Câteva formule practice de cuadratură

Tomul XIII, nr. 8
August 1963

DE
D. V. IONESCU

Comunicare prezentatä de academician T. Popoviciu in spedinla din 18 mai 1963
G. Coulmy [1] a considerat următoarea formulă de cuadratură

\[ \begin{align*} & \int_{x_{n}}^{x_{n}} f(x) \mathrm{d} x=\frac{13 h}{36}\left[f\left(x_{0}\right)+f\left(x_{n}\right)\right]+\frac{7 h}{6}\left[f\left(x_{1}\right)+f\left(x_{n-1}\right)\right]+h\left[f\left(x_{n}\right)+f\left(x_{n-2}\right)\right]+ \\ & \quad+\frac{35 h}{36}\left[f\left(x_{3}\right)+f\left(x_{n-3}\right)\right]+h\left[f\left(x_{4}\right)+f\left(x_{5}\right)+\cdots+f\left(x_{n-4}\right)\right] \tag{C} \end{align*} \]

unde nodurile \(x_{0}, x_{1}, \ldots, x_{n}\) sînt în progresie aritmetică cu rația \(h\), şi a aplicat această formulă in problema mareelor.

Este bine să se compare formula (C) cu următoarele formule de evadratură, bine cunoscute :

\[ \begin{gather*} \int_{x_{0}}^{x_{n}} f(x) \mathrm{d} x=\frac{h}{2}\left[f\left(x_{a}\right)+f\left(x_{n}\right)\right]+h\left[f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots+f\left(x_{n-1}\right)\right] \tag{T}\\ \left.\int_{x_{1}}^{x_{n}} f(x) \mathrm{d} x=\frac{5 h}{12} f\left(x_{0}\right)+f\left(x_{n}\right)\right]+\frac{13 h}{12}\left[f\left(x_{1}\right)+f\left(x_{n-1}\right)+h\left[f\left(x_{2}\right)+f\left(x_{3}\right)+\ldots\right.\right. \\ \left.\ldots+f\left(x_{n-2}\right)\right] \tag{D}\\ \int_{x_{0}}^{x_{n}} f(x) \mathrm{d} x=\frac{3 h}{8}\left[f\left(x_{0}\right)+f\left(x_{n}\right)\right]+\frac{7 h}{6}\left[f\left(x_{1}\right)+f\left(x_{n-1}\right)\right]+ \\ \quad+\frac{23 h}{24}\left[f\left(x_{2}\right)+f\left(x_{n-2}\right)\right]+h\left[f\left(x_{3}\right)+f\left(x_{4}\right)+\ldots+f\left(x_{n-3}\right)\right] \tag{L} \end{gather*} \]

Formula (T) se deduce din formula clasică a trapezului. Formulele (D) şi (L) sînt citate in cartea lui H. Mineur [2] şi sînt numite formulele lui Durand şi Lacroix.

In aceasta lucrare vom da expresile rostului in formule (C), (D) si (L) din care vom trage conclusii cu privire la aplicarea uneia sau alteia in praotica.
I. RESTUL IN FORMULA (C)

Pentru s gåsi restnl ei, se presupune cá functia \(f(x)\) este de class \(C^{9}\left[x_{n}, x_{n}\right]\) si se splical metods intrebuintatas in mod constant in lucrares [3]. Se giseste

\[ \begin{equation*} R_{c}=\int_{e_{c}}^{e^{*}} \varphi(x) f^{\prime \prime}(x) \mathrm{d} x, \tag{1} \end{equation*} \]

unde functia \(\varphi(x)\) coincide pe intervalele \(\left[x_{6}, x_{1}\right],\left[x_{1}, x_{2}\right], \ldots\) cu unnátoarale polinoame de gradul al doiles, pe care le scriem numal pentru prima jumätate a intervalului \(\left[x_{0}, x_{0}\right]\) :
\(\varphi_{1}(x)=\frac{\left(x-x_{0}\right)^{2}}{2}-\frac{13 h}{36}\left(x-x_{0}\right), \quad \varphi_{1}(x)=\frac{\left(x-x_{1}\right)^{\Omega}}{2}-\frac{19 h}{36}\left(x-x_{1}\right)+\frac{5 h^{2}}{36}\),
\(\varphi_{2}(x)=\frac{\left(x-x_{2}\right)^{\prime}}{2}-\frac{19 h}{36}\left(x-x_{2}\right)+\frac{4 h^{\prime}}{36}, \quad \varphi_{k}(x)=\frac{\left(x-x_{k-1}\right)^{\prime}}{2}-\frac{h}{2}\left(x-x_{k-1}\right)+ +\frac{h^{2}}{12}(k=4,5, \ldots)\).
Se constats cå

