Posts by Tiberiu Popoviciu

Tiberiu Popoviciu

Abstract

Original title (in German)

Das Restglied in einigen Formeln der numerischen Integration von Differentialgleichungen

Authors

T. Popoviciu
Institutul de Calcul

Keywords

?

Paper coordinates

T. Popoviciu, Das Restglied in einigen Formeln der numerischen Integration von Differentialgleichungen, Methoden und Verfahren der mathematischen Physik, Band 5, pp. 117-129. B. I.-Hochschulskripten, No. 724 a-b, Bibliographisches Inst., Mannheim, 1971 (in German)

PDF

1971 a -Popoviciu- Methoden und Verfahren der mathematischen Physik – Das restglied in einigen formeln der numerischen Integration von Differentialgleichungen

About this paper

Journal
Publisher Name
DOI
Print ISSN
Online ISSN

[MR0373303, Zbl 0231.65065]

google scholar link

[1] G. Kowalewski, Interpolation and Approximate Quadrature, 1932.
[2] L. Collatz, Numerical Treatment of Differential Equations, 1955.
[3] T. Popoviciu, Les fonctions convexes, Paris, 1945.
[4] T. Popoviciu, “Asupra formei restului în unele formule de aproximare ale analizei,” Lucrările Sesiunii Generale a Academiei RPR, 1950, pp. 183–185.
[5] T. Popoviciu, “Sur le reste dans certaines formules linéaires d’approximation de l’analyse,” Mathematica (Cluj), vol. 1 (24), pp. 95–142, 1959.
[6] T. Popoviciu, “The simplicity of the rest in certain quadrature formulas,” Mathematica (Cluj), vol. 6 (29), pp. 157–184, 1964.

Paper (preprint) in HTML form

1971 a -Popoviciu- Methods and procedures of mathematical physics - The remainder term in some forms
Original text
Rate this translation
Your feedback will be used to help improve Google Translate

MATHEMATICAL PHYSICS

Volume 5 - February 1971

BIBLIOGRAPHICAL INSTITUTE AGMANNHEIM/VIENNA/ZURICH

THE RESIDUE TERM IN SOME FORMULAS OF NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Tiberiu Popoviciu

