O METODĂ PENTRU CONSTRUIREA DE FORMULE DE CUBATURĂ PENTRU FUNCȚIILE DE DOUĂ VARIABILE

Comunicare prezentată la Sesiunea Filialei Cluj a Academiei R.P.R., din 17 aprilie 1957

DE
D. D. STANCU

În această lucrare vom extinde la două variabile o metodă pentru construirea de formule de integrare numerică pe care am expus-o în principiu, în cazul unei singure variabile, în lucrarea [1].

În prima parte a lucrării vom extinde la două variabile formula de interpolare a lui Lagrange-Hermite ¹ ).

§ 1. Asupra unei formule generale de interpolare

1.

Să considerăm o rețea dreptunghiulară de noduri, care se obțin prin intersecția dintre dreptele paralele cu axa $Oy$

	$\displaystyle x=x_{v_{1}}\left(v_{1}\right.$	$\displaystyle=\overline{1,n}),x=\alpha_{v_{2}}\left(v_{2}=\overline{1,i}\right)$		(1)
	$\displaystyle x$	$\displaystyle=a_{p}^{\prime}(p=\overline{1,\rho})$		(2)

și dreptele paralele cu axa $Ox$

	$\displaystyle y=y_{\mu_{1}}\left(\mu_{1}=\overline{1,m}\right),y=\beta_{\mu_{2}}\left(\mu_{2}=\overline{1,k}\right)$		(3)
	$\displaystyle y=b_{q}^{\prime}(q=\overline{1,\sigma})$		(4)

Prin intersecția acestor drepte se obțin $(n+i+\rho)(m+k+\sigma)$ noduri. Despre dreptele (1) și (3) vom presupune că sînt toate distincte, adică numerele reale $x_{\mathrm{v}_{1}}$ , $\alpha_{\nu_{2}}$ și $y_{\mu_{1}},\beta_{\mu_{2}}$ sînt respectiv distincte. Dreptele celelalte pot să nu fie distincte. În mod precis vom presupune că printre dreptele (2) există $r_{\alpha}$ drepte confundate în $x=a_{\alpha}(\alpha=\overline{1,s})$ , iar printre dreptele (4) vom presupune că există $s_{\beta}$ drepte confundate în $y=b_{\beta}(\beta=\overline{1,r})$ . Pe baza notațiilor precedente avem
$r_{1}+r_{2}+\ldots+r_{s}=\rho,s_{1}+s_{2}+\ldots+s_{r}=\sigma$ .
¹ ) Formulei de interpolare a lui Lagrange-Hermite, din cazul unei variabile, i-am consacrat recent lucrarea [2].

În felul acesta se obține o rețea $\mathscr{R}_{N,M}$ cu $(N+1)(M+1)$ noduri, nu toate distincte, unde

N=n+i+\rho-1,\quad M=m+k+\sigma-1

(6)

2.

Formula de interpolare relativă la rețeaua $\Re_{N,M}$ și o funcție $f(x,y)$ definită și derivabilă parțial de un număr suficient de ori pe un domeniu care conține nodurile acestei rețele, se prezintă sub forma
unde

f(x,y)=L_{N,M}(x,y)+R_{N,M}(f;x,y)

(7)

L_{N,M}(x,y)=L\binom{x_{1},\ldots,x_{n},\alpha_{1},\ldots,\alpha_{i},\overbrace{a_{1},\ldots,a_{1}}^{r_{1}},\ldots,\overbrace{a_{s},\ldots,a_{s}}^{r_{s}};f\left\lvert\,\begin{array}[]{l}x\\ y\end{array}\right.)}{y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{\underbrace{1},\ldots,b_{1}},\ldots,{b_{r},\ldots,b_{r}}^{b_{r},\ldots,\underbrace{}_{s_{r}}}}

este polinomul de interpolare de două variabile de gradul ( $N,M$ ) al lui LagrangeHermite relativ la funcția $f(x,y)$ și nodurile rețelei $\Re_{N,M}$ , iar $R_{N,M}(f;x,y)$ este restul acestei formule de interpolare.

Să introducem următoarele notații:

\left.\begin{array}[]{l}h(x)=\prod_{\nu=1}^{n}\left(x-x_{\nu}\right),u(x)=\prod_{\nu=1}^{i}\left(x-\alpha_{\nu}\right),A(x)=\alpha\prod_{\nu=1}^{s}\left(x-a_{\nu}\right)^{r_{\nu}}(\alpha\neq 0),\\ h_{\nu}(x)=\frac{h(x)}{x-x_{\nu}},u_{\nu}(x)=\frac{u(x)}{x-\alpha_{\nu}},A_{\nu}(x)=\frac{A(x)}{\left(x-a_{\nu}\right)^{r_{\nu}}},\\ g(y)=\prod_{\mu=1}^{m}\left(y-y_{\mu}\right),v(y)=\prod_{\mu=1}^{k}\left(y-\beta_{\mu}\right),B(y)=\beta\prod_{\mu=1}^{r}\left(y-b_{\mu}\right)^{s_{\mu}},\\ g_{\mu}(y)=\frac{g(y)}{y-y_{\mu}},v_{\mu}(y)=\frac{\nu(y)}{y-\beta_{\mu}},B_{\mu}(y)=\frac{B(y)}{\left(y-b_{\mu}\right)^{s_{\mu}}}.\end{array}\right\}

Pentru polinomul de interpolare (8) am găsit următoarea expresie

	$\displaystyle L_{N,M}(x,y)=\sum_{\nu=1}^{n}\sum_{\mu=1}^{m}l_{\nu}^{(1)}(x)\omega_{\mu}^{(1)}(y)f\left(x_{\nu},y_{\mu}\right)+\sum_{\nu=1}^{n}\sum_{\mu=1}^{k}l_{\nu}^{(1)}(x)\omega_{\mu}^{(2)}(y)f\left(x_{\nu},\beta_{\mu}\right)+$
	$\displaystyle+\sum_{\nu=1}^{i}\sum_{\mu=1}^{m}l_{\nu}^{(2)}(x)\omega_{\mu}^{(1)}(y)f\left(\alpha_{\nu},y_{\mu}\right)+\sum_{\nu=1}^{i}\sum_{\mu=1}^{k}l_{\nu}^{(2)}(x)\omega_{\mu}^{(2)}(y)f\left(\alpha_{\nu},\beta_{\mu}\right)+$
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r_{i_{1}}-1}\sum_{\mu=1}^{m}l_{i_{1},\nu_{1}}(x)\omega_{\mu}^{(1)}(y)\frac{\partial^{\nu_{1}}f\left(a_{i_{1}},y_{\mu}\right)}{\partial x^{\nu_{1}}}+$
	$\displaystyle+\sum_{\nu=1}^{n}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{s_{j_{1}}-1}l_{\nu}^{(2)}(x)\omega_{j_{1},\mu_{1}}(y)\frac{\partial^{\mu_{1}}f\left(x_{\nu},b_{j_{1}}\right)}{\partial y^{\mu_{1}}}+$		(11)
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r}\sum_{\mu=2}^{m}l_{i_{1},v}(x)\omega_{\mu}^{(2)}(y)\frac{\partial^{\nu_{1}}f\left(a_{i_{1}},\beta_{\mu}\right)}{\partial x^{\nu_{1}}}+$
	$\displaystyle+\sum_{\nu=1}^{i}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{sj_{1}-1}l_{\nu}^{(2)}(x)\omega_{j_{1},\mu_{1}}(y)\frac{\partial^{\mu_{1}}f\left(\alpha_{\nu},b_{j_{1}}\right)}{\partial y^{\mu_{1}}}+$
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r}\sum_{i_{1}=0}^{r}\sum_{\mu_{1}=0}^{s_{j_{1}}-1}l_{j_{1},\nu_{1}}(x)\omega_{j_{1},\mu_{1}}(y)\frac{\partial^{\nu_{1}+\mu_{1}}f\left(a_{i_{1}},b_{j_{1}}\right)}{\partial x^{\nu_{1}}\partial y^{\mu_{1}}}$

