1963-IonescuDV-Some-practical-formulas-of-quadrature-in-Communications-Acad.RPRvol.13nr.8-august1963pp
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COMMUNICATIONS
OF THE ACADEMY OF THE ROMANIAN PEOPLE'S REPUBLIC
Volume XIII, No. 8
August 1963
SOME PRACTICAL QUADRATURE FORMULAS
BY
D. V. IONESCU
Communication presented by academician T. Popoviciu in the journal of May 18, 1963
G. Coulmy [1] considered the following quadrature formula
{:[int_(x_(n))^(x_(n))f(x)dx=(13 h)/(36)[f(x_(0))+f(x_(n))]+(7h)/(6)[f(x_(1))+f(x_(n-1))]+h[f(x_(n))+f(x_(n-2))]+],[(C)quad+(35 h)/(36)[f(x_(3))+f(x_(n-3))]+h[f(x_(4))+f(x_(5))+cdots+f(x_(n-4))]]:}\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*}
where the nodesx_(0),x_(1),dots,x_(n)x_{0}, x_{1}, \ldots, x_{n}are in arithmetic progression with the ratiohh, and applied this formula to the problem of tides.
It is good to compare formula (C) with the following well-known escapement formulas:
{:[(T)int_(x_(0))^(x_(n))f(x)dx=(h)/(2)[f(x_(a))+f(x_(n))]+h[f(x_(1))+f(x_(2))+dots+f(x_(n-1))]],[{:int_(x_(1))^(x_(n))f(x)dx=(5h)/(12)f(x_(0))+f(x_(n))]+(13 h)/(12)[f(x_(1))+f(x_(n-1))+h[f(x_(2))+f(x_(3))+dots:}],[(D){: dots+f(x_(n-2))]],[int_(x_(0))^(x_(n))f(x)dx=(3h)/(8)[f(x_(0))+f(x_(n))]+(7h)/(6)[f(x_(1))+f(x_(n-1))]+],[(L)quad+(23 h)/(24)[f(x_(2))+f(x_(n-2))]+h[f(x_(3))+f(x_(4))+dots+f(x_(n-3))]]:}\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) is derived from the classical trapezoid formula. Formulas (D) and (L) are cited in the book by H. Mineur [2] and are called the Durand and Lacroix formulas.
In this paper we will give the expressions of the ratio in formulas (C), (D) and (L) from which we will draw conclusions regarding the application of one or the other in practice.
I. THE REMAINDER IN FORMULA (C)
To find its remainder, it is assumed that the functionf(x)f(x)it is classyC^(9)[x_(n),x_(n)]C^{9}\left[x_{n}, x_{n}\right]and it is a method used consistently in the work [3]. It is found
{:(1)R_(c)=int_(e_(c))^(e^(**))varphi(x)f^('')(x)dx",":}\begin{equation*}
R_{c}=\int_{e_{c}}^{e^{*}} \varphi(x) f^{\prime \prime}(x) \mathrm{d} x, \tag{1}
\end{equation*}
where the functionvarphi(x)\varphi(x)coincides on the intervals[x_(6),x_(1)],[x_(1),x_(2)],dots\left[x_{6}, x_{1}\right],\left[x_{1}, x_{2}\right], \ldotswith some natural polynomials of the second degree, which we write only for the first half of the interval[x_(0),x_(0)]\left[x_{0}, x_{0}\right] :
varphi_(1)(x)=((x-x_(0))^(2))/(2)-(13 h)/(36)(x-x_(0)),quadvarphi_(1)(x)=((x-x_(1))^(Omega))/(2)-(19 h)/(36)(x-x_(1))+(5h^(2))/(36)\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)=((x-x_(2))^('))/(2)-(19 h)/(36)(x-x_(2))+(4h^('))/(36),quadvarphi_(k)(x)=((x-x_(k-1))^('))/(2)-(h)/(2)(x-x_(k-1))++(h^(2))/(12)(k=4,5,dots)\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).
If you find cå
int_(x_(1))^(alpha_(1))varphi_(1)(x)dx=-(h^(3))/(72),quadint_(x_(1))^(x_(1))varphi_(3)(x)dx=(3h^(3))/(72),quadint_(x_(2))^(y_(2))varphi_(3)(x)dx=(h^(8))/(72),\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)dx=0quad(k=4,5,dots).\int_{k-1}^{x_{k}} \varphi_{k}(x) \mathrm{d} x=0 \quad(k=4,5, \ldots) .