This refers to [ y 1 , y 2 , … , y k + 1 ; f ] y 1 , y 2 , … , y k + 1 ; f [y_(1),y_(2),dots,y_(k+1);f]\left[y_{1}, y_{2}, \ldots, y_{k+1} ; f\right][y1,y2,…,yk+1;f]the divided difference k k kkk-th order of the function f f fff, formed with the nodes y 1 , y 2 , … , y k + 1 y 1 , y 2 , … , y k + 1 y_(1),y_(2),dots,y_(k+1)y_{1}, y_{2}, \ldots, y_{k+1}y1,y2,…,yk+1. These nodes may or may not be different from each other. In the latter case, the divided difference is determined using the derivatives of the function f f fffdefined. For example,
[ y , y , … , y ⏟ k + 1 ; f ] = 1 k ! f ( k ) ( y ) [ y , y , … , y ⏟ k + 1 ; f ] = 1 k ! f ( k ) ( y ) [ubrace(y,y,dots,yubrace)_(k+1);f]=(1)/(k!)f^((k))(y)[\underbrace{y, y, \ldots, y}_{k+1} ; f]=\frac{1}{k!} f^{(k)}(y)[y,y,…,y⏟k+1;f]=1k!f(k)(y)
For simplicity, we denote D k [ f ] D k [ f ] D_(k)[f]D_{k}[f]Dk[f]the divided difference k k kkk-th order of the function f f fff, formed with a certain system of k + 1 k + 1 k+1k+1k+1different nodes from the interval I I III, then relationship (1) can be written as follows:
R [ f ] = R [ x n + 1 ] D n + 1 [ f ] . R [ f ] = R x n + 1 D n + 1 [ f ] . R[f]=R[x^(n+1)]D_(n+1)[f].R[f]=R\left[x^{n+1}\right] D_{n+1}[f] .R[f]=R[xn+1]Dn+1[f].
  1. Is the linear functional R [ f ] R [ f ] R[f]R[f]R[f]of simple form, then it has the degree of accuracy n n nnn, i.e. for all polynomials n n nnn-th degree it is zero, but not for all polynomials ( n + 1 n + 1 n+1n+1n+1)-th degree (the only polynomial with effective degree -1 is the polynomial identical to zero). n n nnnthe integer for which the corresponding relation (1) holds. However, this condition is generally not sufficient, since R [ f ] R [ f ] R[f]R[f]R[f]is of simple form.
In contrast, the following sentence applies:
A linear functional R [ f ] R [ f ] R[f]R[f]R[f]is of simple form if and only if it is an integer n ≥ − 1 n ≥ − 1 n >= -1n \geq-1n≥−1so that R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0is for all convex functions f n f n fnfnfn-th order S S SSS.
A function is called f f fffconvex of order n n nnnon I I III, if their divided difference ( n + 1 n + 1 n+1n+1n+1)-th order, formed with any system of n + 2 n + 2 n+2n+2n+2different points of the interval I I III, is positive. If these divided differences are not negative, we speak of a non-concave function n n nnn-th order. Finally, a function is called f f fffconcave or non-convex of order n n nnn, if − f − f -f-f−fconvex or non-concave of order n n nnnis.
The proof of the above theorem is simple. I have given further related properties in my previous works [3], [4], [5].
Is n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0, the points can ξ v , v = 1 , 2 , … , n + 2 ξ v , v = 1 , 2 , … , n + 2 xi_(v),v=1,2,dots,n+2\xi_{v}, v=1,2, \ldots, n+2ξv,v=1,2,…,n+2in formula (1) even from the interior of the interval I I IIIbe selected.
3. If the divided difference ( n + 1 n + 1 n+1n+1n+1)-th order of the function f f ffflimited, ie
(2) | [ x 1 , x 2 , … , x n + 2 ; f ] | ≦ M ( M finally ) (2) x 1 , x 2 , … , x n + 2 ; f ≦ M ( M  finally  ) {:(2)|[x_(1),x_(2),dots,x_(n+2);f]| <= M quad(M" finite "):}\begin{equation*} \left|\left[x_{1}, x_{2}, \ldots, x_{n+2} ; f\right]\right| \leqq M \quad(M \text { finite }) \tag{2} \end{equation*}(2)|[x1,x2,…,xn+2;f]|≦M(M finally )
for all systems from n + 2 n + 2 n+2n+2n+2different points x 1 , x 2 , … , x n + 2 x 1 , x 2 , … , x n + 2 x_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}x1,x2,…,xn+2out of I I III, one obtains from (1) the following estimate for R [ f ] R [ f ] R[f]R[f]R[f]:
| R [ f ] | ≦ | R [ x n + 1 ] | M . | R [ f ] | ≦ R x n + 1 M . |R[f]| <= |R[x^(n+1)]|M.|R[f]| \leqq\left|R\left[x^{n+1}\right]\right| M .|R[f]|≦|R[xn+1]|M.
Is the function f ( n + 1 ) f ( n + 1 ) f(n+1)f(n+1)f(n+1)-times differentiable, then formula (1) can be written as
