T. Popoviciu, Über die Approximation der Funktionen und der Lösungen einer Gleichung durch quadratische Interpolation, Numerische Methoden der Approximationstheorie, Band 1 (Tagung, Math. Forschungsinst., Oberwolfach, 1971), pp. 155-163. Internat. Schriftenreihe Numer. Math., Band 16, Birkhäuser, Basel, 1972 (in German)
1972 a -Popoviciu- Numer. Meth. Approximation theory - On the approximation of functions
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ON THE APPROXIMATION OF FUNCTIONS AND SOLUTIONS OF AN EQUATION BY QUADRATICAL INTERPOLATION by Tiberiu Popoviciu in Cluj
In the following,f=f(x)f=f(x)always a real-valued function that is on an intervalIIwhose length is nonzero. We will specify the conditions that this function satisfies in the course of the discussion.
Furthermore,[x_(1),x_(2),dots,x_(n+1);f]\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]the divided difference (nn-th order) andL(x_(1),x_(2),dots,x_(n+1);f∣x)L\left(x_{1}, x_{2}, \ldots, x_{n+1} ; f \mid x\right)the interpolation polynomial of LagrangeHermite of the functionffregarding the nodesx_(1),x_(2),dots,x_(n+1)x_{1}, x_{2}, \ldots, x_{n+1}. These nodes may or may not be different from each other. In the latter case, derivatives of the function at the nodes appear in the divided difference and in the interpolation polynomial.
2. We now assume that the functionffsatisfies the following two conditions:
I. the equation
(1)
f(x)=0f(x)=0
has at least one solution in the interior of the interval I.
II.ffis a convex or concave function of 0th, 1st and 2nd order.
A function is called convex, non-concave, non-convex or concave ofnn-th order (n >= -1n \geq-1) if their divided difference (n+1n+1) -th order for each system ofn+2n+2different points of the domain of definition is positive, non-negative, non-positive or negative. In all these cases
, the divided difference for any system ofn+2n+2Points that do not all coincide have the same property, provided that this divided difference exists.
It can be shown that a functionff, which satisfies conditions I and II, is continuous and in the interior of the intervalIIis continuously differentiable, and that equation (1) has exactly one solutionzzhas.
Applies to two interior pointsawayawayof the intervalIIthe inequalitya < z < ba<z<b, we callaaas lower andbbas an upper approximation ofzz. As is well known, using the Regula falsi and the method of RaphsonNewton, one can then obtain better approximations ofzzAnother way to obtain better approximations is given below. OBdA one can assume that the functionffincreasing and convex in the usual sense, according to our terminology convex functionOO-th and 1 -th order. Denotez^('),z^('')z^{\prime}, z^{\prime \prime}the (only) zeros of the polynomialsL(a,b;f∣x)L(a, b ; f \mid x),L(b,b;f∣x)L(b, b ; f \mid x), thena < z^(') < z < z^('') < ba<z^{\prime}<z<z^{\prime \prime}<b.
3. Further approximations for the solutionzzare also those in the open interval]a,b[:}] a, b\left[\right.located (single) zerosz_(1)^('),z_(1)^('')z_{1}^{\prime}, z_{1}^{\prime \prime}of the polynomialsL(a,a,b;f∣x)L(a, a, b ; f \mid x),L(a,b,b;f∣x)L(a, b, b ; f \mid x). Indeed, considering the equations
{:[f(x)-L(a","a","b;f∣x)=(x-a)^(2)(x-b)[a","a","b","x;f]],[f(x)-L(a","b","b;f∣x)=(x-a)(x-b)^(2)[a","b","b","x;f]]:}\begin{aligned}
& f(x)-L(a, a, b ; f \mid x)=(x-a)^{2}(x-b)[a, a, b, x ; f] \\
& f(x)-L(a, b, b ; f \mid x)=(x-a)(x-b)^{2}[a, b, b, x ; f]
\end{aligned}
This givesa < x < ba<x<bthe relationship
{:(2)L(a","b","b;f∣x) < f(x) < L(a","a","b;f∣x)",":}\begin{equation*}
L(a, b, b ; f \mid x)<f(x)<L(a, a, b ; f \mid x), \tag{2}
\end{equation*}
in the case of a convex function of 2nd order, or the relationship
{:(3)L(a","b","b;f∣x) > f(x) > L(a","a","b;f∣x)",":}\begin{equation*}
L(a, b, b ; f \mid x)>f(x)>L(a, a, b ; f \mid x), \tag{3}
\end{equation*}
in the case of a concave function of the second order. This then leads toz_(1)^(') < z < z_(1)^('')z_{1}^{\prime}<z<z_{1}^{\prime \prime}, ifffconvex, orz_(1)^('') < z < z_(1)^(')z_{1}^{\prime \prime}<z<z_{1}^{\prime}, ifffconcave of order 2.