The degree of accuracy of formula (C) is 1, because in formula (1) we have
int_(-4)^(x_(n))varphi(x)dx=(h^(3))/(12).\int_{-4}^{x_{n}} \varphi(x) \mathrm{d} x=\frac{h^{3}}{12} .
From formulas (1) an evaluation of s is deducedR_(c)R_{c}and name
{:(2)|R_(c)| <= K_(c)M_(2)h^(3)",":}\begin{equation*}
\left|R_{c}\right| \leqslant K_{c} M_{2} h^{3}, \tag{2}
\end{equation*}
where 8-noted
{:(3)M_(2)=s u p_([x_(n)=x_(n)])|f^('')(x)|:}\begin{equation*}
M_{2}=\sup _{\left[x_{n}=x_{n}\right]}\left|f^{\prime \prime}(x)\right| \tag{3}
\end{equation*}
and
{:(4)R_(c)h^(s)=int_(alpha_(0))^(x_(n))|varphi(x)|d omega:}\begin{equation*}
R_{c} h^{s}=\int_{\alpha_{0}}^{x_{n}}|\varphi(x)| d \omega \tag{4}
\end{equation*}
some practical formulas for quadrature
Wind calculations can be found
{:[int_(A_(0))^(x_(1))|varphi_(1)(x)|dx=(1)/(34)(711)/(992)h^(8)","quadint_(x_(4))^(x_(2))|varphi_(e)(x)|dx=(1459)/(34992)h^(2)],[int_(x_(4))^(x_(2))|varphi_(1)(x)|dx=(486+73sqrt73)/(34992)h^(3)","quadint_(x_(3))^(z_(4))|varphi_(4)(x)|dx=(sqrt3)/(54)h^(2).]:}\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}
Taking into account the symmetry of the curvey=varphi(x)y=\varphi(x)right sidex=x= got x_(2)x_{2}, we have
int_(pi_(**))^(pi_(**))|varphi(x)|dx=[2(1711)/(34992)+2^(')(1459)/(24 theta92)+2-(486+73//7 bar(3))/(34982)+(n-6)(sqrt3)/(54)|_(A):}\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.and therefore
{:(b)K_(c)=(3650+73sqrt73-1944sqrt3)/(17496)+(n psi bar(3))/(54):}\begin{equation*}
K_{c}=\frac{3650+73 \sqrt{73}-1944 \sqrt{3}}{17496}+\frac{n \psi \overline{3}}{54} \tag{b}
\end{equation*}
- HESTUL IN FORMULA (D)
Assuming that the functionf(x)f(x)it is annoyingC^(2)[x_(01)x_(n)]C^{2}\left[x_{01} x_{n}\right]the rest is deductedR_(D)R_{D}has the form
{:(theta")"R_(D)=int_(n_(0))^(x_(m))psi(x)f^('')(x)dx",":}\begin{equation*}
R_{D}=\int_{n_{0}}^{x_{m}} \psi(x) f^{\prime \prime}(x) \mathrm{d} x, \tag{$\theta$}
\end{equation*}
where the functionpsi(x)\psi(x)coincides on the intervals[x_(0),x_(1)],[x,x_(1)]dots\left[x_{0}, x_{1}\right],\left[x, x_{1}\right] \ldotsI polynomialspsi_(1)(x),psi_(2)(x),dots\psi_{1}(x), \psi_{2}(x), \ldotsgiven by the equations
psi_(1)(x)=((x-x_(0))^(1))/(2)-(5h)/(12)(x-x_(0)),quadpsi_(k)(x)=((x-x_(k-1))^(1))/(2)-(h)/(2)(x-x_(1-1))+(h^(1))/(12)\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).(k=2,3, \cdots) .
It is found that
int_(x_(1))^(x_(1))psi_(1)(x)dx=-(h^(3))/(24),quadint_(k+1)^(x_(k))psi_(1)(x)dx=0quad(k=2,3,dots)\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)
and therefore the result is
int_(x_(0))^(x_(n))psi(x)dx=-(h^(3))/(12),\int_{x_{0}}^{x_{n}} \psi(x) \mathrm{d} x=-\frac{h^{3}}{12},
which means that the degree of accuracy of formula (D) is 1.