(4) R [ f ] = R [ x n + 1 ] f ( n + 1 ) ( ξ ) ( n + 1 ) ! , ξ ∈ I (4) R [ f ] = R x n + 1 f ( n + 1 ) ( ξ ) ( n + 1 ) ! , ξ ∈ I {:(4)R[f]=R[x^(n+1)](f^((n+1))(xi))/((n+1)!)","quad xi in I:}\begin{equation*} R[f]=R\left[x^{n+1}\right] \frac{f^{(n+1)}(\xi)}{(n+1)!}, \quad \xi \in I \tag{4} \end{equation*}(4)R[f]=R[xn+1]f(n+1)(ξ)(n+1)!,ξ∈I
replace. For n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0it is sufficient to assume that the derivative f ( n + 1 ) f ( n + 1 ) f^((n+1))f^{(n+1)}f(n+1)inside of I I IIIexists and in (4) one can then ξ ξ xi\xiξfrom the inside of I I IIIchoose.
Is f ( n + 1 ) f ( n + 1 ) f^((n+1))f^{(n+1)}f(n+1)bounded, then there is a (finite) number M M MMM, so that (2) and therefore also (3) hold. One can even
M = sup | f ( n + 1 ) | ( n + 1 ) ! M = sup f ( n + 1 ) ( n + 1 ) ! M=(s u p|f^((n+1))|)/((n+1)!)M=\frac{\sup \left|f^{(n+1)}\right|}{(n+1)!}M=sup|f(n+1)|(n+1)!
We note
that for formula (1) the existence of the ( n + 1 n + 1 n+1n+1n+1)-th derivative. Therefore, formula (1) is more general than the classical formula (3), which holds when R [ f ] R [ f ] R[f]R[f]R[f]is of simple form and also the ( n + 1 ) ( n + 1 ) (n+1)(n+1)(n+1)-th derivative f ( n + 1 ) f ( n + 1 ) f^((n+1))f^{(n+1)}f(n+1)exists.
A sufficient condition for the estimation (2) and hence also (3) to be valid is that f ( n ) f ( n ) f^((n))f^{(n)}f(n)exists and satisfies an ordinary Lipschitz condition.
4. To determine whether a linear functional R [ f ] R [ f ] R[f]R[f]R[f]of simple form, it is sufficient to apply the previously mentioned sentence. Man mú β β beta\betaβso show that β [ f ] β [ f ] beta[f]\beta[f]β[f]for all convex functions n n nnn-th order S S SSSis different from zero. For this purpose, different representations
of R [ f ] R [ f ] R[f]R[f]R[f]or use different criteria derived from the properties of the convex functions n n nnn-th order. I have given such criteria in my previous works [5], [6].
I would like to discuss one case in more detail, as it has numerous applications in the numerical integration of differential equations.
5. We consider the linear approximation formula
(5) A [ f ] = ∑ i = 1 p ∑ j = 0 k a i k j f ( j ) ( x i ) + R [ f ] (5) A [ f ] = ∑ i = 1 p   ∑ j = 0 k   a i k j f ( j ) x i + R [ f ] {:(5)A[f]=sum_(i=1)^(p)sum_(j=0)^(k)a_(i_(k)j)f^((j))(x_(i))+R[f]:}\begin{equation*} A[f]=\sum_{i=1}^{p} \sum_{j=0}^{k} a_{i_{k} j} f^{(j)}\left(x_{i}\right)+R[f] \tag{5} \end{equation*}(5)A[f]=∑i=1p∑j=0kaikjf(j)(xi)+R[f]
where the linear functional A [ f ] A [ f ] A[f]A[f]A[f]on the crowd S S SSSis defined. The remainder R [ f ] R [ f ] R[f]R[f]R[f]then has the shape given in No. 1.
In formula (5)
1 ∘ . x 1 , x 2 , … , x p p 1 ∘ . x 1 , x 2 , … , x p p 1^(@).x_(1),x_(2),dots,x_(p)quad p1^{\circ} . x_{1}, x_{2}, \ldots, x_{p} \quad p1∘.x1,x2,…,xppdifferent points of the interval I I III.
2 ∘ . k 1 , k 2 , … , k p p 2 ∘ . k 1 , k 2 , … , k p p 2^(@).k_(1),k_(2),dots,k_(p)quad p2^{\circ} . k_{1}, k_{2}, \ldots, k_{p} \quad p2∘.k1,k2,…,kppnatural numbers.
These are the multiplicity orders of the corresponding nodes x 1 , x 2 , … , x p x 1 , x 2 , … , x p x_(1),x_(2),dots,x_(p)x_{1}, x_{2}, \ldots, x_{p}x1,x2,…,xpIf each node is counted as often as its multiplicity order, we have in the whole n + 1 n + 1 n+1n+1n+1Nodes, where
(6) n = k 1 + k 2 + … + k p − 1 (6) n = k 1 + k 2 + … + k p − 1 {:(6)n=k_(1)+k_(2)+dots+k_(p)-1:}\begin{equation*} n=k_{1}+k_{2}+\ldots+k_{p}-1 \tag{6} \end{equation*}(6)n=k1+k2+…+kp−1
is.
3 ∘ . a i , j , j = 0 , 1 , … , k i − 1 , i = 1 , 2 , … , p 3 ∘ . a i , j , j = 0 , 1 , … , k i − 1 , i = 1 , 2 , … , p 3^(@).a_(i,j),j=0,1,dots,k_(i)-1,quad i=1,2,dots,p3^{\circ} . a_{i, j}, j=0,1, \ldots, k_{i}-1, \quad i=1,2, \ldots, p3∘.ai,j,j=0,1,…,ki−1,i=1,2,…,pCoefficients that are independent of the function f f fffThey can be negative, zero or positive,
