+\frac{m^{3}\left(3m^{2}-5\right)}{180}h^{5}f(IV)(\xi)$

Pentru $m=1$ se obține o binecunoscută formulă de tip închis : formula lui Cavalieri-Simpson.

Pentru $m=3,p=2$ se găseşte formula de cuadratură de tip deschis

	$\displaystyle\int_{x_{0}-3h}^{x_{0}+3h}f(x)dx=\frac{h}{30}\left[234f\left(x_{0}\right)-126\overline{f\left(x_{0}+h\right)+f\left(x_{0}-h\right)}+\right.$		(54)
	$\displaystyle\left.\quad+99\overline{f\left(x_{0}+2h\right)+f\left(x_{0}-2h\right)}\right]+\frac{41}{140}h^{7}f(VI)(\xi)$

de grad de exactitate 5.
15. In cazul $\therefore=2$ formula de cubatură (39) se scrie

	$\displaystyle\int_{x_{0}-mh}^{x_{0}+mh}dx\int_{y_{0}-nk}^{y_{0}+nk}f(x,y)dy=hk\left[A_{0}^{0}f\left(x_{0},x_{0}\right)+\sum_{i=1}^{p}A_{i}^{1}S_{i}^{0,1}f\left(x_{0},y_{0}\right)+\right.$
	$\displaystyle\left.+\sum_{j=1}^{q}A_{j}^{2}S_{j}^{0,2}f\left(x_{0},y_{0}\right)+\sum_{i=1}^{p}\sum_{j=1}^{q}A_{i,j}^{1,2}S_{i}^{0,1}S_{j}^{0,2}f\left(x_{0},y_{0}\right)\right]+r_{2},$		(51)

sau mai explicit

\int_{x_{u}-mh}^{x_{0}+mh}dx\int_{y_{0}-nk}^{y_{0}+nk}f(x,y)dy=

		$\displaystyle=hk\left[A_{0}^{0}f\left(x_{0},y_{0}^{\prime}\right)+\sum_{i=1}^{p}A_{i}^{1}\left(f\left(x_{0}+ih,y_{0}\right)+f\left(x_{0}-ih,y_{0}\right)\right)+\right.$
		$\displaystyle\quad+\sum_{j=1}^{q}A_{j}^{2}\left(f\left(x_{0},y_{0}+jk\right)+f\left(x_{0},y_{0}-jk\right)\right)+$
		$\displaystyle+\sum_{i=1}^{p}\sum_{j=1}^{q}A_{i,j}^{1,2}\left(f\left(x_{0}+ih,y_{0}+jk\right)+f\left(x_{0}-ih,y_{0}+jk\right)+\right.$
		$\displaystyle\left.\left.\quad+f\left(x_{0}+ih,y_{0}-jk\right)+i\left(x_{0}-ih,y_{0}-jk\right)\right)\right]+r_{2},$

unde

\begin{array}[]{r}\frac{1}{4}A_{0}^{0}=\frac{(-1)^{p+q}}{(p!)^{2}(q!)^{2}}\int_{0}^{m}\int_{0}^{n}u(x)v(y)dxdy^{\prime}\\ \frac{1}{4}A_{i}^{1}=\frac{(-1)^{p+q-i}}{(p-i)!(p+i)!(q!)^{2}}\int_{0}^{m}\int_{0}^{n}v(y)a_{i}(x)dxdy\\ \frac{1}{4}A_{j}^{2}=\frac{(-1)^{p+q-j}}{(p!)^{2}(q-j)!(q+j)!}\int_{0}^{m}\int_{0}^{n}u(x)b_{j}(y)dxdy\\ \frac{1}{1}A_{i,j}^{1,2}=\frac{(-1)^{p+q-i-j}}{(p-i)!(p+i)!(q-j)!(q+j)!}\int_{0}^{m}\int_{0}^{n}a_{i}(x)b_{j}(y)dxdy\end{array}

iar

\begin{gathered}\|(x)=\prod_{i=1}^{p}\left(x^{2}-i^{2}\right),\quad v(y)=\prod_{j=1}^{q}\left(y^{2}-j^{2}\right)\\ l_{i}(x)=x^{2}\left(x^{2}-1\right)\ldots\left(x^{2}-\bar{i}-1^{2}\right)\left(x^{2}-\overline{i+1^{2}}\right)\ldots\left(x^{2}-p^{2}\right)\\ b_{j}(y)=y^{2}\left(y^{2}-1\right)\ldots\left(y^{2}-\overline{j-1^{2}}\right)\left(y^{2}-\overline{j+1^{2}}\right)\ldots\left(y^{2}-q^{2}\right)\end{gathered}

Restul are următoarea expresie

$\displaystyle\frac{1}{4}r_{2}$	$\displaystyle=\frac{nh^{:n+3}k}{(2p+2)!}D_{\mathrm{E}}^{2p+2}\int_{0}^{m}x^{2}u(x)dx+{}_{(2q+2)!}^{mhk^{2q+3}}D_{\eta}^{2q+2}\int_{0}^{n}y^{2}v(y)dy-$
	$\displaystyle\quad h^{2p+3}k^{2q+3}$	(55)
	$\displaystyle\left(2p+2!(2q+2)!D_{\mathrm{E}}^{2p+2}D_{\eta}^{2q+2}\int_{0}^{m}x^{2}u(x)dx\int_{0}^{n}y^{2}v(y)dy\right.$

16.