From formula (0) it is deduced that
{:(T)|R_(D)| <= K_(D)M_(B)h^(3):}\begin{equation*}
\left|R_{D}\right| \leqslant K_{D} M_{B} h^{3} \tag{T}
\end{equation*}
" ande "K_(b)A^(a)=int_(M_(0))^(x_(0))|Psi(x)|dx.\text { ande } K_{b} A^{\mathrm{a}}=\int_{M_{0}}^{x_{0}}|\Psi(x)| d x .
Doing the calculations, we find
int_(x_(0))^(pi_(1))|psi_(1)(x)|dx=(71k^(8))/(1296),quadint_(alpha_(1))^(pi_(2))|psi_(8)(x)|dx=(V bar(3))/(54)h^(0)\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}
and taking into account the symmetry of the curvey=psi(x)y=\psi(x)right sidex=(x_(0)+x_(0))/(2)x=\frac{x_{0}+x_{0}}{2}HAVE
int_(x_(0))^(x_(0))[psi(x)|dx=[2(71)/(1296)+(pi-2)(V bar(3))/(54)]A^(3)=((71-12 V3)/(648)+(nV bar(3))/(54))h^(3):}\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.
{:(8)K_(D)=(71-12sqrt3)/(648)+(nsqrt3)/(54).:}\begin{equation*}
K_{D}=\frac{71-12 \sqrt{3}}{648}+\frac{n \sqrt{3}}{54} . \tag{8}
\end{equation*}
Presuming that it worksf(x)f(x)it is classyC^(2)[x_(01)x_(p)]C^{2}\left[x_{01} x_{p}\right]we can serialize her crime in the form
{:(9)R_(L)=int_(x_(v))^(x_(n))theta(x)int^('')(x)dx:}\begin{equation*}
R_{L}=\int_{x_{v}}^{x_{n}} \theta(x) \int^{\prime \prime}(x) \mathrm{d} x \tag{9}
\end{equation*}
where the functiontheta(x)\theta(x)coincides on the intervals[x_(0),x_(1)][x_(1),x_(1)],dots\left[x_{0}, x_{1}\right]\left[x_{1}, x_{1}\right], \ldotsI polynomials
{:[theta_(1)(x)=((x-x_(0))^(2))/(2)-(3h)/(8)(x-x_(0))","quadtheta_(3)(x)=((x-x_(1))^(2))/(2)-(13 h)/(24)(x-x_(1))+(h^(2))/(8)],[theta_(k)(x)=((x-x_(k-1))^(2))/(2)-(h)/(2)(x-x_(k-1))+(h^(2))/(12)quad(k=3","4","dots)]:}\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}
we
int_(x_(0))^(x_(1))theta_(1)(x)dx=-(h^(8))/(48),quadint_(x_(1))^(x_(1))theta_(2)(x)dx=(h^(8))/(48)*int_(x_(1))^(x_(1))theta_(3)(x)dx=0.\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 .
Result horse
int_(x_(0))^(z_(u))theta(x)dx=0\int_{x_{0}}^{z_{u}} \theta(x) \mathrm{d} x=0
which means that the degree of enactity of the formula (L) is much greater than 1. It is easily seen that the degree of enactity of the formula (L) is ib.
SOME PRACTICAL FORMULAS OF SQUAREDNESS
603
From formula (9) it follows that
{:[|R_(t)| <= K_(L)M_(2)A^(**)],[K_(t)h^(3)=int_(x_(0))^(x_(**))|theta(x)|dx]:}\begin{gathered}
\left|R_{t}\right| \leqslant K_{L} M_{2} A^{*} \\
K_{t} h^{3}=\int_{x_{0}}^{x_{*}}|\theta(x)| d x
\end{gathered}
Doing the calculations we find
int_(x_(1))^(x_(1))|0_(1)(x)|dx=(19)/(384)h^(8),quadint_(alpha_(1))^(x_(1))|theta_(2)(x)|dx=(341)/(10368)h^(2),int_(alpha_(0))^(alpha_(1))|theta_(1)(x)|dx=(sqrt3)/(54)h^(1)\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}.