4. It is of course important to use the obtained approximate valuesz^('),z^('')z^{\prime}, z^{\prime \prime}andz_(1)^('),z_(1)^('')z_{1}^{\prime}, z_{1}^{\prime \prime}to compare. For this purpose, we distinguish two cases.
Isf^(')f^{\prime}a convex function of 2nd order, then fora < x < ba<x<bfrom (2) and from
{:[L(a","a","b;f∣x)-L(a","b;f∣x)=(x-a)(x-b)[a","a","b;f]],[L(a","b","b;f∣x)-L(b","b;f∣x)=(x-b)^(2)[a","b","b;f]]:}\begin{aligned}
& L(a, a, b ; f \mid x)-L(a, b ; f \mid x)=(x-a)(x-b)[a, a, b ; f] \\
& L(a, b, b ; f \mid x)-L(b, b ; f \mid x)=(x-b)^{2}[a, b, b ; f]
\end{aligned}
the inequalities
L(a,a,b;f∣x) < L(a,b;f∣x),quad L(b,b;f∣x) < L(a,b,b;f∣x).L(a, a, b ; f \mid x)<L(a, b ; f \mid x), \quad L(b, b ; f \mid x)<L(a, b, b ; f \mid x) .
So there isz^(') < z_(1)^(') < z < z_(1)^('') < z^('')z^{\prime}<z_{1}^{\prime}<z<z_{1}^{\prime \prime}<z^{\prime \prime}, i.e.z_(1)^(')z_{1}^{\prime},z_(1)^('')z_{1}^{\prime \prime}are better approximations thanz^('),z^('')z^{\prime}, z^{\prime \prime}.
Isffa concave function of 2nd order, then from (3) and from
L(a,b,b;f∣x)-L(a,b;f∣x)=(x-a)(x-b)[a,b,b;f]L(a, b, b ; f \mid x)-L(a, b ; f \mid x)=(x-a)(x-b)[a, b, b ; f]
the relationshipz^(') < z_(1)^(n) < z.quadz_(1)^(n)z^{\prime}<z_{1}^{n}<z . \quad z_{1}^{n}is therefore a better lower approximation forzzasz^(')z^{\prime}5.