\[ \int_{x_{1}}^{\alpha_{1}} \varphi_{1}(x) \mathrm{d} x=-\frac{h^{3}}{72}, \quad \int_{x_{1}}^{x_{1}} \varphi_{3}(x) \mathrm{d} x=\frac{3 h^{3}}{72}, \quad \int_{x_{2}}^{y_{2}} \varphi_{3}(x) \mathrm{d} x=\frac{h^{8}}{72}, \]

\[ \int_{k-1}^{x_{k}} \varphi_{k}(x) \mathrm{d} x=0 \quad(k=4,5, \ldots) . \]

Gradul de exactitate al formules (C) este 1, deoarece in formula (1) avem

\[ \int_{-4}^{x_{n}} \varphi(x) \mathrm{d} x=\frac{h^{3}}{12} . \]

Din formule (1) se deduce o evaluare s lui \(R_{c}\) şi snume

\[ \begin{equation*} \left|R_{c}\right| \leq K_{c} M_{2} h^{3}, \tag{2} \end{equation*} \]

unde 8-a notat

\[ \begin{equation*} M_{2}=\sup _{\left[x_{n}=x_{n}\right]}\left|f^{\prime \prime}(x)\right| \tag{3} \end{equation*} \]

iar

\[ \begin{equation*} R_{c} h^{s}=\int_{\alpha_{0}}^{x_{n}}|\varphi(x)| d \omega \tag{4} \end{equation*} \]

citeva formule practice de cuadratura
Wind calculele se gaseste

\[ \begin{aligned} & \int_{A_{0}}^{x_{1}}\left|\varphi_{1}(x)\right| \mathrm{d} x=\frac{1}{34} \frac{711}{992} h^{8}, \quad \int_{x_{4}}^{x_{2}}\left|\varphi_{\mathrm{e}}(x)\right| \mathrm{d} x=\frac{1459}{34992} h^{2} \\ & \int_{x_{4}}^{x_{2}}\left|\varphi_{1}(x)\right| \mathrm{d} x=\frac{486+73 \sqrt{73}}{34992} h^{3}, \quad \int_{x_{3}}^{z_{4}}\left|\varphi_{4}(x)\right| \mathrm{d} x=\frac{\sqrt{3}}{54} h^{2} . \end{aligned} \]

Tinind seama de simetria curbei \(y=\varphi(x)\) faţd de dreapta \(x=\) got \(x_{2}\), avem
\(\int_{\pi_{*}}^{\pi_{*}}|\varphi(x)| \mathrm{d} x=\left[2 \frac{1711}{34992}+2^{\prime} \frac{1459}{24 \theta 92}+2-\frac{486+73 / 7 \overline{3}}{34982}+\left.(n-6) \frac{\sqrt{3}}{54}\right|_{\mathrm{A}}\right.\) si prin urmare

\[ \begin{equation*} K_{c}=\frac{3650+73 \sqrt{73}-1944 \sqrt{3}}{17496}+\frac{n \psi \overline{3}}{54} \tag{b} \end{equation*} \]

Presupunind că functia \(f(x)\) este de ulasa \(C^{2}\left[x_{01} x_{n}\right]\) se deduce eă restul \(R_{D}\) are forma

\[ \begin{equation*} R_{D}=\int_{n_{0}}^{x_{m}} \psi(x) f^{\prime \prime}(x) \mathrm{d} x, \tag{$\theta$} \end{equation*} \]

unde functia \(\psi(x)\) coincide pe intervalele \(\left[x_{0}, x_{1}\right],\left[x, x_{1}\right] \ldots\) eu polinoamele \(\psi_{1}(x), \psi_{2}(x), \ldots\) date de ecuatiile
\(\psi_{1}(x)=\frac{\left(x-x_{0}\right)^{1}}{2}-\frac{5 h}{12}\left(x-x_{0}\right), \quad \psi_{k}(x)=\frac{\left(x-x_{k-1}\right)^{1}}{2}-\frac{h}{2}\left(x-x_{1-1}\right)+\frac{h^{1}}{12}\)

\[ (k=2,3, \cdots) . \]