4 ∘ ∘ ^(@){ }^{\circ}∘. f ( j ) ( f ( ∘ ) = f ) f ( j ) f ( ∘ ) = f quadf^((j))quad(f^((@))=f)\quad f^{(j)} \quad\left(f^{(\circ)}=f\right)f(j)(f(∘)=f)the j j jjj-th derivative of the function f f fff.
If the remainder of formula (5) for each polynomial n n nnn-th degree equals zero, where n n nnnhas the same value as in (6), the coefficients a i , j a i , j a_(i,j)a_{i, j}ai,juniquely determined. It is obtained when f f fffthrough the f f fffand the nodes x i x i x_(i)x_{i}xiwith the multiplicity orders k i k i k_(i)k_{i}kicorresponding polynomial L L LLLby Lagrange-Hermite. In this case,
(7) R [ f ] = A [ ∏ i = 1 p ( x − x i ) k i [ z 1 , z 2 , … , z n + 1 , x ; f ] ] (7) R [ f ] = A ∏ i = 1 p   x − x i k i z 1 , z 2 , … , z n + 1 , x ; f {:(7)R[f]=A[prod_(i=1)^(p)(x-x_(i))^(k)^(i)[z_(1),z_(2),dots,z_(n+1),x;f]]:}\begin{equation*} R[f]=A\left[\prod_{i=1}^{p}\left(x-x_{i}\right)^{k}{ }^{i}\left[z_{1}, z_{2}, \ldots, z_{n+1}, x ; f\right]\right] \tag{7} \end{equation*}(7)R[f]=A[∏i=1p(x−xi)ki[z1,z2,…,zn+1,x;f]]
where z 1 , z 2 , … , z n + 1 z 1 , z 2 , … , z n + 1 z_(1),z_(2),dots,z_(n+1)z_{1}, z_{2}, \ldots, z_{n+1}z1,z2,…,zn+1the n + 1 n + 1 n+1n+1n+1Nodes (different from each other or not). Looking at the right side of formula (7), it turns out that the argument of A [ f ] A [ f ] A[f]A[f]A[f]in this formula with the remainder f − L f − L f-Lf-Lf−Lthe interpolation formula used f ≈ L f ≈ L f~~Lf \approx Lf≈Lof Lagrange-Hermite. Takes x x xxxthe value of one of the nodes, then this remainder is zero. This must be taken into account when interpreting A [ f − L ] A [ f − L ] A[f-L]A[f-L]A[f−L]in formula (7) must always be taken into account.
6. In many important cases, the remainder R [ f ] R [ f ] R[f]R[f]R[f]the formula (5) not only the degree of accuracy given by (6) n n nnn, but is also of simple form.
The functional A [ f ] A [ f ] A[f]A[f]A[f]is positive with respect to a subinterval I ∗ I ∗ I^(**)I^{*}I∗from I I III, if A [ f ] > 0 A [ f ] > 0 A[f] > 0A[f]>0A[f]>0is for each function f f fffwhich is on the interval I ∗ I ∗ I^(**)I^{*}I∗, except for a finite number ( ≥ 0 ≥ 0 >= 0\geq 0≥0) of points from I ∗ I ∗ I^(**)I^{*}I∗, is positive. It is always assumed that I ∗ I ∗ I^(**)I^{*}I∗contains at least two points.
If we consider formula (7), the following property results:
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. Is the linear functional A [ f ] A [ f ] A[f]A[f]A[f]positive with respect to the subinterval I ∗ I ∗ I^(**)I^{*}I∗from I I III,
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘. are all multiplicity orders k i k i k_(i)k_{i}kifrom nodes inside I ∗ I ∗ I^(**)I^{*}I∗even numbers (this applies, for example, if the interior of I ∗ I ∗ I^(**)I^{*}I∗does not contain a node),
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘. is the remaining term R [ f ] R [ f ] R[f]R[f]R[f]the formula (5) for each polynomial n n nnn-th degree is zero, then this remainder has R [ f ] R [ f ] R[f]R[f]R[f]the degree of accuracy n n nnnand is of simple form.
As an example, let us take
A [ f ¯ ] = ∫ a b ω ( x ) f ( x ) d x A [ f ¯ ] = ∫ a b   ω ( x ) f ( x ) d x A[ bar(f)]=int_(a)^(b)omega(x)f(x)dxA[\bar{f}]=\int_{a}^{b} \omega(x) f(x) d xA[f¯]=∫abω(x)f(x)dx
where a , b ( a < b ) a , b ( a < b ) a,b(a < b)a, b(a<b)a,b(a<b)two finite points of the interval I I IIIare, ω ω omega\omegaωone on the completed interval [ a , b ] [ a , b ] [a,b][a, b][a,b]nonnegative continuous functions that are not identically equal to zero. The property formulated above can then be applied to the remainder of the quadrature formula