What are the approximate valuesz_(1)^(')z_{1}^{\prime}andz^('')z^{\prime \prime}fromzzAs far as the equations are concerned, they cannot generally be compared in the case of a concave function of the second order. This is shown by the following considerations. From the equations
{:[(4)L(a","a","b;f∣x)-L(b","b;f∣x)=],[=(x-b){(x-a)[a","a","b;f]-(b-a)[a","b","b;f]}=],[=(x-b){(x-b)[a,a,b;f]-(b-a)^(2)[a,a,b,b;f]}]:}\begin{align*}
L(a, a, b ; f \mid x) & -L(b, b ; f \mid x)= \tag{4}\\
& =(x-b)\{(x-a)[a, a, b ; f]-(b-a)[a, b, b ; f]\}= \\
& =(x-b)\left\{(x-b)[a, a, b ; f]-(b-a)^{2}[a, a, b, b ; f]\right\}
\end{align*}
It follows that the differenceL(a,a,b;f∣x)-L(b,b;f∣x)quadL(a, a, b ; f \mid x)-L(b, b ; f \mid x) \quadin a neighborhood of the pointaa(right ofaa) positive and in a neighborhood of the pointbb(left ofb)b)is negative. Therefore, the polynomial (4) has exactly one zeroxi\xiin the interval]a,b[] a, b[. If you setm=L(b,b;f∣xi)m=L(b, b ; f \mid \xi), thenf(a) < m < f(b)f(a)<m<f(b). If you now choose a constantkk, so thatf(a) < k < f(b)f(a)<k<f(b)holds, then one immediately notices that the functionvarphi(x)=f(x)-k\varphi(x)=f(x)-ksatisfies the same conditions I and II as the functionff. Furthermore,
L(a,a,b;varphi∣x)-L(b,b;varphi∣x)=L(a,a,b;f∣x)-L(b,b;f∣x)L(a, a, b ; \varphi \mid x)-L(b, b ; \varphi \mid x)=L(a, a, b ; f \mid x)-L(b, b ; f \mid x)
and the functionvarphi\varphicorresponding valuesz_(1)^('),z^('')z_{1}^{\prime}, z^{\prime \prime}are both left ofxi\xi, if
{:(5)f(a) < k < m:}\begin{equation*}
f(a)<k<m \tag{5}
\end{equation*}
is, or both to the right ofxi\xi, if
{:(6)m < k < f(b):}\begin{equation*}
m<k<f(b) \tag{6}
\end{equation*}
If we now consider the sign of the difference (4), it follows that for the functionvarphi\varphithe relationshipz < z_(1)^(') < z^('')z<z_{1}^{\prime}<z^{\prime \prime}applies ifkksatisfies condition (5) andz < z^('') < z_(1)^(')z<z^{\prime \prime}<z_{1}^{\prime}, ifkkthe condition (6) is satisfied. In the casek=mk=misz_(1)^(')=z^('')z_{1}^{\prime}=z^{\prime \prime}for the functionvarphi\varphi6.
If conditions I and II are met, the polynomialL(a,a,b;f∣x)+L(a,b,b;f∣x)quadL(a, a, b ; f \mid x)+L(a, b, b ; f \mid x) \quadalso a single zeroz_(1)z_{1}in the interval ]a,ba, b[. The numberz_(1)z_{1}is located strictly betweenz_(1)^('),z_(1)^('')z_{1}^{\prime}, z_{1}^{\prime \prime}and is therefore a better approximation ofzzas the worst of the valuesz_(1)^('),z_(1)^('')z_{1}^{\prime}, z_{1}^{\prime \prime}. If one considers that
{:[2f(x)-L(a","a","b;f∣x)-L(a","b","b;f∣x)=],[=(x-a)(x-b){(x-a)[a","a","b","x;f]+(x-b)[a","b","b","x;f]}]:}\begin{aligned}
2 f(x) & -L(a, a, b ; f \mid x)-L(a, b, b ; f \mid x)= \\
& =(x-a)(x-b)\{(x-a)[a, a, b, x ; f]+(x-b)[a, b, b, x ; f]\}
\end{aligned}
, as well as the properties of the convex functions of 2nd order, it follows that the left side has exactly one zeroeta\etain the interval]a,b[] a, b[If you now choose a constantkk, so thatf(a) < k < f(b)f(a)<k<f(b)holds, then the functionvarphi(x)=f(x)-k\varphi(x)=f(x)-kthe same conditions I and II as the functionff. Isk!=n=f(eta)k \neq n=f(\eta), the functionvarphi\varphicorresponding pointsz,z_(1)z, z_{1}both left or both right ofeta\eta. Furthermore,f(a) < n < f(b)f(a)<n<f(b). If we assume thatffis a convex function of 2nd order, then the functionvarphi\varphi,
{:[n < k < f(b)Longrightarrowz_(1) < z","],[f(a) < k < n Longrightarrow z < z_(1).]:}\begin{aligned}
& n<k<f(b) \Longrightarrow z_{1}<z, \\
& f(a)<k<n \Longrightarrow z<z_{1} .