unde

	$\displaystyle l_{\nu}^{(1)}(x)=\frac{h_{\nu}(x)u(x)A(x)}{h_{\nu}\left(x_{\nu}\right)u\left(x_{\nu}\right)A\left(x_{\nu}\right)},l_{\nu}^{(2)}(x)=\frac{h(x)u_{\nu}(x)A(x)}{h\left(\alpha_{\nu}\right)u_{\nu}\left(\alpha_{\nu}\right)A\left(\alpha_{\nu}\right)},$		(12)
	$\displaystyle l_{i_{1},v_{1}}(x)=\sum_{\gamma=0}^{ri-\nu_{1}-1}\frac{\left(x-a_{i_{1}}\right)^{\nu_{1}}}{\nu_{1}!}\left[\frac{\left(x-a_{i_{1}}\right)^{\gamma}}{\gamma!}\left(\frac{1}{C_{i_{1}}(x)}\right)_{a_{i_{1}}}^{(\gamma)}\right]C_{i_{1}}(x),$		(13)
	$\displaystyle\omega_{\mu}^{(1)}(y)=\frac{g_{\mu}(y)v(y)B(y)}{g\left(y_{\mu}\right)v\left(y_{\mu}\right)B\left(y_{\mu}\right)},\omega_{\mu}^{(2)}(y)=\frac{g(y)v_{\mu}(y)B(y)}{g\left(\beta_{\mu}\right)v_{\mu}\left(\beta_{\mu}\right)B\left(\beta_{\mu}\right)},$		(14)
	$\displaystyle\omega_{j_{1},u_{1}},(y)\sum_{\delta=0}^{s_{j_{1}}-\mu_{1}-1}\frac{\left(y-b_{j_{1}}\right)^{\mu_{1}}}{\mu_{1}!}\left[\frac{\left(y-b_{j_{1}}\right)^{\delta}}{\delta!}\left(\frac{1}{D_{\nu_{1}}(y)}\right)_{b_{j_{1}}}^{(\delta)}\right]D_{j_{1}}(y),$		(15)

iar

C_{i_{1}}(x)=h(x)u(x)A_{i_{1}}(x),\quad D_{v_{1}}(y)=g(y)v(y)B_{v_{1}}(y)

(16)

Restul formulei de interpolare (7) se poate exprima cu ajutorul diferent,elor divizate parțiale, astfel

	$\displaystyle R_{N,M}(f;x,y)=h(x)u(x)\widetilde{A}(x)\left[x,x_{1},\ldots,x_{n},\alpha_{1},\ldots,\alpha_{i},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s};f\right]+$
	$\displaystyle+g(y)v(y)\widetilde{B}(y)\left[y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};f\right]-$
	$\displaystyle-h(x)u(x)\widetilde{A}(x)g(y)v(y)\widetilde{B}(y)\left[\begin{array}[]{l}x,x_{1},\ldots,x_{n},\alpha_{1},\ldots,\alpha_{i},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s}\\ y,y_{1},\ldots,y_{m},\ldots,\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r}\end{array}\right]\cdot(17)$		(17)

3.

Acest rest se poate pune sub o formă remarcabilă în ipoteza că $f(x,y)$ admite derivatele parțiale de ordinele $N+1$ în raport cu $x$ și $M+1$ în raport cu $y$ , pe cel mai mic dreptunghi care conține nodurile rețelei $\mathcal{R}_{N,M}$ .

Ne vom folosi de cîteva formule de suprapunere și de medie referitoare la diferențele divizate parțiale de care ne-am ocupat în lucrările [3], [4]. Avem succesiv
$R_{N,M}(f;x,y)=h(x)u(x)\widetilde{A}(x)\left[x,x_{1},\ldots,x_{n},\alpha_{1},\ldots,\alpha_{i},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s};f\right]+$

+g(y)v(y)\widetilde{B}(y)\left[y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};f\right]-

$-h(x)u(x)\widetilde{A}(x)g(y)v(y)\widetilde{B}(y)\left[y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};\right.$

		$\displaystyle\left.\left[x,x_{1},\ldots,x_{n},\alpha_{1},\ldots,\alpha_{i},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s};f\right]\right]=\frac{h(x)u(x)\widetilde{A}(x)}{(N+1)!}\frac{\partial^{N+1}f(\xi,y),}{\partial\xi^{N+1}}+$
		$\displaystyle\quad+g(y)v(y)\widetilde{B}(y)\left[y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};f\right]-$
		$\displaystyle-\frac{h(x)u(x)\widetilde{A}(x)g(y)v(y)\widetilde{B}(y)}{(N+1)!}\left[y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r},\right.$

¹ ) Fie $P_{m}(t)$ un polinom de gradul efectiv $m$ ; prin $\widetilde{P}_{m}(t)$ vom înţelege produsul dintre o constantă $C$ și $P_{m}(t)$ astfel încît $CP_{m}(t)$ să aibă coeficientul lui $t^{m}$ egal cu 1 .

\begin{gathered}\left.\frac{\partial^{N+1}f(\xi,y)}{\partial\xi^{N+1}}\right]=\frac{h(x)u(x)\widetilde{A}(x)}{(N+1)!}\frac{\partial^{N+1}f(\xi,y)}{\partial\xi^{N+1}}+\\ +g(y)v(y)\widetilde{B}(y)\left[y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{1},\ldots b_{r},\ldots,b_{r};f(x,y)-\right.\\ \left.-\frac{h(x)u(x)\widetilde{A}(x)}{(N+1)!}\frac{\partial^{N+1}f(\xi,y)}{\partial\xi^{N+1}}\right]=\frac{h(x)u(x)\widetilde{A}(x)}{(N+1)!}\frac{\partial^{N+1}f(\xi,y)}{\partial\xi^{N+1}}+\\ +\frac{g(y)v(y)\widetilde{B}(y)}{(M+1)!}\left(\frac{\partial^{M+1}f(x,\eta)}{\partial\eta^{M+1}}-\frac{h(x)u(x)\widetilde{A}(x)}{(N+1)!}\frac{\partial^{N+M+2}f\left(\xi,r_{1}\right)}{\partial\xi^{N+1}\partial\eta^{M+1}}\right)\end{gathered}

În felul acesta am obținut pentru restul formulei de interpolare (7) următoarea expresie

	$\displaystyle R_{N,M}(f;x,y)$	$\displaystyle=\frac{h(x)u(x)\widetilde{A}(x)}{(N+1)!}\frac{\partial^{N+1}f(\xi,y)}{\partial\xi^{N+1}}+\frac{g(y)v(y)\widetilde{B}(y)}{(M+1)}\frac{\partial^{M+1}(x,\eta)}{\partial\eta^{M+1}}-$
		$\displaystyle-\frac{h(x)u(x)\widetilde{A}(x)g(y)v(y)\widetilde{B}(y)}{(N+1)!(M+1)!}\frac{\delta^{N+M+2}f(\xi,\eta)}{\partial\xi^{N+1}\delta\eta^{M+1}}$		(18)

unde $\xi$ și $\eta$ sînt valori din cele mai mici intervale care îi conțin respectiv pe $x,x_{1},\ldots,x_{n},\alpha_{1},\ldots,\alpha_{i},a_{1},\ldots,a_{s}\stackrel{{\scriptstyle\text{ si }}}{{\mathrm{i}}}y,y_{1},\ldots,y_{m},\beta_{1},\ldots,\beta_{k},b_{1},\ldots,b_{r}$ .

Subliniem că valorile $\xi$ și $\eta$ care intervin mai sus sînt aceleași în toți termenii restului.

Rezultatele precedente se pot extinde fără nici o greutate la trei și mai multe variabile."

§ 2. Formule de integrare numerică de grad înalt de exactitate

4.

Folosind formula de interpolare (7) vom construi o formulă generală de cubatură pentru calculul numeric al integralei duble

I=\iint_{D}p(x,y)f(x,y)dxdy

(19)

unde $D$ este dreptunghiul definit de inegalitățile

a\leqq x\leqq b,\quad c\leqq y\leqq d,

(20)

iar $p(x,y)$ e o funcție dată, nenegativă și integrabilă în $D$ .