Se constată ca

\[ \int_{x_{1}}^{x_{1}} \psi_{1}(x) \mathrm{d} x=-\frac{h^{3}}{24}, \quad \int_{k+1}^{x_{k}} \psi_{1}(x) \mathrm{d} x=0 \quad(k=2,3, \ldots) \]

si de sici rezultä es

\[ \int_{x_{0}}^{x_{n}} \psi(x) \mathrm{d} x=-\frac{h^{3}}{12}, \]

cea ce inseamná ch gradul de exactitate al formulei (D) este 1.
Din formula (0) se deduce ed

\[ \begin{equation*} \left|R_{D}\right| \leq K_{D} M_{B} h^{3} \tag{T} \end{equation*} \]

\[ \text { ande } K_{b} A^{\mathrm{a}}=\int_{M_{0}}^{x_{0}}|\Psi(x)| d x . \]

Facind calculale se gäseste

\[ \int_{x_{0}}^{\pi_{1}}\left|\psi_{1}(x)\right| d x=\frac{71 k^{8}}{1296}, \quad \int_{\alpha_{1}}^{\pi_{2}}\left|\psi_{8}(x)\right| d x=\frac{V \overline{3}}{54} h^{0} \]

şi tinind seamà de simetria curbei \(y=\psi(x)\) fafa de dreapta \(x=\frac{x_{0}+x_{0}}{2}\) avem

\[ \int_{x_{0}}^{x_{0}}\left[\psi(x) \left\lvert\, \mathrm{d} x=\left[2 \frac{71}{1296}+(\pi-2) \frac{V \overline{3}}{54}\right] A^{3}=\left(\frac{71-12 V 3}{648}+\frac{n V \overline{3}}{54}\right) h^{3}\right.\right. \]

\[ \begin{equation*} K_{D}=\frac{71-12 \sqrt{3}}{648}+\frac{n \sqrt{3}}{54} . \tag{8} \end{equation*} \]

Presupunind ca functis \(f(x)\) este de class \(C^{2}\left[x_{01} x_{p}\right]\) putem serie reatul ei sub forma

\[ \begin{equation*} R_{L}=\int_{x_{v}}^{x_{n}} \theta(x) \int^{\prime \prime}(x) \mathrm{d} x \tag{9} \end{equation*} \]

unde functia \(\theta(x)\) coincide pe intervalele \(\left[x_{0}, x_{1}\right]\left[x_{1}, x_{1}\right], \ldots\) eu polinosmele

\[ \begin{gathered} \theta_{1}(x)=\frac{\left(x-x_{0}\right)^{2}}{2}-\frac{3 h}{8}\left(x-x_{0}\right), \quad \theta_{3}(x)=\frac{\left(x-x_{1}\right)^{2}}{2}-\frac{13 h}{24}\left(x-x_{1}\right)+\frac{h^{2}}{8} \\ \theta_{k}(x)=\frac{\left(x-x_{k-1}\right)^{2}}{2}-\frac{h}{2}\left(x-x_{k-1}\right)+\frac{h^{2}}{12} \quad(k=3,4, \ldots) \end{gathered} \]

Avem

\[ \int_{x_{0}}^{x_{1}} \theta_{1}(x) \mathrm{d} x=-\frac{h^{8}}{48}, \quad \int_{x_{1}}^{x_{1}} \theta_{2}(x) \mathrm{d} x=\frac{h^{8}}{48} \cdot \int_{x_{1}}^{x_{1}} \theta_{3}(x) \mathrm{d} x=0 . \]

Rezulta cal

\[ \int_{x_{0}}^{z_{u}} \theta(x) \mathrm{d} x=0 \]

ceea ce inseamna cá gradul de enactitate al formulei (L) este mui mare ca 1. Färä greutate se constatä cl gradul de enactitate al formulei (L) este ib.