Dacă în (52) se ia $p=q=0$ se obține următoarea formulă de cubatură de tip deschis de grad de exactitate $(1,1)$

\int_{x_{n}-mh}^{x_{0}+mh}dx\int_{y_{n}-nk}^{y_{n}+nk}f\left(x,ydy^{\prime}=4mnhkf\left(x_{0},y_{0}\right)=\rho\right.

unde

\rho=\frac{2}{3}m^{3}nh^{3}kD_{\xi}^{2}+\frac{2}{3}mn^{3}hk^{3}D_{\eta}^{2}-\frac{1}{9}m^{3}n^{3}h^{3}k^{3}D_{\xi}^{2}D_{\xi}^{2}

Se obține o formulă de cubatură importantă dacă se ia $p=q=1$

\begin{gathered}\int_{x_{0}-mh}^{x_{0}+mh}dx\int_{y_{0}-nk}^{y_{0}+nk}f(x,y)dy=\\ =\frac{nmhk}{9}\left[4\left(m^{2}-3\right)\left(n^{2}-3\right)f\left(x_{0},y_{0}\right)-2m^{2}\left(n^{3}-3\right)\right.\\ \cdot\left(f\left(x_{0}+h,y_{0}\right)+f\left(x_{0}-h,y_{0}\right)\right)-2n^{2}\left(m^{2}-3\right)\left(f\left(x_{0},y_{0}+k\right)+f\left(x_{0},y_{0}-k\right)\right)+\\ +m^{2}n^{2}\left(f\left(x_{0}+h,y_{0}+k\right)\right.\\ \left.+f\left(x_{0}-h,y_{0}+k\right)+f\left(x_{0}+h,y_{0}-k\right)+f\left(x_{0}-h,y_{0}-k\right)\right]+\rho\end{gathered}

unde restul are expresia

\begin{gathered}\rho=\frac{h^{5}km^{3}n\left(3m^{2}-5\right)}{90}D_{\xi}^{4}+\frac{hk^{5}mn^{3}\left(3n^{2}-5\right)}{90}D_{\eta}^{4}-\\ -\frac{h^{5}k^{5}m^{3}n^{3}\left(3m^{2}-5\right)\left(3n^{2}-5\right)}{144}D_{\xi}^{4}D_{\eta}^{4}.\end{gathered}

Făcînd mai sus $m=n=1$ se ajunge la formula de cubatură a lui Cavalieri-Simpson pentru două variabile

\begin{gathered}\qquad\int_{x_{o}-h}^{x_{o}+h}dx\int_{y_{o}-k}^{y_{o}+k}f(x,y)dy=\\ =\frac{hk}{9}\left[f\left(x_{0}+h,y_{0}+k\right)+f\left(x_{0}-h,y_{0}+k\right)+f\left(x_{0}+h,y_{0}-k\right)+f\left(x_{0}-h,y_{0}-k\right)\right.\\ \left.+4\left(f\left(x_{0}+h,y_{0}\right)+f\left(x_{0}-h,y_{0}\right)+f\left(x_{0},y_{0}+k\right)+f\left(x_{0},y_{0}-k\right)\right)+16f\left(x_{0},y_{0}\right)\right]+\rho\\ \text{ unde }\end{gathered}

\rho=-\frac{hk}{45}\left[h^{4}D_{\xi}^{4}+k^{4}D_{\eta}^{4}+\frac{1}{180}h^{4}k^{4}D_{\xi}^{4}D_{\eta}^{4}\right].

Am regăsit astfel pe altă cale expresia (25) a restului formulei lui Cavalieri-Simpson.

Pentru $n=n=2$ avem

\begin{gathered}\int_{x_{0}-2h}^{x_{0}+2h}dx\int_{y_{0}-2k}^{y_{0}+2k}f(x,y)dy=\\ =\frac{16hk}{9}\left\{f\left(x_{0},y_{0}\right)-2\left[f\left(x_{0}+h,y_{0}\right)+f\left(x_{0}-h,y_{0}\right)+f\left(x_{0},y_{0}+k\right)+f\left(x_{0},y_{0}-k\right)\right]+\right.\\ \left.+4\left[f\left(x_{0}+h,y_{0}+k\right)+f\left(x_{0}-h,y_{0}+k\right)+f\left(x_{0}+h,y_{0}-k\right)+f\left(x_{0}-h,y_{0}-k\right)\right]\right\}+\rho\end{gathered}

\rho=\frac{56}{45}h^{5}kD_{\xi}^{4}+\frac{56}{45}hk^{5}D_{\eta}^{4}-\frac{196}{9}h^{5}k^{5}D_{\xi}^{4}D_{\eta}^{4}.