Formula de cubatură care se obține este

	$\displaystyle I=\sum_{\nu=1}^{n}\sum_{\mu=1}^{m}A_{\nu,\mu}f\left(x_{\nu},y_{\mu}\right)+\sum_{\nu=1}^{n}\sum_{\mu=1}^{k}B_{\nu,\mu}f\left(x_{\nu},\beta_{\mu}\right)+$
	$\displaystyle+\sum_{\nu=1}^{i}\sum_{\mu=1}^{m}C_{\nu,\mu}f\left(\alpha_{\nu},y_{\mu}\right)+\sum_{\nu=1}^{i}\sum_{\mu=1}^{k}D_{\nu,\mu}f\left(\alpha_{\nu},\beta_{\mu}\right)+$
	$\displaystyle+\sum_{\nu=1}^{i}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{sj_{1}-1}E_{\nu,j_{1},\mu_{1}}\frac{\partial^{\mu_{1}}f\left(\alpha_{\nu},b_{j_{1}}\right)}{\partial y^{\mu_{1}}}+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r_{i_{1}}-1}\sum_{\mu=1}^{k}F_{i_{1},\nu_{1},\mu}\frac{\partial^{\nu_{1}}f\left(a_{i_{1}},\beta_{\mu}\right)}{\partial x^{\nu_{1}}}+$		(21)
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r_{t_{1}}-1}\sum_{\mu=1}^{m}G_{i_{1},\nu_{1},\mu}\frac{\partial^{\nu_{1}}f\left(a_{i_{1}},y_{\mu}\right)}{\partial x^{\nu_{1}}}+\sum_{\nu=1}^{n}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{s_{1}-1}H_{\nu,j_{1},\mu_{1}}\frac{\partial^{\mu_{1}}f\left(x_{\nu},b_{j_{1}}\right)}{\partial y^{\mu_{1}}}+$
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r_{i_{1}}-1}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{s}s_{j_{1}-1}I_{i_{1},\nu_{1},j_{1},\mu_{1}}\frac{\partial^{\nu_{1}+\mu_{1}}f\left(a_{i_{1},}b_{j_{1}}\right)}{\partial x^{\nu_{1}}\partial y^{\mu_{1}}}+\rho[f]$

expresiile coeficienților căreia se pot scrie imediat pe baza formulelor (7) şi (11).
5. Să presupunem că $p(x,y)=p_{1}(x)p_{2}(y)$ . Considerînd mai general că dreptunghiul $D$ poate fi și infinit, vom presupune că există « momentele»

\begin{gathered}c_{n}=\int_{a}^{b}p_{1}(x)x^{n}dx,d_{m}=\int_{a}^{d}p_{2}(y)y^{m}dy\\ (n,m=0,1,2,\ldots)\end{gathered}

și că $c_{0}>0,d_{0}>0$ .
Vom încerca acum să determinăm pe $x_{\nu}(\nu=\overline{1,n})$ și $y_{\mu}(\mu=\overline{1,m})$ astfel încît în membrul al doilea al formulei (21) să dispară sumele duble : a doua, a treia, a patra, a cincea și a șasea. În mod precis vom demonstra următoarea:

Teoremă. Conditia necesară și suficientă ca în formula (21), unde $i=n$ , $k=m$ , să avem, oricare ar fi $\alpha_{1},\ldots,\alpha_{n},\beta_{1},\ldots,\beta_{m}$ ,

\left.\begin{array}[]{c}B_{\nu,\mu}=0(\nu=\overline{1,n};\mu=\overline{1,m}),C_{\nu,\mu}=0(\nu=\overline{1,n};\mu=\overline{1,m}),\\ D_{\nu,\mu}=0(\nu=\overline{1,n};\mu=\overline{1,m}),E_{\nu,i_{1},\mu_{1}}=0\left(\nu=\overline{1,n};j_{1}=\overline{1,r};\mu_{1}=0,s_{j_{1}}-1\right)\\ F_{i_{1},\nu_{1},\mu}=0\left(i_{1}=\overline{1,s};\nu_{1}=\overline{0,r_{i_{1}}-1};\mu=\overline{1,m}\right),\end{array}\right\}

este ca $x_{\nu}(\nu=\overline{1,n}),y_{\nu}(\mu=\overline{1,m})$ să fie rădăcinile reale și distincte, respectiv ale polinoamelor ortogonale
unde $\left\{\Phi_{n}(x)\right\}$ este sistemul de polinoame ortogonale relative la intervalul ( $a,b$ și ponderea $p_{1}(x)$ , iar $\left\{\Psi_{m}(y)\right\}$ este sistemul de polinoame ortogonale relative la intervalul ( $c,d$ ) şi ponderea $p_{2}(y)$ .

Demonstrația este imediată. Avînd în vedere expresiile coeficienților de la (22) se găsește că este necesar și suficient ca

\int_{a}^{b}p_{1}(x)A(x)h(x)P(x)dx=0,\int_{c}^{d}p_{2}(y)B(y)g(y)Q(y)dy=0

unde $P(x)$ și $Q(y)$ sînt polinoame oarecare de grad $n-1$ , respectiv $m-1$ .

Se știe că polinoamele $h(x)$ și $g(y)$ , care verifică aceste condiții, sînt date de formulele (23) și (24) ale lui Christoffel.
6. Alegînd valorile $x_{1},\ldots,x_{n}$ și $y_{1},\ldots,y_{m}$ să fie respectiv rădăcinile polinoamelor (23) și (24), formula de cubatură (21) se reduce la următoarea

	$\displaystyle\iint_{D}p(x,y)f(x,y)dxdy=\sum_{\nu=1}^{n}\sum_{\mu=1}^{m}A_{\nu,\mu}f\left(x_{\nu},y_{\mu}\right)+$
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r_{i_{1}}-1}\sum_{\mu=1}^{m}G_{i_{1},j_{1},\mu_{2}}\frac{\partial^{\nu_{1}}f\left(a_{i_{1}},y_{\mu}\right)}{\partial x^{\nu_{1}}}+\sum_{\nu=1}^{n}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{s_{j_{1}}-1}H_{\nu,j_{1},\mu_{1}}\frac{\partial^{\mu_{1}}f\left(x_{\nu},b_{j_{1}}\right)}{\partial y^{\mu_{1}}}+$		(25)
	$\displaystyle+\sum_{i_{1}=1}^{s}\sum_{\nu_{1}=0}^{r_{i_{1}}-1}\sum_{j_{1}=1}^{r}\sum_{\mu_{1}=0}^{s_{j_{1}}-1}I_{i_{1},\nu_{1}j_{1},\mu_{1}}\frac{\partial^{\nu_{1}+\mu_{1}}f\left(a_{i_{1}},b_{j_{1}}\right)}{\partial x^{\nu_{1}}\partial y^{\mu_{1}}}+\rho[f]$

Pentru coeficienții acestei formule de cubatură am obținut următoarele expresii

\left.\begin{array}[]{rl}A_{\nu,\mu}&=A_{\nu}^{\prime}B_{\mu}^{\prime}=A_{\nu}^{\prime\prime}B_{\mu}^{\prime\prime}=A_{\nu}^{\prime\prime\prime}B_{\mu}^{\prime\prime\prime}\\ G_{i_{1},\nu_{1},\mu}&=C_{i_{1},\nu_{1}}B_{\mu}^{\prime}=C_{i_{1},\nu_{1}}B_{\mu}^{\prime\prime}=C_{i_{1},\nu_{1}}B_{\mu}^{\prime\prime\prime}\\ H_{\nu,j_{1},\mu_{1}}&=A_{\nu}^{\prime}D_{j_{1},\mu_{1}}=A_{\nu}^{\prime\prime}D_{j_{1},\mu_{1}}=A_{\nu}^{\prime\prime\prime}D_{j_{1},\mu_{1}}\end{array}\right\}

unde
unde

\left.\begin{array}[]{c}A_{\nu}^{\prime}==\int_{a}^{b}p_{1}(x)\frac{h_{\nu}(x)A(x)}{h_{\nu}\left(x_{\nu}\right)A\left(x_{\nu}\right)}dx,A_{\nu}^{\prime\prime}=\int_{a}^{b}p_{1}(x)\left(\frac{h_{\nu}(x)}{h_{\nu}\left(x_{\nu}\right)}\right)^{2}\frac{A(x)}{A\left(x_{\nu}\right)}dx\\ A_{\nu}^{\prime\prime\prime}=\frac{c_{n+\rho}\gamma_{n-1}^{2}G_{n}^{2}}{\alpha c_{n-1}}\cdot\frac{1}{A\left(x_{\nu}\right)Q_{n}^{\prime}\left(x_{\nu}\right)Q_{n-1}\left(x_{\nu}\right)},\\ B_{\mu}^{\prime}=\int_{c}^{a}p_{2}(y)\frac{g_{\mu}(y)B(y)}{g_{\mu}\left(y_{\mu}\right)B\left(y_{\mu}\right)}dy,B_{\nu\mu}^{\prime\prime}=\int_{c}^{a}p_{2}(y)\left(\frac{g_{\mu}\left(y^{\prime}\right)}{g_{\mu}\left(y_{\mu}\right)}\right)^{2}\frac{B(y)}{B\left(y_{\mu}\right)}dy\end{array}\right\}