(8) ∫ a b ω ( x ) f ( x ) d x = ∑ i = 1 p ∑ j = 0 k i − 1 a i , j f ( j ) ( x i ) + R [ f ] (8) ∫ a b   ω ( x ) f ( x ) d x = ∑ i = 1 p   ∑ j = 0 k i − 1   a i , j f ( j ) x i + R [ f ] {:(8)int_(a)^(b)omega(x)f(x)dx=sum_(i=1)^(p)sum_(j=0)^(k_(i)-1)a_(i,j)f^((j))(x_(i))+R[f]:}\begin{equation*} \int_{a}^{b} \omega(x) f(x) d x=\sum_{i=1}^{p} \sum_{j=0}^{k_{i}-1} a_{i, j} f^{(j)}\left(x_{i}\right)+R[f] \tag{8} \end{equation*}(8)∫abω(x)f(x)dx=∑i=1p∑j=0ki−1ai,jf(j)(xi)+R[f]
7.
Many formulas for the numerical integration of differential equations have the form (8) and satisfy the given conditions when considering antiderivatives of different orders of the function f f fffThrough
such transformations, the rest remains of a simple form.
This claim results from the following sentences:
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. If the derivative f ′ f ′ f^(')f^{\prime}f′a function f f fffconvex of order n n nnn, then f f fffconvex of order n + 1 n + 1 n+1n+1n+1.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘. Has R ∗ [ f ] R ∗ [ f ] R^(**)[f]R^{*}[f]R∗[f]the degree of accuracy n + 1 n + 1 n+1n+1n+1( n ≥ − 1 n ≥ − 1 n >= -1n \geq-1n≥−1) and is of simple form, where R ∗ [ f ] = R [ f ′ ] R ∗ [ f ] = R f ′ R^(**)[f]=R[f^(')]R^{*}[f]=R\left[f^{\prime}\right]R∗[f]=R[f′]is, then R [ f ] R [ f ] R[f]R[f]R[f]the degree of accuracy n n nnnand is also of simple form.
For further details, please refer to my work [5].
When applying formulas for numerical integration, after such a transformation, the sum on the right-hand side of formula (8) is usually approximated by the left-hand side. In such a modification, the remainder term R [ f ] R [ f ] R[f]R[f]R[f]through − R [ f ] − R [ f ] -R[f]-R[f]−R[f]replaced, but neither the degree of accuracy nor the simple form changes.
To give some concrete examples, I would like to refer to the excellent book by L.COLLATZ [1], in which a wealth of such formulas is given.
Example 1. Adams’ formulas for numerical integration have the form (8) if one adds to the antiderivative of f f fffIn this case, in the corresponding open interval, there is ] a , b [ ] a , b ]a,b[:}] a, b\left[\right.]a,b[no points x i x i x_(i)x_{i}xi. Adams' integration formulas therefore have a remainder term of simple form.
Example 2. According to G.KOWALEWSKI [2], the difference n n nnn-th order ( n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0)
Δ h n f ( a ) = ∑ v = 0 n ( − 1 ) n − v ( n v ) f ( a + v h ) = ∫ a a + n h ϕ ( x ) f ( n ) ( x ) d x Δ h n f ( a ) = ∑ v = 0 n   ( − 1 ) n − v ( n v ) f ( a + v h ) = ∫ a a + n h   ϕ ( x ) f ( n ) ( x ) d x Delta_(h)^(n)f(a)=sum_(v=0)^(n)(-1)^(n-v)((n)/(v))f(a+vh)=int_(a)^(a+nh)phi(x)f^((n))(x)dx\Delta_{h}^{n} f(a)=\sum_{v=0}^{n}(-1)^{n-v}\binom{n}{v} f(a+v h)=\int_{a}^{a+n h} \phi(x) f^{(n)}(x) d xΔhnf(a)=∑v=0n(−1)n−v(nv)f(a+vh)=∫aa+nhϕ(x)f(n)(x)dx
if one assumes that the n n nnn-th derivative of the function f f fffis continuous. ϕ ϕ phi\phiϕdenotes a continuous (not identically vanishing) function that is piecewise polynomial and non-negative. ϕ ϕ phi\phiϕis, using today's terminology, a spline function.
One can then show that the remainder of the formulas
y 0 ′ ′ ≈ 1 h 2 ( y − 1 − 2 y 0 + y 1 ) y − 1 ′ ′ + 10 y 0 ′ ′ + y 1 ′ ′ ≈ 12 h 2 ( y − 1 − 2 y 0 + y 1 ) y 1 ′ ′ + y 2 ′ ′ ≈ 2 h 3 ( y 3 − 3 y 2 + 3 y 1 − y 0 ) y 0 ′ ′ ≈ 1 h 2 y − 1 − 2 y 0 + y 1 y − 1 ′ ′ + 10 y 0 ′ ′ + y 1 ′ ′ ≈ 12 h 2 y − 1 − 2 y 0 + y 1 y 1 ′ ′ + y 2 ′ ′ ≈ 2 h 3 y 3 − 3 y 2 + 3 y 1 − y 0 {:[y_(0)^('')~~(1)/(h^(2))(y_(-1)-2y_(0)+y_(1))],[y_(-1)^('')+10y_(0)^('')+y_(1)^('')~~(12)/(h^(2))(y_(-1)-2y_(0)+y_(1))],[y_(1)^('')+y_(2)^('')~~(2)/(h^(3))(y_(3)-3y_(2)+3y_(1)-y_(0))]:}\begin{aligned} & y_{0}^{\prime \prime} \approx \frac{1}{h^{2}}\left(y_{-1}-2 y_{0}+y_{1}\right) \\ & y_{-1}^{\prime \prime}+10 y_{0}^{\prime \prime}+y_{1}^{\prime \prime} \approx \frac{12}{h^{2}}\left(y_{-1}-2 y_{0}+y_{1}\right) \\ & y_{1}^{\prime \prime}+y_{2}^{\prime \prime} \approx \frac{2}{h^{3}}\left(y_{3}-3 y_{2}+3 y_{1}-y_{0}\right) \end{aligned}y0′′≈1h2(y−1−2y0+y1)y−1′′+10y0′′+y1′′≈12h2(y−1−2y0+y1)y1′′+y2′′≈2h3(y3−3y2+3y1−y0)