Pentru $p=q=2$ formula (52) devine, luînd pentru simplificare $x_{0}=y_{0}=0$

	$\displaystyle\quad\int_{-mh}^{mh}\int_{-nk}^{nk}f(x,y)dxdy=$
	$\displaystyle=\frac{nmhk}{32400}\left\{C_{00}f(0,0)+C_{10}[f(h,0)+f(-h,0)]+C_{20}[f(2h,0)+f(-2h,0)]\right.$		(56)
	$\displaystyle+C_{01}[f(0,k)+f(0,-k)]+C_{62}[f(0,2k)+f(0,-2k)]+C_{11}[f(h,h)+f(-h,k)$
	$\displaystyle+f(h,-k)+f(-h,-k)]+C_{12}[f(h,2k)+f(-h,2k)+f(h,-2k)+f(-h,-2k)]$
	$\displaystyle+C_{21}[f(2h,k)+f(-2h,k)+f(2h,-k)+f(-2h,-k)]+C_{22}[f(2h,2k)$
	$\displaystyle+f(-2h,2k)+f(2h,-2k)+f(-2h,-2k)]\}+\rho.$

unde

	$\displaystyle\rho$	$\displaystyle=\frac{nmhk}{3780}\left[Ah^{6}D_{\xi}^{6}+Bk^{6}D_{\eta}^{6}-Ch^{6}k^{6}D_{\xi}^{6}D_{\eta}^{6}\right]$
	$\displaystyle C_{00}$	$\displaystyle=6\left(3m^{4}-25m^{2}+60\right)\left(3n^{4}-25n^{2}+60\right)$
	$\displaystyle C_{10}$	$\displaystyle=-4m^{2}\left(3m^{2}-20\right)\left(2n^{4}-25n^{2}+60\right)$
	$\displaystyle C_{20}$	$\displaystyle=6m^{2}\left(3m^{2}-5\right)\left(3n^{4}-25n^{2}+60\right)$
	$\displaystyle C_{01}$	$\displaystyle=-4n^{2}\left(3n^{2}-20\right)\left(3m^{4}-25m^{2}+60\right)$
	$\displaystyle C_{02}$	$\displaystyle=6n^{2}\left(3n^{2}-5\right)\left(3m^{4}-25n^{2}+60\right)$
	$\displaystyle C_{11}$	$\displaystyle=6m^{2}n^{2}\left(3m^{2}-20\right)\left(3n^{2}-20\right)$
	$\displaystyle C_{12}$	$\displaystyle=-4m^{2}n^{2}\left(3m^{2}-20\right)\left(3n^{2}-5\right)$
	$\displaystyle C_{21}$	$\displaystyle=-4m^{2}n^{2}\left(3m^{2}-5\right)\left(3n^{2}-20\right)$
	$\displaystyle C_{22}$	$\displaystyle=m^{2}n^{2}\left(3m^{2}-5\right)\left(3n^{2}-5\right)$

iar

		$\displaystyle A=m^{2}\left(3m^{4}-21m^{2}+28\right)$
		$\displaystyle B=n^{2}\left(3n^{4}-21n^{2}+28\right)$
		$\displaystyle C=\frac{1}{15120}m^{2}n^{2}\left(3m^{4}-21m^{2}+28\right)\left(3n^{4}-21n^{2}+28\right)$

Dacă în (56) se face $n=m=2$ se obține următoarea formulă de cubatură de tip închis, care are gradul parțial de exactitate $(5,5)$ :
$\int_{-2h}^{2h}\int_{-2k}^{2k}f(x,y)dxdy=\frac{4hk}{2025}\{144f(0,0)+384[f(-h,0)+f(h,0)+f(0,-k)+f(0,k)]+84[f(-2h,0)+f(0,-2k)+f(0,2k)+f(2h,0)]+1024[f(-h,-k)+f(-h,k)++f(h,-k)+f(h,k)]+224[f(-2h,-k)+f(-2h,k)+f(-h,-2k)+f(-h,2k)++f(h,-2k)+f(h,2k)+f(2h,-k)+f(2h,k)]+49[f(-2h,-2k)+f(-2h,2k)+f(2h,-2k)+f(2h,2k)]\}+\mathrm{p}$ .
cu

\rho=-\frac{32hk}{945}\left[h^{6}D_{\xi}^{6}+k^{6}D_{\eta}^{6}+\frac{2}{945}h^{6}k^{6}D_{\xi}^{6}D_{\eta}^{6}\right].