Mai sus am folosit și următoarele notații:
$1^{\circ}c_{k}$ e coeficientul lui $x^{k}$ din polinomul ortogonal $\Phi_{r}(x)$ , iar $d_{r}$ e coeficientul lui $y^{r}$ din polinomul ortogonal $\Psi_{r}(y)$ .
$2^{\circ}G_{n}$ e minorul elementului $\Phi_{n+\rho}(x)$ din determinantul de la (23), iar $L_{m}$ minorul elementului $\Psi_{m+\sigma}^{\prime}(y)$ din determinantul de la (24).
$3^{\circ}$

\gamma_{n}^{2}=\int_{a}^{b}p_{1}(x)\Phi_{n}^{2}(x)dx,\delta_{m}^{2}=\int_{c}^{a}p_{2}(y)\Psi_{m}^{2}(y)dy

$4^{\circ}$

	$\displaystyle L_{i,v_{1}}^{(1)}(x)$	$\displaystyle=\sum_{\rho_{1}=0}^{r_{i_{1}}-v_{1}-1}\frac{\left(x-a_{i_{1}}\right)^{v_{1}}}{v_{1}!}\left[\frac{(x-a_{i_{1}})^{\rho_{1}}}{\rho_{1}!}\left(\left.\left.\frac{1}{E_{i_{1}}(x)}\right\|_{a_{i_{1}}}^{\left(\rho_{1}\right)}\right\rvert\,E_{i_{1}}(x),E_{i_{1}}(x)=h(x)A_{i_{1}}(x),\right.\right.$
	$\displaystyle L_{j_{1},\mu_{1}}^{(2)}(y)$	$\displaystyle=\sum_{\sigma_{1}=0}^{s_{i_{1}}-\mu_{1}-1}\frac{\left(y-b_{j_{1}}\right)^{\mu_{1}}}{\mu_{1}!}\left[\left.\frac{\left(y-b_{j_{1}}\right)^{\sigma_{1}}}{\sigma_{1}!}\left(\frac{1}{F_{j_{1}}(y)}\right)_{b_{j_{1}}}^{\left(\sigma_{1}\right)}\right\|_{j_{1}}(y),F_{j_{1}}(y)=g(y)B_{j_{1}}(y).\right.$

7.

Formula de cubatură (25) are gradul de exactitate ( $2n+\rho-1,2m+\sigma-1$ ), adică $\rho[f]\equiv 0$ în cazul cînd $f(x,y)$ este un polinom de gradul $2n+\rho-1$ în raport cu $x$ și de gradul $2m+\sigma-1$ în raport cu $y$ .

Pentru a găsi expresia restului formulei (25) vom lua $i=n$ și $k=m$ şi vom considera cazuł limită

h(x)\equiv u(x)\equiv\widetilde{Q}_{n}(x),\quad g(y)\equiv v(y)\equiv\widetilde{R}_{m}(y)

(28)

Vom pleca de la formula (17), care în acest caz se scrie

R_{N,M}(f;x,y)=r_{1}(x,y)+r_{2}(x,y)-r_{3}(x,y)

unde

\begin{gathered}r_{1}(x,y)=\widetilde{Q}_{n}^{2}(x)\widetilde{A}(x)\left[x,x_{1},x_{1},x_{2},x_{2},\ldots,x_{n},x_{n},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s};f\right],\\ r_{2}(x,y)=\widetilde{R}_{m}^{2}(y)\widetilde{B}(y)\left[y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};f\right],\\ r_{3}(x,y)=\widetilde{Q}_{n}^{2}(x)\widetilde{A}(x)\widetilde{R}^{2}(y)\widetilde{B}(y)\left[\begin{array}[]{l}x,x_{1},x_{1},x_{2},x_{2},\ldots,x_{n},x_{n},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s}\\ y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},h_{1},\ldots,b_{1},\ldots,b_{s},\ldots,b_{s}\end{array}\right].\end{gathered}

Pe baza unei proprietăți de suprapunere a diferențelor divizate parțiale, de care ne-am mai folosit în această lucrare, și unor formule de medie bine cunoscute putem scrie succesiv

\begin{gathered}\iint p(x,y)\left[r_{1}(x,y)-r_{3}(x,y)\right]dxdy=\\ =\int_{c}^{d}p_{2}(y)\left\{\int_{a}^{b}p_{1}(x)\widetilde{Q}_{n}^{2}(x)A(x)\left[x,x_{1},x_{1},x_{2},x_{2},\ldots,x_{n},x_{n},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s};f(x,y)-\right.\right.\\ \left.\left.-\widetilde{R}_{m}^{2}(y)\widetilde{B}(y)\left[y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};f(x,y)\right]\right]dx\right\}dy=\\ =\int_{c}^{d}p_{2}(y)\left\{\left[\xi^{\prime},x_{1},x_{1},x_{2},x_{2},\ldots,x_{n},x_{n},a_{1},\ldots,a_{1},\ldots,a_{s},\ldots,a_{s};f\left(\xi^{\prime},y\right)-\right.\right.\\ \left.\left.-\widetilde{Q}_{m}^{2}(y)\widetilde{B}(y)\left[y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};f\left(\xi^{\prime},y\right)\right]\right]A_{1}\right\}dy=\\ =A_{1}\int_{c}^{d}p_{2}(y)\left\{\frac{1}{(2n+\rho)!}\frac{\partial^{2n+\rho}f(\xi,y)}{\partial\xi^{2n+\rho}}-\frac{\widetilde{R}_{m}^{2}(y)\widetilde{B}(y)}{(2n+\rho)!}\left[y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots\right.\right.\\ \left.\left.\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};\frac{\partial^{2n+\rho}(\xi,y)}{\partial\xi^{2n+\rho}}\right]\right\}dy=\frac{A_{1}B_{2}}{(2n+\rho)!}\frac{\partial^{2n+\rho}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n+\rho}}-\end{gathered}

$-\int_{c}^{d}p_{2}(y)\frac{\widetilde{R}_{m}^{2}(y)\widetilde{B}(y)}{(2n+\rho)!}\left[y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};\frac{\partial^{2n+\rho}f(\xi,y)}{\partial\xi^{2n+\rho}}\right]dy$ ,
unde

A_{1}=\int_{a}^{b}p_{1}(x)\widetilde{Q}_{n}^{2}(x)\widetilde{A}(x)dx,\widetilde{B}_{2}=\int_{c}^{d}p_{2}(y)dy

(29)

iar $\xi\in(a,b)$ și $\eta\in(c,d)$ .

Cu acestea avem

		$\displaystyle\rho[f]=\frac{A_{1}B_{2}}{(2n+\rho)!}\frac{\partial^{2n+\rho}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n+\rho}}+$
		$\displaystyle+\int_{c}^{d}p_{2}(y)\widetilde{R}_{m}^{2}(y)\widetilde{B}(y)\left[y,y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};\right.$
		$\displaystyle\left.\int_{a}^{b}p_{1}(x)f(x,y)dx-\frac{A_{1}}{(2n+\rho)!}\frac{\partial^{2n+\rho}f(\xi,y)}{\partial\xi^{2n+\rho}}\right]dy+\frac{A_{1}B_{2}}{(2n+\rho)!}\frac{\partial^{2n+\rho}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n+\rho}}+$
		$\displaystyle+A_{2}\left[\eta^{\prime},y_{1},y_{1},y_{2},y_{2},\ldots,y_{m},y_{m},b_{1},\ldots,b_{1},\ldots,b_{r},\ldots,b_{r};\int_{a}^{b}p_{1}(x)f\left(x,\eta^{\prime}\right)dx-\right.$
		$\displaystyle\left.-\frac{A_{1}}{(2n+\rho)!}\frac{\partial^{2n+\rho}f\left(\xi,\eta^{\prime}\right)}{\partial\xi^{2n+\rho}}\right]$

Rezultă că restul formulei generale de cubatură (25) se poate pune sub forma

	$\displaystyle\rho[f]=$	$\displaystyle\frac{A_{1}B_{2}}{(2n+\rho)!}\frac{\partial^{2n+\rho}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n+\rho}}+\frac{A_{2}B_{1}}{(2m+\sigma)}\frac{\partial^{2m+\rho}f\left(\xi_{1},\eta\right)}{\partial\nu^{2m+\rho}}-$
		$\displaystyle-\frac{A_{1}A_{2}}{(2n+\rho)!(2m+\sigma)!}\frac{\delta^{2n+2m+\rho+\sigma}f(\xi,\eta)}{\partial\xi^{2n+\rho}\partial\eta^{2m+\sigma}}$		(30)

unde $\xi_{1}\in(a,b),\eta\in(c,d)$ , iar alături de notațiile de la (29) am mai folosit și următoarele

A_{2}=\int_{c}^{d}p_{2}(y)\widetilde{R}_{m}^{2}(y)\widetilde{B}(y)dy,B_{1}=\int_{a}^{b}p_{1}(x)dx

(31)

8.