with the accuracy levels 3, 5 and 6 respectively is of simple form, and is equal to − 2 h 2 D 4 [ f ] , 36 h 4 D 6 [ f ] − 2 h 2 D 4 [ f ] , 36 h 4 D 6 [ f ] -2h^(2)D_(4)[f],36h^(4)D_(6)[f]-2 h^{2} D_{4}[f], 36 h^{4} D_{6}[f]−2h2D4[f],36h4D6[f], − 42 h 4 D 7 [ f ] − 42 h 4 D 7 [ f ] -42h^(4)D_(7)[f]-42 h^{4} D_{7}[f]−42h4D7[f]
The terms used are those of L.Collatz .
For example, to obtain the second of the considered formulas, it is sufficient to use the Gaussian quadrature formula
∫ a a + 2 π ϕ ( x ) f ( x ) d x = h 2 12 [ f ( a ) + lof ( a + h ) + f ( a + 2 h ) ] + R [ f ] ∫ a a + 2 π   ϕ ( x ) f ( x ) d x = h 2 12 [ f ( a ) + lof ( a + h ) + f ( a + 2 h ) ] + R [ f ] int_(a)^(a+2pi)phi(x)f(x)dx=(h^(2))/(12)[f(a)+lof(a+h)+f(a+2h)]+R[f]\int_{a}^{a+2 \pi} \phi(x) f(x) d x=\frac{h^{2}}{12}[f(a)+\operatorname{lof}(a+h)+f(a+2 h)]+R[f]∫aa+2πϕ(x)f(x)dx=h212[f(a)+lof(a+h)+f(a+2h)]+R[f]
to be applied for the function f = y ′ ′ f = y ′ ′ f=y^('')f=y^{\prime \prime}f=y′′, where
ϕ ( x ) = { x − a , für x ∈ [ a , a + h ] a + 2 h − x , für x ∈ [ a + h , a + 2 h ] ϕ ( x ) = x − a ,       für  x ∈ [ a , a + h ] a + 2 h − x ,       für  x ∈ [ a + h , a + 2 h ] phi(x)={[x-a","," für "x in[a","a+h]],[a+2h-x","," für "x in[a+h","a+2h]]:}\phi(x)= \begin{cases}x-a, & \text { für } x \in[a, a+h] \\ a+2 h-x, & \text { für } x \in[a+h, a+2 h]\end{cases}ϕ(x)={x−a, for x∈[a,a+h]a+2h−x, for x∈[a+h,a+2h]
The same procedure is used for the other two formulas.
Example 3. If we consider the formula
11 f ( a ) + 27 f ( a + h ) − 27 f ( a + 2 h ) − 11 f ( a + 3 h ) = ∫ a a + 3 h ϕ ( x ) f ′ ( x ) d x 11 f ( a ) + 27 f ( a + h ) − 27 f ( a + 2 h ) − 11 f ( a + 3 h ) = ∫ a a + 3 h   ϕ ( x ) f ′ ( x ) d x 11 f(a)+27 f(a+h)-27 f(a+2h)-11 f(a+3h)=int_(a)^(a+3h)phi(x)f^(')(x)dx11 f(a)+27 f(a+h)-27 f(a+2 h)-11 f(a+3 h)=\int_{a}^{a+3 h} \phi(x) f^{\prime}(x) d x11f(a)+27f(a+h)−27f(a+2h)−11f(a+3h)=∫aa+3hϕ(x)f′(x)dx
where ϕ ϕ phi\phiϕis a positive and piecewise constant function, it follows that β β beta\betaβthe remainder of the formula
y − 1 ′ + 9 y 0 ′ + 9 y 1 ′ + y 2 ′ = 1 3 h ( 11 y 2 + 27 y 1 − 27 y 0 − 11 y − 1 ) + 36 h 6 D 7 [ f ] y − 1 ′ + 9 y 0 ′ + 9 y 1 ′ + y 2 ′ = 1 3 h 11 y 2 + 27 y 1 − 27 y 0 − 11 y − 1 + 36 h 6 D 7 [ f ] y_(-1)^(')+9y_(0)^(')+9y_(1)^(')+y_(2)^(')=(1)/(3h)(11_(y_(2))+27y_(1)-27y_(0)-11_(y_(-1)))+36h^(6)D_(7)[f]y_{-1}^{\prime}+9 y_{0}^{\prime}+9 y_{1}^{\prime}+y_{2}^{\prime}=\frac{1}{3 h}\left(11_{y_{2}}+27 y_{1}-27 y_{0}-11_{y_{-1}}\right)+36 h^{6} D_{7}[f]y−1′+9y0′+9y1′+y2′=13h(11y2+27y1−27y0−11y−1)+36h6D7[f]
with the degree of accuracy 6 is of simple form.
8. The question naturally arises as to what the structure of a linear functional R [ f ] R [ f ] R[f]R[f]R[f]which is not of simple form. In the case of the remainder terms of formulas for the numerical integration of differential equations, this question can be easily answered. Such a remainder term has the form
(9) R [ f ] = ∑ i = 1 p ∑ j = 0 k i − 1 b i , j f ( j ) ( x i ) (9) R [ f ] = ∑ i = 1 p   ∑ j = 0 k i − 1   b i , j f ( j ) x i {:(9)R[f]=sum_(i=1)^(p)sum_(j=0)^(k_(i)^(-1))b_(i,j^(f))^((j))(x_(i)):}\begin{equation*} R[f]=\sum_{i=1}^{p} \sum_{j=0}^{k_{i}^{-1}} b_{i, j^{f}}{ }^{(j)}\left(x_{i}\right) \tag{9} \end{equation*}(9)R[f]=∑i=1p∑j=0ki−1bi,jf(j)(xi)
You can x 1 < x 2 < … < x p x 1 < x 2 < … < x p x_(1) < x_(2) < dots < x_(p)x_{1}<x_{2}<\ldots<x_{p}x1<x2<…<xpassume; the coefficients b i , j b i , j b_(i,j)b_{i, j}bi,jare of the function f f fffindependent.
Is the degree of accuracy of the linear functional (9) n ≥ max ( r 1 − 2 , r 2 − 2 , … , r p − 2 ) n ≥ max r 1 − 2 , r 2 − 2 , … , r p − 2 n >= max(r_(1)-2,r_(2)-2,dots,r_(p)-2)n \geq \max \left(r_{1}-2, r_{2}-2, \ldots, r_{p}-2\right)n≥max(r1−2,r2−2,…,rp−2), then we have a formula of the form