Pentru $m=n=3$ se obține formula de cubatură de tip deschis

		$\displaystyle\quad\int_{-3h}^{3h}\int_{-3k}^{3k}f(x,y)dxdy=\frac{9hk}{100}\{76f(0,0)-64[f(h,0)+f(-h,0)+f(0,k)$
		$\displaystyle+f(0,-k)]+86[f(0,2k)+f(0,-2k)+f(2h,0)]+96[f(h,k)+$
		$\displaystyle+f(-h,k)+f(h,-k)+f(-h,-k)]-54[f(h,2k)+f(-h,2k)+f(h,-2k)$
		$\displaystyle+f(-h,-2k)+f(2h,k)+f(-2h,k)+f(2h,-k)+f(-2h,-k)]+21\quad[f(2h,2k)$
		$\displaystyle+f(-2h,2k)+f(2h,-2k)+f(-2h,-2k)]+p$

unde

\rho=\frac{123}{70}hk\left[h^{6}D_{\xi}^{6}+k^{6}D_{\eta}^{6}-\frac{41}{840}h^{6}k^{6}D_{\xi}^{6}D_{\eta}^{6}\right]

Pentru $m=n=2$ din (56) se obține o formulă de cubatură cu noduri în afara domeniulıu de integrare, care merită să fie menționată.
17. In cazul $s=3$ formula de cubatură (39) se scrie

\begin{gathered}\int_{x_{0}-m_{1}h_{1}}^{x_{0}+m_{1}h_{1}}dx\int_{y_{0}-m_{2}h_{2}}^{y_{0}+m_{2}h_{2}}dy\int_{z_{0}-m_{3}h_{3}}^{z_{0}+m_{3}h_{3}}f(x,y,z)dz=\\ =h_{1}h_{2}h_{3}\left[A_{0}^{0}f\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{1}=1}^{p_{1}}A_{j_{1}}^{1}S_{j_{1}}^{0,1}f\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{2}=1}^{p_{2}}A_{j_{2}}^{2}S_{j_{2}}^{0,2}f\left(x_{0},y_{0},z_{0}\right)\right.\\ +\sum_{j_{3}=1}^{p_{3}}A_{j_{3}}^{3}S_{j_{3}}^{0,3}/\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{1}=1}^{p_{1}}\sum_{j_{2}=1}^{p_{0}}A_{j_{1}j_{2}}^{1,2}S_{j_{1}}^{0,1}S_{j_{2}}^{0,2}f\left(x_{0},y_{0},z_{0}\right)+\\ +\sum_{j_{1}=1}^{p_{1}}\sum_{j_{3}=1}^{p_{3}}A_{j_{1}j_{3}}^{1,3}S_{j_{1}}^{0,1}S_{j_{3}}^{0,3}f\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{2}=1}^{p_{2}}\sum_{j_{3}=1}^{p_{3}}A_{j_{2}j_{3}}^{2,3}S_{j_{2}}^{0,2}S_{j_{3}}^{0,3}f\left(x_{0},y_{0}z_{0}\right)\\ \left.+\sum_{j_{1}=1}^{p_{1}}\sum_{j_{2}=1}^{p_{3}}\sum_{j_{3}=1}^{p_{4}}A_{j_{1}j_{2}j_{3}}^{1,2,3}S_{j_{1}}^{0,1}S_{j_{2}}^{0,2}S_{j_{3}}^{0,3}f\left(x_{0},y_{0},z_{0}\right)\right]+\gamma_{3},\end{gathered}

minde

$\displaystyle S_{j_{1}}^{0,1}f\left(x_{0},y_{0},z_{0}\right)$	$\displaystyle=f\left(x_{0}+j_{1}h_{1},y_{0},z_{0}\right)+f\left(x_{0}-j_{1}h_{1},y_{0},z_{0}\right)$	(58)
$\displaystyle S_{j_{2}}^{0,2}f\left(x_{0},y_{0},z_{0}\right)$	$\displaystyle=f\left(x_{0},y_{0}+j_{2}h_{2},z_{0}\right)+f\left(x_{0},y_{0}-j_{2}h_{2},z_{0}\right)$
$\displaystyle S_{j_{3}}^{0,3}f\left(x_{0},y_{0},z_{0}\right)$	$\displaystyle=f\left(x_{0},y_{0},z_{0}+j_{3}h_{3}\right)+/\left(x_{0},y_{0},z_{0}-j_{3}h_{3}\right)$

iar restul $r_{3}$ are expresia de la (42) cu modificarea notat,iei deja folosită

\begin{array}[]{lll}t^{1}=x,&t^{2}=y,&t^{3}=z\\ t_{0}^{1}=x_{0},&t_{0}^{2}=y_{6},&t_{0}^{3}=z_{0}\end{array}