CTEVA FORMULE PRACTICE DE CUADRATURA
603
pin formuls ( 9 ) se deduce cá

\[ \begin{gathered} \left|R_{t}\right| \leq K_{L} M_{2} A^{*} \\ K_{t} h^{3}=\int_{x_{0}}^{x_{*}}|\theta(x)| d x \end{gathered} \]

Facind calculele se gáseste
\(\int_{x_{1}}^{x_{1}}\left|0_{1}(x)\right| \mathrm{d} x=\frac{19}{384} h^{8}, \quad \int_{\alpha_{1}}^{x_{1}}\left|\theta_{2}(x)\right| \mathrm{d} x=\frac{341}{10368} h^{2}, \int_{\alpha_{0}}^{\alpha_{1}}\left|\theta_{1}(x)\right| \mathrm{d} x=\frac{\sqrt{3}}{54} h^{1}\).
Regultă că

\[ \begin{gathered} \int_{x_{n}}^{x_{n}}|\theta(x)| d x=2\left(\frac{19}{384}+\frac{341}{10368}\right) h^{3}+(n-4) \frac{\sqrt{3}}{54} h^{2}=\frac{427}{2592} h^{2}+ \\ +(n-4) \frac{\sqrt{3}}{54} h^{2} \end{gathered} \]

aj decl

\[ \begin{equation*} K_{L}=\frac{427-192 \sqrt{3}}{\overline{2} 592}+n \frac{\sqrt{3}}{54} . \tag{11} \end{equation*} \]

Dupå cum se stie in formula (T), presupunind ch functiaff(x) este de class \(C^{1}\left[x_{0}, x_{n}\right]\), evem

\[ \begin{equation*} R_{T}=\int_{x_{0}}^{a_{x}} x(x) f^{\prime \prime}(x) d x \tag{12} \end{equation*} \]

unde functia \(\chi(x)\) coincide pe intervalele \(\left[x_{0}, x_{1}\right],\left[x_{1}, x_{0}\right], \ldots\) cu polinoamele

\[ \chi_{1}(x)=\frac{\left(x-x_{0}\right)\left(x-x_{1}\right)}{2}, \quad \chi_{1}(x)=\frac{\left(x-x_{1}\right)\left(x-x_{1}\right)}{2}, \ldots \]

So constates că

\[ \begin{equation*} \left(R_{T}\right) \leq K_{T} M_{2} h^{\mathrm{a}} \tag{13} \end{equation*} \]

unde

\[ \begin{equation*} K_{r}=\frac{n}{12} . \tag{(14} \end{equation*} \]

So constata cal avem

\[ \begin{gathered} K_{C}-K_{L}=\frac{1095+292|73-2592| 3}{69984}=0,0157 \ldots, \\ K_{p}-K_{c}=1620+3-73 / 73-1739=0,0253 . \end{gathered} \]

De asemenea, se constata ca arem

\[ K_{r}-K_{D}=\frac{(54-12 \mid 3)(n-2)+(37-12 \mid \overline{3})}{648}>0 \quad(n \geq 2) \]

de unde rezultä că avem

\[ \begin{equation*} \boldsymbol{K}_{t}<\boldsymbol{K}_{r}<\boldsymbol{K}_{\mathrm{D}}<\boldsymbol{K}_{T} . \tag{15} \end{equation*} \]

In concluxie, am determinat si evaluat resturile in formulele (C), (D), (L). Din inegalitapile (15) rezults cá ces mai avantajoasa este formula (L), dups care urmeass imedist, cu o foarte mica diferentā pentru coeficiantul \(K\), formula (C). Ces mai neavantajoss dintre toate formulele considerate in aceasta nots este formula (T).

Vom de intro nond nots o metods pentru se construi formulele (C), (D), (L) jrecum ai altele analoge.

Academia R.P.R. Filiala Cluj,
Institutului de calcul

On détermine les restes des formules de quadrature (C), (D), (L), (T), par les formules (1), (6), (9), (12), et le s évaluations (2), (7), (10), (13), caractériséca par les nombres \(K_{G}, K_{D}, K_{L}, K_{T}\) donnés par les formules

Cateva formule practice de cuadratura
895
(1), (14). On démontre les inégalitérs, (15), d'où í résulte que la (1). Ja (1) est la plus aventageuse, suivie de la formule (C), et que la forficulo (T) est la plus désavantageuse.

BIBLIOGRAPTE
G Goblarl Operalions sur les commes experimeniales, C.F. Aced. Sel. Fatii, 1058, e46, 1799-1800.
Minkull, Technique de Caleut namérique, Libralite Polylechnique Ch. Béranger, Paris, 1052, 244.
3. D. V. IONFBCT, CDOdraturi numerice, Edit. Lehnica, Bucurest, 1957.