În cazul particular

\rho=0,\sigma=0

(32)

formula generală de cubatură (25) se reduce la formula de tip Gauss

\iint_{D}p(x,y)f(x,y)dxdy=\sum_{\nu=1}^{n}\sum_{\mu=1}^{m}A_{\nu,\mu}^{\prime}f\left(x_{\nu},y_{\mu}\right)+\rho_{1}[f]

(33)

unde

A_{\nu,\mu}^{\prime}=\frac{1}{\sqrt{\lambda_{n}\gamma_{m}}\Phi_{n}^{\prime}\left(x_{\nu}\right)\psi_{m}^{\prime}\left(y_{\mu}\right)\Phi_{n-1}\left(x_{\nu}\right)\psi_{m-1}\left(y_{\mu}\right)},

		$\displaystyle\left.\lambda_{n}=\left(\int_{a}^{b}p_{1}(x)\widetilde{\Phi}_{n}^{2}(x)dx\right)\right):\left(\int_{a}^{b}p_{1}(x)\widetilde{\Phi}_{n-1}^{2}(x)dx\right)$
		$\displaystyle\gamma_{m}=\left(\int_{c}^{d}p_{2}(y)\widetilde{\Psi}_{m}^{2}(y)dy\right):\left(\int_{c}^{d}p_{2}(y)\widetilde{\Psi}_{m-1}^{2}(y)dy\right)$

Pe baza formulei (30), restul formulei (33) are următoarea expresie

	$\displaystyle\rho_{1}(f)=$	$\displaystyle\frac{A_{1}^{\prime}B_{2}}{(2n)!}\frac{\partial^{2n}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n}}+\frac{A_{2}^{\prime}B_{1}}{(2m)!}\frac{\partial^{2m}f\left(\xi_{1},\eta\right)}{\partial\eta^{2m}}-$		(34)
		$\displaystyle-\frac{A_{1}^{\prime}A_{2}^{\prime}}{(2n)!(2m)!}\frac{\partial^{2n+2m}f(\xi,\eta),}{\partial\xi^{2n}\partial\eta^{2m}},$

unde

A_{1}^{\prime}=\int_{a}^{b}p_{1}(x)\widetilde{\Phi}_{n}^{2}(x)dx,A_{2}^{\prime}=\int_{c}^{d}p_{2}(y)\widetilde{\psi}_{m}^{2}(y)dy

iar $B_{1}$ și $B_{2}$ au expresiile de la (31) și (29).
9. Dacă $p(x,y)\equiv 1,a=c=-1,b=d=1$ , formula (33) se reduce la următoarea

\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\sum_{\nu=1}^{n}\sum_{\mu=1}^{m}A_{\nu,\mu}^{\prime\prime}f\left(x_{\nu},y_{\mu}\right)+\rho_{2}[f]

(35)

unde

A_{\nu,\mu}=\frac{4}{nmP_{n}^{\prime}\left(x_{\nu}\right)P_{n-1}\left(x_{\nu}\right)P_{m}^{\prime}\left(y_{m}\right)P_{m-1}^{\prime}\left(y_{m}\right)},

iar

\begin{gathered}\rho_{2}[f]=\frac{4^{n+1}}{2n+1}\cdot\frac{m!}{[(n+1)(n+2)\ldots 2n]^{3}}\frac{\partial^{2n}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n}}+\\ +\frac{4^{m+1}}{2m+1}\frac{m!}{[(m+1)(m+2)\ldots 2m]^{3}}\frac{\partial^{2m}f\left(\xi_{1},\eta_{1}\right)}{\partial\eta^{2m}}-\\ -\frac{4^{n+m+1}}{(2n+1)(2m+1)}\cdot\frac{n!m!}{[(n+1)(n+2)\ldots 2n]^{3}[(m+1)(m+2)\ldots 2m]^{3}}\frac{\partial^{2n+2m}f\left(\xi,\eta_{1}\right)}{\partial\xi^{2n}\partial\eta^{2m}}\end{gathered}

De această formulă de cubatură ne-am ocupat și în lucrarea [4].
$\left.{}^{1}\right)$ Cu $P_{r}(t)$ am notat polinomul lui Legendre: $\frac{1}{2^{r}\cdot r!}\left[\left(t^{2}-1\right)^{r}\right]^{(r)}$ .

§ 3. Cazuri particulare ale formulei generale de cubatură

10.

Folosind rezultatele din paragraful precedent se pot construi efectiv, cu ușurință, multe formule de cubatură care sînt simple și au gradul înalt de exactitate. În cazul

	$\displaystyle p_{1}(x)=(1-x)^{\alpha}(1+x)^{\beta},p_{2}(y)=(1-y)^{\alpha^{\prime}}(1+y)^{\beta^{\prime}}\left(\alpha,\alpha^{\prime},\beta,\beta^{\prime}>-1\right)$		(32)
	$\displaystyle A(x)=1-x^{\prime},B(y)=1-y^{2},a=c=-1,b=d=1$

Pentru

x_{1}=\frac{\beta-\alpha}{\alpha+\beta+4},y_{1}=\frac{\beta^{\prime}-\alpha^{\prime}}{\alpha^{\prime}+\beta^{\prime}+4}.

și $n=m=1$ , se obțin valorile

\alpha=\alpha^{\prime}=\frac{1}{2},\beta=\beta^{\prime}=-\frac{1}{2}

(33)

se obține formula de cubatură

	$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}\sqrt{\frac{(1-x)(1-y)}{(1+x)(1+y)}}f(x,y)dxdy=\frac{\pi^{2}}{3600}\{625f(-1,1)+$
	$\displaystyle+75[f(-1,1)+f(1,-1)]+9f(1,1)+800\left[f\left(-1,-\frac{1}{4}\right)+f\left(-\frac{1}{4},-1\right)\right]+$
	$\displaystyle\left.+96\left[f\left(1,-\frac{1}{4}\right)+f\left(-\frac{1}{4},1\right)\right]+1024f\left(-\frac{1}{4},-\frac{1}{4}\right)\right\}-$
	$\displaystyle-\frac{\pi^{2}}{256}\left[\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\partial\eta^{4}}+\frac{1}{256}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\tau_{1}^{4}}\right]$		(34)

\alpha=\beta,\alpha^{\prime}=\beta^{\prime},a=c=-1,b=d=+1

(35)

coordonatele nodurilor formulei de cubatură (25) se află rezolvînd ecuațiile

\left.\begin{array}[]{l}(\alpha+n+1)(\alpha+n+2)J_{n}^{(\alpha,\alpha)}(x)-(n+1)(n+2)J_{n+2}^{(\alpha,\alpha)}(x)=0\\ \left(\alpha^{\prime}+m+1\right)\left(\alpha^{\prime}+m+2\right)J_{m}^{\left(\alpha^{\prime},\alpha^{\prime}\right)}(y)-(m+1)(m+2)J_{m+2}^{\left(\alpha^{\prime},\alpha^{\prime}\right)}(y)=0,\end{array}\right\},

unde $\mathrm{cu}J_{\nu}^{(p,p)}(t)$ am notat polinomul lui Jacobi

J_{\nu}^{(p,p)}(t)=\frac{1}{2^{\nu}\cdot\nu!}\left(t^{2}-1\right)^{-p}\frac{d^{\nu}}{dt^{\nu}}\left(t^{2}-1\right)^{\nu+p}