Coeficienții formulei (57) au expresiile
$\frac{1}{8}A_{0}^{0}=\frac{(-1)^{p_{1}+p_{2}+p_{3}}}{\left(p_{1}!\right)^{2}\left(p_{2}!\right)^{2}\left(p_{3}!\right)}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)u_{2}(y)u_{3}(z)dxdydz$
$\frac{1}{8}A_{j_{1}}^{1}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{1}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}!\right)^{2}\left(p_{3}!\right)^{2}}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}v_{j_{1}}^{1}(x)u_{2}(y)u_{3}(z)dxdydz$
$\frac{1}{8}A_{j_{2}}^{2}=\frac{(-1)^{p_{1}}+p_{2}+p_{3}-j_{2}}{\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{1}!\right)^{2}\left(p_{3}!\right)^{2}}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)v_{j_{3}}^{2}(y)u_{3}(z)dxdzdy$
$\frac{1}{8}A_{j_{3}}^{3}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{3}}}{\left(p_{3}-j_{3}\right)!\left(p_{3}+j_{3}\right)!\left(p_{1}!\right)^{2}\left(p_{2}!\right)^{2}}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)u_{2}(y)v_{j_{3}}^{3}(z)dxdydz$
$\frac{1}{8}A_{j_{1}j_{2}}^{1,2}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{1}-j_{2}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{3}!\right)^{2}}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}v_{j_{1}}^{1}(x)v_{j_{2}}^{2}(y)u_{3}(z)dxdydz

$\frac{1}{8}A_{j_{1}j_{3}}^{1,3}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{1}-j_{3}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}!\right)^{2}\left(p_{3}-j_{3}\right)!\left(p_{3}+j_{3}\right)!}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m^{3}}v_{j_{1}}^{1}(x)u_{2}(y)v_{j_{3}}^{3}(z)dxdydz

$\frac{1}{8}A_{j_{2}j_{3}}^{2,3}=\frac{(-1)^{p_{1}}+p_{2}+p_{3}-j_{2}-j_{3}}{\left(p_{1}!\right)^{2}\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{3}-j_{3}\right)!\left(p_{3}+i_{3}\right)!}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)v_{j_{2}}^{2}(y)v_{j_{3}}^{3}(z)dxdydz

$\frac{1}{8}A_{j_{1}j_{2}j_{2}}^{1,2,3}=\frac{(-1)^{p_{1}}+p_{2}+p_{3}-j_{1}-j_{2}-j_{3}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{3}-j_{3}\right)!\left(p_{3}+j_{3}\right)!}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}v_{j_{1}}^{1}(x)v_{j_{2}}^{2}(y)v_{j_{3}}^{3}(z)dxdydz

18.

Ne vom opri acum asupra unor cazuri particulare importante ale acestei formule.

Pentru $p_{1}=p_{2}=p_{3}=0$ se obține o formulă de cubatură care foloseste un singur nod și are gradul partial de exactitate ( $1,1,1$ )

\iiint_{D}f(x,y,z)dxdydz=8m_{1}m_{2}m_{3}h_{1}h_{2}h_{3}f\left(x_{0},y_{0},z_{0}\right)+\rho

(59)

unde $D$ este paralelipipedul

x_{0}-m_{1}h_{1}\leqq x\leqq x_{0}+m_{1}h_{1},y_{0}-m_{2}h_{2}\leqq y\leqq y_{0}+m_{2}h_{2},z_{0}-m_{3}h_{3}\leqq z\leqq z_{0}+m_{3}h_{3}

(60)

iar restul are expresia.
$\rho=\frac{m_{1}m_{2}m_{3}h_{1}h_{2}h_{3}}{3}\left\lceil 4m_{1}^{2}h_{1}^{2}D_{\xi}^{2}+4m_{2}^{2}h_{2}^{2}D_{\eta}^{2}+4m_{3}^{2}h_{3}^{3}D_{\xi}^{2}-\frac{2}{3}m_{1}^{2}m_{2}^{2}h_{1}^{2}h_{2}^{2}D_{\xi}^{2}D_{\eta}^{2}-\right.$
$\left.-\frac{2}{3}m_{1}^{2}m_{3}^{2}h_{1}^{2}h_{3}^{2}D_{\xi}^{2}D_{\zeta}^{2}-\frac{2}{3}m_{2}^{2}m_{3}^{2}h_{2}^{2}h_{3}^{2}D_{\eta}^{2}D_{\xi}^{2}+\frac{1}{9}m_{1}^{2}m_{2}^{2}m_{3}^{2}h_{1}^{2}h_{2}^{2}h_{3}^{2}D_{\xi}^{2}D_{\eta}^{2}D_{\xi}^{2}\right]$ .
Se observă că unicul nod pe care e definită această formulă se găseste în centrul de greutate al domeniului $D$ , presupus omogen. Această formulă e de tip Gauss, întrucît folosește minimul de noduri posibil.
19. Făcînd în formula (57) $p=p_{2}=p_{3}=1$ , se obține formula de cubatură