R [ f ] = ∑ i = 1 r − n − 1 λ i [ y i , y i + 1 , … , y i + n + 1 ; f ] , R [ f ] = ∑ i = 1 r − n − 1   λ i y i , y i + 1 , … , y i + n + 1 ; f , R[f]=sum_(i=1)^(r-n-1)lambda_(i)[y_(i),y_(i+1),dots,y_(i+n+1);f],R[f]=\sum_{i=1}^{r-n-1} \lambda_{i}\left[y_{i}, y_{i+1}, \ldots, y_{i+n+1} ; f\right],R[f]=∑i=1r−n−1λi[yi,yi+1,…,yi+n+1;f],
where r = r 1 + r 2 + … + r p , y r 1 + r 2 + … + r i − 1 + i + s = x i , ( y 1 + s = x 1 ) r = r 1 + r 2 + … + r p , y r 1 + r 2 + … + r i − 1 + i + s = x i , y 1 + s = x 1 r=r_(1)+r_(2)+dots+r_(p),y_(r_(1)+r_(2))+dots+r_(i-1)+i+s=x_(i),(y_(1+s)=x_(1))r=r_{1}+r_{2}+\ldots+r_{p}, y_{r_{1}+r_{2}}+\ldots+r_{i-1}+i+s=x_{i},\left(y_{1+s}=x_{1}\right)r=r1+r2+…+rp,yr1+r2+…+ri−1+i+s=xi,(y1+s=x1),
s = 0 , 1 , … , r i − 1 , i = 1 , 2 , … , p s = 0 , 1 , … , r i − 1 , i = 1 , 2 , … , p s=0,quad1,dots,r_(i)^(-1),quad i=1,2,dots,ps=0, \quad 1, \ldots, r_{i}^{-1}, \quad i=1,2, \ldots, ps=0,1,…,ri−1,i=1,2,…,p
The sum of the f f fffindependent coefficients λ i λ i lambda_(i)\lambda_{i}λiis not equal to zero and according to the mean value theorem for divided differences the equality
R [ f ] = λ [ ξ 1 , ξ 2 , … , ξ n + 2 ; f ] + μ [ ξ 1 ′ , ξ 2 ′ , … , ξ n + 2 ′ ; f ] . R [ f ] = λ ξ 1 , ξ 2 , … , ξ n + 2 ; f + μ ξ 1 ′ , ξ 2 ′ , … , ξ n + 2 ′ ; f . R[f]=lambda[xi_(1),xi_(2),dots,xi_(n+2);f]+mu[xi_(1)^('),xi_(2)^('),dots,xi_(n+2)^(');f].R[f]=\lambda\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n+2} ; f\right]+\mu\left[\xi_{1}^{\prime}, \xi_{2}^{\prime}, \ldots, \xi_{n+2}^{\prime} ; f\right] .R[f]=λ[ξ1,ξ2,…,ξn+2;f]+μ[ξ1′,ξ2′,…,ξn+2′;f].
λ , μ λ , μ lambda,mu\lambda, \muλ,μare two of f f fffindependent coefficients ( λ + μ = ∑ i = 1 r − η − 1 λ i = R [ x n + 1 ] ) λ + μ = ∑ i = 1 r − η − 1   λ i = R x n + 1 (lambda+mu=sum_(i=1)^(r-eta-1)lambda_(i)=R[x^(n+1)])\left(\lambda+\mu=\sum_{i=1}^{r-\eta-1} \lambda_{i}=R\left[x^{n+1}\right]\right)(λ+μ=∑i=1r−n−1λi=R[xn+1])and ξ ν , ξ ν ′ ξ ν , ξ ν ′ xi_(nu),xi_(nu)^(')\xi_{\nu}, \xi_{\nu}^{\prime}ξn,ξn′two systems of n + 2 n + 2 n+2n+2n+2from different points of the interval I I III9.
Do the coefficients λ i λ i lambda_(i)\lambda_{i}λiall have the same sign, then the linear functional (9) is of simple form. In the opposite case, it may no longer be of simple form. For example, if λ 1 λ 2 < 0 λ 1 λ 2 < 0 lambda_(1)lambda_(2) < 0\lambda_{1} \lambda_{2}<0λ1λ2<0or λ 1 λ r − n − 1 < 0 λ 1 λ r − n − 1 < 0 lambda_(1)lambda_(r-n-1) < 0\lambda_{1} \lambda_{r-n-1}<0λ1λr−n−1<0, then one can say with certainty that R [ f ] R [ f ] R[f]R[f]R[f]is not of simple form [5].
Example 4. It holds
3 f 0 − 10 f 1 + 12 f 2 − 6 f 3 + f 4 + 2 f 1 ′ ′ = − 6 [ 0 , 1 , 1 , 1 , 1 , 2 ; f ] + + 12 [ 1 , 1 , 1 , 1 , 2 , 3 ; f ] + 54 [ 1 , 1 , 1 , 2 , 3 , 4 ; f ] 3 f 0 − 10 f 1 + 12 f 2 − 6 f 3 + f 4 + 2 f 1 ′ ′ = − 6 [ 0 , 1 , 1 , 1 , 1 , 2 ; f ] + + 12 [ 1 , 1 , 1 , 1 , 2 , 3 ; f ] + 54 [ 1 , 1 , 1 , 2 , 3 , 4 ; f ] {:[3f_(0)-10f_(1)+12f_(2)-6f_(3)+f_(4)+2f_(1)^('')=-6[0","1","1","1","1","2;f]+],[+12[1","1","1","1","2","3;f]+54[1","1","1","2","3","4;f]]:}\begin{array}{r} 3 f_{0}-10 f_{1}+12 f_{2}-6 f_{3}+f_{4}+2 f_{1}^{\prime \prime}=-6[0,1,1,1,1,2 ; f]+ \\ +12[1,1,1,1,2,3 ; f]+54[1,1,1,2,3,4 ; f] \end{array}3f0−10f1+12f2−6f3+f4+2f1′′=−6[0,1,1,1,1,2;f]++12[1,1,1,1,2,3;f]+54[1,1,1,2,3,4;f]
where f i ( j ) = f ( j ) ( i ) f i ( j ) = f ( j ) ( i ) f_(i)^((j))=f^((j))(i)f_{i}^{(j)}=f^{(j)}(i)fi(j)=f(j)(i)is.
One can now easily determine that the remainder of the non-symmetric formula (in Collatz notation)
y 0 ′ ′ = 1 2 h 3 ( − 3 y − 1 + 10 y 0 − 12 y 1 + 6 y 2 − y 3 ) y 0 ′ ′ = 1 2 h 3 − 3 y − 1 + 10 y 0 − 12 y 1 + 6 y 2 − y 3 y_(0)^('')=(1)/(2h^(3))(-3y_(-1)+10y_(0)-12y_(1)+6y_(2)-y_(3))y_{0}^{\prime \prime}=\frac{1}{2 h^{3}}\left(-3 y_{-1}+10 y_{0}-12 y_{1}+6 y_{2}-y_{3}\right)y0′′=12h3(−3y−1+10y0−12y1+6y2−y3)
with accuracy level 4 is not of simple form.