În cazul $n=m=1$ se obține formula de cubatură

	$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}\left(1-x^{\prime}\right)^{\alpha}(1-y)^{\alpha^{\prime}}f(x,y)dxdy=4^{\alpha+\alpha^{\prime}+1}\frac{\Gamma(\alpha+1)\Gamma(\alpha+2)\Gamma\left(\alpha^{\prime}+1\right)\Gamma\left(\alpha^{\prime}+2\right)}{\Gamma(2\alpha+4)\Gamma\left(2\alpha^{\prime}+4\right)}$
	$\displaystyle\{f(-1,-1)+f(-1,1)+f(1,-1)+f(1,1)+4[(\alpha+1)f(0,-1)+f(0,1)+$
	$\displaystyle\left.\left.\quad+\left(\alpha^{\prime}+1\right)f(-1,0)+f(1,0)\right]+16(\alpha+1)\left(\alpha^{\prime}+1\right)f(0,0)\right\}-$		(37)
	$\displaystyle\quad-\frac{\left\langle{}^{\alpha+\alpha^{\prime}+1}\right.}{3}\left[\frac{\Gamma(\alpha+2)\Gamma(\alpha+3)\Gamma\left(\alpha^{\prime}+1\right)^{2}}{\Gamma(2\alpha+6)}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\right.$
	$\displaystyle\quad+\frac{\Gamma(\alpha+1)^{2}\Gamma\left(\alpha^{\prime}+2\right)\Gamma\left(\alpha^{\prime}+3\right)}{\Gamma\left(2\alpha^{\prime}+6\right)}\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\partial\eta^{4}}+$
	$\displaystyle\left.+\frac{\Gamma(\alpha+2)\Gamma(\alpha+3)\Gamma\left(\alpha^{\prime}+2\right)\Gamma\left(\alpha^{\prime}+3\right)}{3\Gamma(2\alpha+6)\Gamma\left(2\alpha^{\prime}+6\right)}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}\right]$

De aici, pentru $\alpha=\alpha^{\prime}=0$ , se obține formula de integrare numerică a lui Cavalieri-Simpson pentru două variabile

	$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{9}\{f(-1,-1)+f(-1,1)+f(1,-1)+f(1,1)+$
	$\displaystyle+4[f(-1,0)+f(0,-1)+f(1,0)+f(0,1)]+16f(0,0)\}-$		(38)
	$\displaystyle-\frac{1}{45}\left[\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\delta\eta^{4}}+\frac{1}{180}\frac{\partial^{8}f(\xi,\eta)}{\gamma_{1}^{4}\xi\partial\eta^{4}}\right]$

Într-un caz mai general ne-am ocupat de această formulă în lucrarea [5].
Pentru $\alpha=\alpha^{\prime}=-\frac{1}{2}$ formula (37) devine

	$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}\frac{f(x,y)dxdy}{\sqrt{\left(1-x^{2}\right)\left(1-y^{2}\right)}}=\frac{\pi^{2}}{16}\{f(-1,-1)+f(-1,1)+f(1,-1)+f(1,1)+$
	$\displaystyle+2[f(-1,0)+f(0,-1)+f(1,0)+f(0,1)]+4f(0,0)\}-$		(39)
	$\displaystyle-\frac{\pi^{2}}{192}\left[\frac{\partial^{4}f\left(\xi,\gamma_{1}\right)}{\partial\xi^{4}}+\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\partial\eta^{4}}+\frac{1}{192}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}\right]$

12.

In cazul $p(x,y)=1,A(x)=1-x^{2},B(y)=1-y^{2},n=m=2$ și $n=m=3$ se obțin următoarele formule de cubatură de grad de exactitate $(5,5)$ , respectiv $(7,7)$

	$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{36}\{f(-1,-1)+f(-1,1)+f(1,-1)+f(1,1)+$
	$\displaystyle\quad+5\left[f\left(\left(-1,-\frac{1}{\sqrt{5}}\right)+f\left(-\frac{1}{\sqrt{5}},-1\right)+f\left(-1,\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}},1\right)+\right.\right.$
	$\displaystyle\left.\quad+f\left(1,-\frac{1}{\sqrt{5}}\right)+f\left(-\frac{1}{\sqrt{5}},1\right)+f\left(1,\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}},1\right)\right]+$		(40)
	$\displaystyle\left.+25\left[f\left(-\frac{1}{\sqrt{5}},-\frac{1}{\sqrt{5}}\right)+f\left(-\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}},-\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}}\right)\right]\right\}-$
	$\displaystyle\quad-\frac{4}{23625}\left[\frac{\partial^{6}f\left(\xi,\eta_{1}\right)}{\partial\xi^{6}}+\frac{\partial^{6}f\left(\xi_{1},\eta\right)}{\partial\eta^{6}}+\frac{1}{23625}\frac{\partial^{12}f(\xi,\eta)}{\partial\xi^{6}\partial\eta^{6}}\right],$
	$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{8100}\{81[f(-1,-1)+f(-1,1)+f(1,-1)+$
	$\displaystyle\quad+f(1,1)]+576f[(-1,0)+f(0,-1+f(1,0)+f(0,1)]+$
	$\displaystyle 4096f(0,0)+441\left[f\left(-1,-\sqrt{\frac{3}{7}}\right)+f\left(-\sqrt{\frac{3}{7}},-1\right)+f\left(-1,\sqrt{\frac{3}{7}}\right)+\right.$
	$\displaystyle\left.\quad+f\left(\sqrt{\frac{3}{7}},-1\right)+f\left(1,-\sqrt{\frac{3}{7}}\right)+f\left(-\sqrt{\frac{3}{7}},1\right)+f(1,]\sqrt{\frac{3}{7}}\right)+$
	$\displaystyle+f\left(\sqrt{\frac{3}{7}},1\right)\left\lvert\,+3136\left[f\left(0,-\sqrt{\frac{3}{7}}\right)+f\left(-\sqrt{\frac{3}{7}},0\right)+f\left(0,\sqrt{\frac{3}{7}}\right)+\right.\right.$		(41)
	$\displaystyle\left.+f\left(\sqrt{\frac{3}{7}},0\right)\right]+2401\left[f\left(-\sqrt{\frac{3}{7}},-\sqrt{\frac{3}{7}}\right)+f\left(-\sqrt{\frac{3}{7}},\sqrt{\frac{3}{7}}\right)+\right.$
	$\displaystyle\left.\left.+f\left(\sqrt{\frac{3}{7}},-\sqrt{\frac{3}{7}}\right)+f\left(\sqrt{\frac{3}{7}},\sqrt{\frac{3}{7}}\right)\right]\right\}-$
	$\displaystyle-\frac{1}{1389150}\left[\frac{\partial^{8}f\left(\xi,\eta_{1}\right)}{\partial\xi^{8}}+\frac{\partial^{8}f\left(\xi_{1},\eta\right)}{\partial\eta^{8}}+\frac{1}{5556600}\frac{\partial^{16}f(\xi,\eta)}{\partial\xi^{8}\partial\eta^{8}}\right]$

		$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{540}\{9[f(-1,-1)+f(-1,1)+f(1,-1)+$
		$\displaystyle+f(1,1)]+5\left[f\left(-\frac{1}{\sqrt{5}},-1\right)+f\left(\frac{1}{\sqrt{5}},-1\right)+f\left(-\frac{1}{\sqrt{5}},1\right)+f\left(\frac{1}{\sqrt{5}},1\right)\right]+$
		$\displaystyle+9\left[f\left(-1,-\sqrt{\frac{3}{7}}\right)+f\left(1,-\sqrt{\frac{3}{7}}\right)+f\left(-1,\sqrt{\frac{3}{7}}\right)+f\left(1,\sqrt{\frac{3}{7}}\right)\right]+$
		$\displaystyle+4[f(-1,0)+f(1,0)]+20\left[f\left(-\frac{1}{\sqrt{5}},0\right)+f\left(\frac{1}{\sqrt{5}},0\right)\right]+$
		$\displaystyle+45\left[f\left(-\frac{1}{\sqrt{5}},-\sqrt{\frac{3}{7}}\right)+f\left(\frac{1}{\sqrt{5}},-\sqrt{\frac{3}{7}}\right)+f\left(-\frac{1}{\sqrt{5}},\sqrt{\frac{3}{7}}\right)+\right.$
		$\displaystyle\left.\left.+f\left(\frac{1}{\sqrt{5}},\sqrt{\frac{3}{7}}\right)\right]\right\}-\left[\frac{4}{23625}\frac{\partial^{6}f\left(\xi,r_{1}\right)}{\partial\xi^{6}}+\frac{1}{1389150}\frac{\partial^{8}f\left(\xi_{1},\eta\right)}{\partial\eta^{8}}+\right.$
		$\displaystyle\left.+\frac{1}{32818668750}\frac{\partial^{14}f\left(\xi,\xi^{7}\right)}{\partial\xi^{6}\partial\eta^{8}}\right].$