	$\displaystyle\iiint_{D}f(x,y,z)dxdydz=$
	$\displaystyle=\frac{m_{1}m_{2}m_{3}h_{1}h_{2}h_{3}}{27}\int-8\left(m_{1}^{2}-3\right)\left(m_{2}^{2}-3\right)\left(m_{3}^{2}-3\right)f\left(x_{0},y_{0},z_{0}\right)+4m_{1}^{2}\left(m_{2}^{2}-3\right)\left(m_{3}^{2}-3\right)$
	$\displaystyle.S_{1}^{0,1}f\left(x_{0}y_{0},z_{0}\right)+4\left(m_{1}^{2}-3\right)m_{2}^{2}\left(m_{3}^{2}-3\right)S_{1}^{0,2}f\left(x_{0},y_{0},z_{0}\right)+4\left(m_{1}^{2}-3\right)\left(m_{2}^{2}-3\right)m_{3}^{2}$
	$\displaystyle.S_{1}^{0,3}f\left(x_{0},y_{0},z_{0}\right)-2m_{1}^{2}m_{2}^{2}\left(m_{3}^{2}-3\right)S_{1}^{0,1}S_{1}^{0,2}f\left(x_{0},y_{0},z_{0}\right)-2m_{1}^{2}\left(m_{2}^{2}-3\right)m_{3}^{2}$		(60)
	$\displaystyle.S_{1}^{0,1}S_{1}^{0,3}f\left(x_{0},y_{0},z_{0}\right)-2\left(m_{1}^{2}-3\right)m_{2}^{2}m_{3}^{2}S_{1}^{0,2}S_{1}^{0,3}f\left(x_{0},y_{0},z_{0}\right)+m_{1}^{2}m_{2}^{2}m_{3}^{2}$

\begin{gathered}\rho=\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}m_{3}h_{1}^{5}h_{2}h_{3}}{45}D_{5}^{4}+\frac{m_{1}m_{2}^{3}\left(3m_{2}^{2}-5\right)m_{3}h_{1}h_{2}^{5}h_{3}}{45}D_{\eta}^{4}\\ +\frac{m_{1}m_{2}m_{3}^{3}\left(3m_{3}^{2}-5\right)h_{1}h_{2}h_{3}^{5}}{45}D_{5}^{4}-\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}^{3}\left(3m_{2}^{2}-5\right)m_{3}}{16200}h_{1}^{5}h_{2}^{5}h_{3}D_{5}^{4}D_{\eta}^{4}\\ -\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}m_{3}^{3}\left(3m_{3}^{2}-5\right)}{16200}h_{1}^{5}h_{2}h_{3}^{5}D_{\xi}^{4}D_{5}^{4}-\frac{m_{1}m_{2}^{3}\left(3m_{2}^{2}-5\right)m_{3}^{3}\left(3m_{3}^{2}-5\right)}{16200}\\ \cdot h_{1}h_{2}^{5}h_{3}^{5}D_{\eta}^{4}D_{5}^{4}+\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}^{2}\left(3m_{2}^{2}-5\right)m_{3}^{3}\left(3m_{3}^{2}-5\right)}{5832000}h_{1}^{5}h_{2}^{5}h_{3}^{5}D_{5}^{4}D_{\eta}^{4}D_{5}^{4}.\end{gathered}

Din aceasta vom obține imediat următoarea formulă de cubatură care reprezintă extinderea formulei lui Cavalieri-Simpson la trei variabile

		$\displaystyle\quad\int_{x_{0}-h_{1}}^{x_{0}+h_{1}}dx\int_{y_{0}-h_{2}}^{y_{0}+h_{2}}dy\int_{z_{0}-h_{3}}^{z_{0}+h_{3}}f(x,y,z)dz=$
		$\displaystyle=\frac{h_{1}h_{2}h_{3}}{27}\left\{f\left(x_{0}+h_{1},y_{0}+h_{2},z_{0}+h_{3}\right)+f\left(x_{0}+h_{1},y_{0}+h_{2},z_{0}-h_{3}\right)+f\left(x_{0}-h_{1},y_{0}+h_{2}z_{0}+h_{3}\right)\right.$
		$\displaystyle+f\left(x_{0}-h_{1},y_{0}+h_{2},z_{0}-h_{3}\right)+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}+h_{3}\right)+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}-h_{3}\right)$
		$\displaystyle+f\left(x_{0}-h_{1},y_{0}-h_{2},z_{0}+h_{3}\right)+f\left(x_{0}-h_{1},y_{0}-h_{2},z_{0}-h_{3}\right)+4\left[f\left(x_{0},y_{0}+h_{2},z_{0}+h_{3}\right)+\right.$
		$\displaystyle+f\left(x_{0},y_{0}+h_{2},z_{0}-h_{3}\right)+f\left(x_{0},y_{0}-h_{2},z_{0}+h_{3}\right)+f\left(x_{0},y_{0}-h_{2},z_{0}-h_{3}\right)+$
		$\displaystyle+f\left(x_{0}+h_{1},y_{0},z_{0}+h_{3}\right)+f\left(x_{0}+h_{1},y_{0},z_{0}-h_{3}\right)+f\left(x_{0}-h_{1},y_{0},z_{0}+h_{3}\right)+$
		$\displaystyle+f\left(x_{0}-h_{1},y_{0},z_{0}-h_{3}\right)+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}\right)+f\left(x_{0}-h_{1},y_{0}+h_{2},z_{0}\right)$
		$\displaystyle\left.+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}\right)+f\left(x_{0}-h_{1},y_{0}-h_{2},z_{0}\right)\right]+6\left[f\left(x_{0},y_{0},z_{0}+h_{3}\right)+\right.$
		$\displaystyle+f\left(x_{0},y_{0},t_{0}-h_{3}\right)\Omega f\left(x_{0},y_{0}+h_{2},z_{0}\right)+f\left(x_{0},y_{0}-h_{2},z_{0}\right)+f\left(x_{0}+h_{1},y_{0},z_{0}\right)+$
		$\displaystyle\left.\left.+f\left(x_{0}-h_{1},y_{0},z_{0}\right)\right]+64f\left(x_{0},y_{0},z_{0}\right)\right\}+\rho$