It is a rule that
{:[int_(x_(n))^(x_(n))|theta(x)|dx=2((19)/(384)+(341)/(10368))h^(3)+(n-4)(sqrt3)/(54)h^(2)=(427)/(2592)h^(2)+],[+(n-4)(sqrt3)/(54)h^(2)]:}\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}
I declare
{:(11)K_(L)=(427-192sqrt3)/( bar(2)592)+n(sqrt3)/(54).:}\begin{equation*}
K_{L}=\frac{427-192 \sqrt{3}}{\overline{2} 592}+n \frac{\sqrt{3}}{54} . \tag{11}
\end{equation*}
- COMPARISON OF NUMBERSK_(C),K_(D),K_(L)K_{C}, K_{D}, K_{L}, WITH THE NUMBERK_(", ")K_{\text {, }}CORRESPONDING TO THE FORMULA (T)
As is known in formula (T), assuming that the function aff(x) is of classC^(1)[x_(0),x_(n)]C^{1}\left[x_{0}, x_{n}\right], evem
{:(12)R_(T)=int_(x_(0))^(a_(x))x(x)f^('')(x)dx:}\begin{equation*}
R_{T}=\int_{x_{0}}^{a_{x}} x(x) f^{\prime \prime}(x) d x \tag{12}
\end{equation*}
where the functionchi(x)\chi(x)coincides on the intervals[x_(0),x_(1)],[x_(1),x_(0)],dots\left[x_{0}, x_{1}\right],\left[x_{1}, x_{0}\right], \ldotswith polynomials
chi_(1)(x)=((x-x_(0))(x-x_(1)))/(2),quadchi_(1)(x)=((x-x_(1))(x-x_(1)))/(2),dots\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
It is noted that
{:(13)(R_(T)) <= K_(T)M_(2)h^(a):}\begin{equation*}
\left(R_{T}\right) \leqslant K_{T} M_{2} h^{\mathrm{a}} \tag{13}
\end{equation*}
where
{:((14)K_(r)=(n)/( 12).:}\begin{equation*}
K_{r}=\frac{n}{12} . \tag{(14}
\end{equation*}
Let's see what we have.
{:[K_(C)-K_(L)=(1095+292|73-2592|3)/(69984)=0","0157 dots","],[K_(p)-K_(c)=1620+3-73//73-1739=0","0253.]:}\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}
It is also found that we have
K_(r)-K_(D)=((54-12∣3)(n-2)+(37-12∣ bar(3)))/(648) > 0quad(n >= 2)K_{r}-K_{D}=\frac{(54-12 \mid 3)(n-2)+(37-12 \mid \overline{3})}{648}>0 \quad(n \geqslant 2)
from which it follows that we have
{:(15)K_(t) < K_(r) < K_(D) < K_(T).:}\begin{equation*}
\boldsymbol{K}_{t}<\boldsymbol{K}_{r}<\boldsymbol{K}_{\mathrm{D}}<\boldsymbol{K}_{T} . \tag{15}
\end{equation*}
In conclusion, we have determined and evaluated the residues in formulas (C), (D), (L). From inequalities (15) it follows that the more advantageous is formula (L), which immediately follows, with a very small difference for the coefficientKK, formula (C). The most disadvantageous of all the formulas considered in this note is formula (T).
We will now introduce a method for constructing formulas (C), (D), (L) and other analogous ones.
RPR Academy Cluj Branch,
Computing Institute
SUMMARY
The remainders of the quadrature formulas (C), (D), (L), (T) are determined by the formulas (1), (6), (9), (12), and the evaluations (2), (7), (10), (13), characterized by the numbersK_(G),K_(D),K_(L),K_(T)K_{G}, K_{D}, K_{L}, K_{T}given by the formulas
Cateva formula practice of cuadratura
895
(1), (14). We demonstrate the inequalities, (15), from which it follows that (1). Ja (1) is the most advantageous, followed by formula (C), and that formula (T) is the most disadvantageous.
BIBLIOGRAPHY
G Goblarl Operalions sur les commes experimeniales, CF Aced. Sel. Fatii, 1058, e46, 1799-1800.
Minkull, Technique de Caleut namérique, Libralite Polylechnique Ch. Béranger, Paris, 1052, 244.
3. DV IONFBCT, CDOdraturi numerice, Edit. Lehnica, Bucurest, 1957.