LITERATURE

COLLATZ, L.
[1] Numerical treatment of differential equations. 1955.
KOWALEWSKI, G.
[2] Interpolation and approximate quadrature. 1932.
POPOVICIU, T.
[3] Les fonctions convexes. Paris 1945.
[4] Asupra formei restului in unele formule de approximare ale analizei. Lucrările Ses.Gen.Stii.ale Acad.RPR, din 1950, 183-185.
[5] Sur le reste dans certaines formulas linéaires d'approximation de l'analyse. MATHEMATICA (Cluj) l(24), 95-142 (1959).
[6] The simplicity of the rest in certain quadrature formulas. MATHEMATICA (Cluj) 6(29), 157-184 (1964).
Address:
Prof. Tiberiu Popoviciu
Institutul de Calcul
Str. Republicii, 37
Cluj, Romania
(Received July 30, 1970).

Related Posts

On Certain Multiplicative Arithmetic Functions

Abstract AuthorsTiberiu Popoviciu Institutul de Calcul Keywords? Paper coordinatesT. Popoviciu, Sur certaines fonctions arithmétiques multiplicatives, Mathematica (Cluj), 13(36) (1971) no. 2,…

On certain mean value formulas

Abstract English translation of the titleOn certain mean value formulas AuthorsTiberiu Popoviciu Institutul de Calcul Keywords? Paper coordinatesT. Popoviciu, Asupra unor…

Generalization of a Property of Farey Sequences

Abstract AuthorsTiberiu Popoviciu Institutul de Calcul Keywords? Paper coordinatesT. Popoviciu, Généralisation d’une propriété des suites de Farey, Rev. Roumaine Math. Pures…

On an Inequality Between Mean Values

Abstract AuthorsTiberiu Popoviciu Institutul de Calcul Keywords? Paper coordinatesT. Popoviciu, Sur une inégalité entre des valeurs moyennes, Univ. Beograd. Publ. Elektrotehn. Fak.…