unde

\begin{gathered}\rho=-\frac{h_{1}h_{2}h_{3}}{45}\left[2h_{1}^{4}D_{\xi}^{4}+2h_{2}^{4}D_{\eta}^{4}+2h_{3}^{4}D_{\xi}^{4}+\frac{1}{90}h_{1}^{4}h_{2}^{4}D_{\xi}^{4}D_{\eta}^{4}+\frac{1}{90}h_{1}^{4}h_{3}^{4}D_{\xi}^{4}D_{\xi}^{4}+\right.\\ \left.+\frac{1}{90}h_{2}^{4}h_{3}^{4}D_{\eta}^{4}D_{\xi}^{4}+\frac{1}{16200}h_{1}^{4}h_{2}^{4}h_{3}^{4}D_{\xi}^{4}D_{\eta}^{4}D_{\xi}^{4}\right]\end{gathered}

Dacă în formula (60) facem $m_{1}=m_{2}=m_{3}=2$ se obține următoarea formulă de cubatură de tip deschis, care utilizează același număr de noduri ca și formula (61) și are la fel gradul parțial de exactitate ( $3,3,3$ )

	$\displaystyle\quad\int_{-2h_{1}}^{2h_{1}}\int_{-2h_{2}}^{2h_{2}}\int_{-2h_{3}}^{2h_{3}}f(x,y,z)dxdydz=$
	$\displaystyle=\frac{64h_{1}h_{2}h_{3}}{27}\left\{-f(0,0,0)+2\left[f\left(0,0,h_{3}\right)+f\left(0,0,-h_{3}\right)+f\left(0,h_{2},0\right)+f\left(0,-h_{2},0\right)+\right.\right.$
	$\displaystyle\left.+f\left(h_{1},0,0\right)+f\left(-h_{1},0,0\right)\right]-4\left[f\left(0,h_{2},h_{3}\right)+f\left(0,h_{2},-h_{3}\right)+f\left(0,-h_{2},h_{3}\right)\right.$		(62)
	$\displaystyle+f\left(0,-h_{2},-h_{3}\right)+f\left(h_{1},0,h_{3}\right)+f\left(h_{1},0,-h_{3}\right)+f\left(-h_{1},0,h_{3}\right)+f\left(-h_{1},0,-h_{3}\right)+$
	$\displaystyle\left.+f\left(h_{1},h_{2},0\right)+f\left(-h_{1},h_{2},0\right)+f\left(h_{1},-h_{2},0\right)+f\left(-h_{1},-h_{2},0\right)\right]+8\left[f\left(h_{1},h_{2},h_{3}\right)+\right.$
	$\displaystyle+f\left(h_{1},h_{2},-h_{3}\right)+f\left(-h_{1},h_{2},h_{3}\right)+f\left(-h_{1},h_{2},-h_{3}\right)+f\left(h_{1},-h_{2},h_{3}\right)+f\left(h_{1},-h_{2},-h_{3}\right)+$
	$\displaystyle\left.\left.+f\left(-h_{1},-h_{2},h_{3}\right)+f\left(-h_{1},-h_{2},-h_{3}\right)\right]\right\}+\rho,$

\begin{gathered}\rho=\frac{1792}{45}h_{1}h_{2}h_{3}\left[4\left(h_{1}^{4}D_{\xi}^{4}+h_{2}^{4}D_{\eta}^{4}+h_{3}^{4}D_{\xi}^{4}\right)-\frac{98}{45}\left(h_{1}^{4}h_{2}^{4}D_{\xi}^{4}D_{\eta}^{4}+h_{1}^{4}h_{3}^{4}D_{\xi}^{4}D_{\xi}^{4}+\right.\right.\\ \left.\left.+h_{2}^{4}h_{3}^{4}D_{\eta}^{4}D_{\xi}^{4}\right)+\frac{7}{2025}h_{1}^{4}h_{2}^{4}h_{3}^{4}D_{\xi}^{4}D_{\eta}^{4}D_{\xi}^{4}\right].\end{gathered}

Alte formule de integrare numerică au fost date în lucrarea [4].