Dacă $p(x,y)=1,a=c=-1,b=d=1$ , iar $p=q=1$ sau $p=q=2$ , avem formulele de cubatură de gradele de exactitate $(3,3)$ , respectiv $(7,7)$

\begin{gathered}\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{9}\left\{36f(0,0)+6\left[f_{x^{2}}^{\prime\prime}(0,0)+f_{y^{2}}^{\prime\prime}(0,0)\right]+f_{x^{2}y^{2}}^{(4)}(0,0)\right\}+\\ +\frac{1}{30}\left[\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\partial\eta^{4}}-\frac{1}{120}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}\right]\\ \int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{140625}\left\{207936f(0,0)+67032\left[f\left(0,-\sqrt{\frac{5}{7}}\right)+\right.\right.\\ \left.+f\left(-\sqrt{\frac{5}{7}},0\right)+f\left(0,\sqrt{\frac{5}{7}}\right)+f\left(\sqrt{\frac{5}{7}},0\right)\right]+21609\left[f\left(-\sqrt{\frac{5}{7}},-\sqrt{\frac{5}{7}}\right)+\right.\\ \left.+f\left(-\sqrt{\frac{5}{7}},\sqrt{\frac{5}{7}}\right)+f\left(\sqrt{\frac{5}{7}},-\sqrt{\frac{5}{7}}\right)+f\left(\sqrt{\frac{5}{7}},\sqrt{\frac{5}{7}}\right)\right\rvert\,+\\ +9120\left[f_{x^{2}}^{\prime\prime}(0,0)+f_{y^{2}}^{\prime\prime}(0,0)\right]+2940\left[f_{x^{2}}^{\prime\prime}\left(0,-\sqrt{\frac{5}{7}}\right)+f_{x^{2}}^{\prime\prime}\left(0,\sqrt{\frac{5}{7}}\right)+\right.\\ \left.\left.+f_{y^{2}}^{\prime\prime}\left(-\sqrt{\frac{5}{7}},0\right)+f_{y^{2}}^{\prime\prime}\left(\sqrt{\frac{5}{7}},0\right)\right]+400f_{x^{2}y^{2}(0,0)}^{(4)}\right\}+\frac{1}{1111320}\left[\frac{\partial^{8}f\left(\xi,\eta_{1}\right)}{\partial\xi^{8}}+\right.\\ \left.+\frac{\partial^{8}f\left(\xi_{1},\eta\right)}{\partial\eta_{1}^{8}}-\frac{1}{4445280}\frac{\partial^{16}f(\xi,\eta)}{\partial\xi^{8}\partial y_{1}^{8}}\right].\end{gathered}

14.

In cazul $A(x)=x^{4},B(y)=y^{4},n=m=3,a=c=-1,b=d=1$ , $p(x,y)=1$ , se obține formula de cubatură de gradul de exactitate $(9,9)$ următoare

		$\displaystyle\int_{-1}^{+1}\int_{-1}^{+1}f(x,y)dxdy=\frac{1}{1297080225}\left\{2516025600f(0,0)+548499600\left[f\left(0,-\frac{\sqrt{7}}{3}\right)+\right.\right.$
		$\displaystyle\left.\quad+f\left(-\frac{\sqrt{7}}{3},0\right)+f\left(0,\frac{\sqrt{7}}{3}\right)+f\left(\frac{\sqrt{7}}{3},0\right)\right]+19574225\left[f\left(-\frac{\sqrt{7}}{3},-\frac{\sqrt{7}}{3}\right)+\right.$
		$\displaystyle\left.\quad+f\left(-\frac{\sqrt{7}}{3},\frac{\sqrt{7}}{3}\right)+f\left(\frac{\sqrt{7}}{3},-\frac{\sqrt{7}}{3}\right)+f\left(\frac{\sqrt{7}}{3},\frac{\sqrt{7}}{3}\right)\right]+75560000\left[f_{x^{2}}^{\prime\prime}(0,0)+\right.$
		$\displaystyle\left.\left.\quad+f_{y^{2}}^{\prime\prime}0,0\right)\right]+457840\left[f_{x^{4}}^{(4)}(0,0)+f_{y^{4}}^{(4)}(0,0)\right]+2250000f_{x^{2}y^{2}}^{(4)}(0,0)+$
		$\displaystyle\quad+71500\left[f_{x^{2}y^{4}}^{(6)}(0,0)+f_{x^{4}y^{2}}^{(6)}(0,0]+2401f_{x^{4}y^{4}}^{(8)}(0,0)+\right.$
		$\displaystyle\quad+8272500\left[f_{x^{2}}^{\prime\prime}\left(0,-\frac{\sqrt{7}}{3}\right)+f_{x^{2}}^{\prime\prime}\left(0,\frac{\sqrt{7}}{3}\right)+f_{y^{2}}^{\prime\prime}\left(-\frac{\sqrt{7}}{3},0\right)+f_{y^{2}}^{\prime\prime}\left(\frac{\sqrt{7}}{3},0\right)\right]+$
		$\displaystyle\left.\quad+535815\left[f_{x^{4}}^{(4)}\left(0,-\frac{\sqrt{7}}{3}\right)+f_{x^{4}}^{(4)}\left(0,\frac{\sqrt{7}}{3}\right)+f_{y^{4}}^{(4)}\left(-\frac{\sqrt{7}}{3},0\right)+f_{y^{4}}^{(4)}\left(\frac{\sqrt{7}}{3},0\right)\right]\right\}+$
		$\displaystyle\quad+\frac{1}{202078800}\left[\frac{\partial^{\mathbf{10}}f\left(\xi,\eta_{1}\right)}{\partial\xi^{\mathbf{10}}}+\frac{\partial^{\mathbf{10}}f\left(\xi_{1},\eta\right)}{\partial\eta^{\mathbf{10}}}-\frac{1}{808315200}\frac{\partial^{\mathbf{20}}f(\xi,\eta)}{\partial\xi^{\mathbf{10}}\partial\eta^{\mathbf{10}}}\right].$

15.

Presupunînd că $A(x)=x^{4},B(y)=y^{4},n=m=3,p(x,y)=e^{-x^{2}-y^{2}}$ , $a=c=-\infty,b=d=+\infty$ , se ajunge la formula de cubatură de gradul de exactitate $(9,9)$ următoare