§. 4. Formula de integrare numerică a lui Cavalieri-Simpson in $E_{s}$

20.

In încheiere vom da, sub formă explicită, două din formulele de cubatură mai importante deduse deja în cazurile $s=1,2$ şi 3 .

Astfel avem formula de cubatură de grad de exactitate $(1,1,\ldots,1)$

\iint\ldots\int_{D}f(M)dM=2^{s}m_{1}\ldots m_{s}h_{1}\ldots h_{s}f\left(M_{0}\right)+\rho

unde $D$ este hiperparalelipipedul

t_{0}^{i}-m_{i}h_{i}\leqq t^{i}\leqq t_{0}^{i}+m_{i}h_{i}\quad(i=\overline{1,s})

iar restul are expresia

		$\displaystyle\rho=\frac{m_{1}m_{2}\ldots m_{s}h_{1}\ldots h_{s}}{3}\left[2^{s-1}m_{1}^{2}h_{1}^{2}D_{\xi_{1}}^{2}+\cdots+2^{s-1}m_{s}^{2}h_{s}^{2}D_{\xi_{s}}^{2}-2^{s-2}\frac{m_{1}^{2}m_{2}^{2}h_{1}^{2}h_{2}^{2}}{3}D_{\xi_{1}}^{2}D_{\xi_{2}}^{2}-\right.$
		$\displaystyle\quad\quad\ldots-2^{s-2}\frac{m_{s-1}^{2}m_{s}^{2}h_{s-1}^{2}h_{s}}{3}D_{\xi_{s-1}}^{2}D_{\xi_{s}}^{2}$
		$\displaystyle\left.+\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots h_{s}^{2}D_{\xi_{1}}^{2}\ldots D_{\xi_{3}}^{2}\right].$

21.

Dacă în formula (39) se face

p_{1}=p_{2}=\ldots=p_{s}=m_{1}=m_{2}=\ldots=m_{s}=1

se obține următoarea formulă de cubatură, de grad parțial de exactitate $(3,3,\ldots,3)$ , care reprezintă extinderea formulei lui Cavalieri-Sompson în $E_{s}$

CONTRIBUTIONS À L’INTÉGRATION NUMÉRIQUE DES FONCTIONS DE PLUSIEURS VARIABLES

(Résumé)
En utilisant certaines formules d’interpolation pour les fonctions de plusieurs variables, on construit plusieurs formules pour le calcul approché des intégrales multiples définies. Pour chaque formule donnée on établit l’expression du reste.

Dans le premier paragraphe, après quelques considérations générales sur l’intégration numérique des fonctions de plusieurs variables, on déduit, en particulier, la formule de Cavalieri-Simpson pour deux variables. A cette occasion on donne aussi une expression précise du reste (25) de cette formule.

Dans le second paragraphe est construite une formule de cubature (39) pour les intégrales s-uples. Au (42) on établit l’expression du reste de cette formule.

Dans le troisième paragraphe on déduit sous une forme explicite, de (39), une série de formules d’intégration numérique poúr les intégrales simple, double et triple.

Dans le dernier paragraphe on donne effectivement deux formules de cubature pour les intégrales s-uples: la formule (62) qui utilise un seul noeud et la formule (64) qui représente la généralisation de la formule de quadrature de Cavalieri-Simpson.

Contribuţii la integrarea numerică a funcţiilor de mai multe variabile

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CONTRIBUTTII LA INTEGRAREA NUMERICĂ A FUNCTYILOR DE MAI MULTE VARIABILE

§ 1. Considerațiuni generale

§ 2. Unele formule practice de cubatură pentru integralele $s$ -uple

§ 3. Cazuri particulare importante ale formulelor de cubatură precedente

§. 4. Formula de integrare numerică a lui Cavalieri-Simpson in $E_{s}$

CONTRIBUTIONS À L’INTÉGRATION NUMÉRIQUE DES FONCTIONS DE PLUSIEURS VARIABLES

Related Posts

Contribuţii la integrarea numerică a funcţiilor de mai multe variabile

Rezumat

Autor(i)

Cuvinte cheie

PDF

Citați articolul în forma

Despre acest articol

Revista

Editura

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Referințe bibliografice

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CONTRIBUTTII LA INTEGRAREA NUMERICĂ A FUNCTYILOR DE MAI MULTE VARIABILE

§ 1. Considerațiuni generale

§ 2. Unele formule practice de cubatură pentru integralele ss-uple

§ 3. Cazuri particulare importante ale formulelor de cubatură precedente

§. 4. Formula de integrare numerică a lui Cavalieri-Simpson in EsE_{s}

CONTRIBUTIONS À L’INTÉGRATION NUMÉRIQUE DES FONCTIONS DE PLUSIEURS VARIABLES

Related Posts

§ 2. Unele formule practice de cubatură pentru integralele $s$ -uple

§. 4. Formula de integrare numerică a lui Cavalieri-Simpson in $E_{s}$