		$\displaystyle\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-x^{2}-y^{2}}f(x,y)dxdy=\frac{\pi}{271063296}\{47873536f(0,0)+667840[f(0,-$
		$\displaystyle\left.\left.-\sqrt{\frac{7}{2}}\right)+f\left(-\sqrt{\frac{7}{2}},0\right)+f\left(0,\sqrt{\frac{7}{2}}\right)+f\left(\sqrt{\frac{7}{2}},0\right)\right]+4964864\left[f_{x^{2}}^{\prime\prime}(0,0)+\right.$
		$\displaystyle\left.+f_{y^{2}}^{\prime\prime}(0,0)\right]+314368\left[f_{x^{4}}^{(4)}(0,0)+f_{y^{4}}^{(4)}(0,0)\right]+028160\left[f_{x^{2}}^{\prime\prime}\left(0,-\sqrt{\frac{7}{2}}\right)+\right.$
		$\displaystyle\left.+f_{x^{2}}^{\prime\prime}\left(0,\sqrt{\frac{7}{2}}\right)+f_{y^{2}}^{\prime\prime}\left(-\sqrt{\frac{7}{2}},0\right)+f_{y^{2}}^{\prime\prime}\left(\sqrt{\frac{7}{2}},0\right)\right]+2920\left[f_{x^{4}}^{(4)}\left(0,-\sqrt{\frac{7}{2}}\right)+\right.$
		$\displaystyle\left.+f_{x^{4}}^{(4)}\left(0,\sqrt{\frac{7}{2}}\right)+f_{y^{4}}^{(4)}\left(-\sqrt{\frac{7}{2}},0\right)+f_{y^{4}}^{(4)}\left(\sqrt{\frac{7}{2}},0\right)\right]+156736f_{x^{2}y^{2}}^{(4)}(0,0)+$
		$\displaystyle+19832\left[f_{x^{2}y^{4}}^{(6)}(0,0)+f_{x^{4}y^{2}}^{(6)}(0,0)\right]+1609f_{x^{4}y^{4}}^{(8)}(0,0)+29600\left[f\left(-\sqrt{\frac{7}{2}}\right.\right.$
		$\displaystyle\left.\left.\left.\left.-\sqrt{\frac{7}{2}}\right)+f\left(-\sqrt{\frac{7}{2}},\sqrt{\frac{7}{2}}\right)+f\left(\sqrt{\frac{7}{2}},-\sqrt{\frac{7}{2}}\right)+f\sqrt{\frac{7}{2}},\sqrt{\frac{7}{2}}\right)\right]\right\}+$
		$\displaystyle+\frac{\pi}{552960}\left[\frac{\partial^{\mathbf{10}}f\left(\xi,\eta_{1}\right)}{\partial\xi^{\mathbf{10}}}+\frac{\partial^{\mathbf{10}}f\left(\xi_{1},\eta\right)}{\partial\eta^{\mathbf{10}}}-\frac{1}{552960}\frac{\partial^{\mathbf{20}}f\left(\xi,\eta^{2}\right)}{\partial\xi^{\mathbf{10}}\partial\eta^{\mathbf{10}}}\right].$

16.

In cazul $p(x,y)=1,A(x)=x^{\prime},B(y)=y^{4},n=m=3$ , se obține următoarea formulă de cubatură de gradul de exactitate $(7,9)$

		$\displaystyle\int_{-1}^{+1}\int_{+1}^{+1}f(x,y)dxdy=\frac{1}{13505625}\left\{22872960f(0,0)+4986360\left[f\left(0,-\frac{\sqrt{7}}{3}\right)+\right.\right.$
		$\displaystyle\quad+f\left(0,\frac{\sqrt{7}}{3}\right]+373520\left\lceil f\left(-\frac{\sqrt{5}}{7},0\right)+f\left(\sqrt{\frac{5}{7}},0\right)+\right.$
		$\displaystyle\quad+607445\left[f\left(-\sqrt{\frac{5}{7}},-\frac{\sqrt{7}}{3}\right)+f\left(-\sqrt{\frac{5}{7}},\frac{\sqrt{7}}{3}\right)+f\left(\sqrt{\frac{5}{7}},-\frac{\sqrt{7}}{3}\right)+\right.$
		$\displaystyle\left.+f\left(\sqrt{\frac{5}{7}},\frac{\sqrt{7}}{3}\right)\right]+003200f_{x^{2}}^{\prime\prime}(0,0)+596000f_{y^{2}}^{\prime\prime}(0,0)+18700\left[f_{x^{2}}^{\prime\prime}(0\right.$
		$\displaystyle\left.\left.-\frac{\sqrt{7}}{3}\right)+f_{x^{2}}^{\prime\prime}\left(0,\frac{\sqrt{7}}{3}\right)\right]+14500\left[f_{y^{2}}^{\prime\prime}\left(-\sqrt{\frac{5}{7}},0\right)+f_{y^{2}}^{\prime\prime}\left(\sqrt{\frac{5}{7}},0\right)\right]+$
		$\displaystyle+0000f_{x^{2}y^{2}}^{(4)}(0,0)+2344f_{y^{4}}^{(4)}(0,0)+203\left[f_{y^{2}}^{(4)}\left(-\sqrt{\frac{5}{7}},0\right)+\right.$
		$\displaystyle\left.\left.+f_{y^{1}}^{(4)}\left(\sqrt{\frac{5}{7}},0\right)\right]+980f_{x^{2}y^{4}}^{(6)}(0,0)\right\}+\frac{1}{1111320}\frac{\partial f^{8}\left(\xi,\eta_{1}\right)}{\partial\xi^{8}}+$
		$\displaystyle\frac{1}{202078800}\frac{\partial^{10}f\left(\xi_{1},\eta\right)}{\partial\gamma_{1}^{10}}-\frac{1}{186810534187200}\frac{\partial^{18}f(\xi,\eta)}{\partial\xi^{8}\partial\eta_{1}^{10}}.$

17.

Formulele de cubatură de la aliniatele $13-16$ sînt de tip Gauss, în sensul că ele folosesc un număr minim de noduri. De pildă ultima formulă, care are gradul de exactitate $(7,9)$ , are 20 de termeni, în timp ce formula corespunzătoare a lui Cotes, de același grad de exactitate, conține 80 de termeni.

Institutul de calcul al Academiei R.P.R., Filiala Cluj

UNE MÉTHODE DE CONSTRUCTION DES FORMULES DE CUBATURE POUR LES FONCTIONS DE DEUX VARIABLES

RÉSUMÉ

L’auteur étend au cas de deux variables une méthode de construction des formules de quadrature qu’il a exposée dans son travail [1]. Dans le § 1, la formule d’interpolation de Lagrange-Hermite est étendue à deux variables; on a obtenu la formule (7), où le polynôme d’interpolation $L_{N,M}(x,y)$ est donné en (11), tandis que le reste de la formule est donné en (17); pour ce reste la formule (18) est encore donnée où les valeurs $\xi$ et $\eta$ sont les mêmes dans tous les termes.

Dans le § 2, à l’aide de la formule d’interpolation (7) pour le calcul numérique de l’intégrale double (19), l’auteur obtient la formule de cubature (21). En prenant les racines des polynômes orthogonaux (23) et (24), pour valeurs $x_{1},x_{2},\ldots$ , $x_{n}$ et $y_{1},y_{2},\ldots,y_{m}$ , la formule de cubature (21) se réduit à la formule (25), quels que soient les paramètres $\alpha_{1},\ldots,\alpha_{i},\beta_{1},\ldots,\beta_{k}$ , où $i\leqslant n,k\leqslant m$ . Pour le calcul des coefficients de cette formule, les formules (26) sont données. Il est démontré que, en prenant $i=n$ et $k=m$ et en considérant le cas limite (28), le reste de la formule de cubature (25) peut être exprimé par la formule (30).

Dans le § 3, l’auteur construit effectivement plusieurs formules de cubature, en partant de la formule générale (25). Il faut surtout mentionner les formules de cubature des alinéas $n^{0}13$ à 16 , qui présentent un degré d’exactitude maximum; elles contiennent quatre fois moins de termes que les formules d’un même degré d’exactitude, mais avec des nœuds pris au hasard.

BIBLIOGRAFIE

1.

D. D. Stancu, O metodă pentru construirea de formule de cuadratură de grad inalt de exactitate. Comunicările Acad. R.P.R., 8,4 (1958).
2.
- •
  
  Asupra formulei de interpolare a lui Hermite și a unor aplicatii ale acesteia. Studii și cercetări de matematică, Cluj 8,3-4 (1957).
3.
- •
  
  Consideratii asupra interpolării polinominale a functiilor de mai multe variabile. Bul. Univ. «Babes-Bolyai» din Cluj, Seria Științele naturii, 1,1-1 (1957).
4.
- •
  
  Generalizarea unor formule de interpolare pentru functile de mai multe variabile si unele consideratii asupra formulei de integrare numerică a lui Gauss. Bul. Știinţ. Acad. R.P.R., Secția de stiinte matematice și fizice, 9,2 (1957).
5.
- •
  
  Contribuți la integrarea numerică a functiilor de mai multe variabile. Studii și cercetări matematica, Cluj, 8,1-2 (1957).

A method for constructing cubature formulas for functions of two variables

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O METODĂ PENTRU CONSTRUIREA DE FORMULE DE CUBATURĂ PENTRU FUNCȚIILE DE DOUĂ VARIABILE

§ 1. Asupra unei formule generale de interpolare

§ 2. Formule de integrare numerică de grad înalt de exactitate

§ 3. Cazuri particulare ale formulei generale de cubatură

UNE MÉTHODE DE CONSTRUCTION DES FORMULES DE CUBATURE POUR LES FONCTIONS DE DEUX VARIABLES

RÉSUMÉ

BIBLIOGRAFIE

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