\end{aligned}
In the case of a concave function of the second order, the symbol < must be replaced by > on the right-hand side of these two formulas.
7. Similar considerations can be made if one assumes that the functionfffalling and concave, falling and convex, or rising and concave of the first order. These cases can be reduced to the case under study
by applying the obtained results to the functions-f(x),f((a+b)/(2)-x)-f(x), f\left(\frac{a+b}{2}-x\right), or-f((a+b)/(2)-x)-f\left(\frac{a+b}{2}-x\right)applies.
Condition II can be weakened. Instead of the requirement thatffconvex fromo-,1-o-, 1-, and 2 -th order, one can assume thatffnon-concave or non-convex ofo-,1o-, 1- and 2 -th order. Some or all of thez,z^('),z^(''),z_(1)^('),z_(1)^(''),z_(1)z, z^{\prime}, z^{\prime \prime}, z_{1}^{\prime}, z_{1}^{\prime \prime}, z_{1}Proven inequalities can then be transformed into equations.
Regarding condition II, it can be noted that a functionffwith positive, non-negative, non-positive or negative (n+1n+1)-th derivative convex, non-concave, non-convex or concave ofnn-th order. On the other hand, eachIIconvex, non-concave, non-convex or concave functionnn-th order (n > 1n>1) inside the intervalII(n-1) times (continuously) differentiable.
Numerical example: Given the functionf(x)=x^(3)-x-1f(x)=x^{3}-x-1. One can immediately see that this function is a convex function in an appropriately chosen interval containing the points 1 and 2o-o-, 1st and 2nd order. Becausef(1)f(2) < 0f(1) f(2)<0has the equation (1) between the points1(=a)1(=a)and2(=b)2(=b)a root. Applying the regula falsi and the Raphson-Newton method, one obtains the approximate values(7)/(6)=1,16^(˙),(17)/(11)=1,5^(˙)4^(˙)\frac{7}{6}=1,1 \dot{6}, \frac{17}{11}=1, \dot{5} \dot{4}for this root. The polynomialsL(1,1,2;f∣x)L(1,1,2 ; f \mid x),L(1,2,2;f∣x)L(1,2,2 ; f \mid x)are in this case4x^(2)-6x+1,5x^(2)-9x+34 x^{2}-6 x+1,5 x^{2}-9 x+3and provide the approximate values(3+sqrt5)/(4) > 1,3,(9+sqrt21)/(10) < 1,36\frac{3+\sqrt{5}}{4}>1,3, \frac{9+\sqrt{21}}{10}<1,36.
Accordingly ,1,3dots1,3 \ldotsthe value of the positive root of the equationx^(3)-x-1=0x^{3}-x-1=0, where one decimal place is exact. The zero(4)/(3)=1,3^(˙)\frac{4}{3}=1, \dot{3}of the polynomialL(1,1,2;f∣x)+L(1,2,2;f∣x)L(1,1,2 ; f \mid x)+L(1,2,2 ; f \mid x), which is located between 1 and 2, leads us to the same result.
8. The relations (2) and (3) show that the polynomialsL(a,b,b;f∣x)L(a, b, b ; f \mid x),L(a,a_,b;f∣x)quadL(a, \underline{a}, b ; f \mid x) \quadin the case of a convex or concave function of 2nd order the functionffin the interval]a,b[] a, b[from both below and above. For example, let us assume thatffin the intervalIIconvex of 2nd order, then it follows thatffconstantly and withinIIis (continuously) differentiable.
Arec,a,b,dc, a, b, dfour inner points ofIIwithc <= a < b <= dc \leq a<b \leq d, it follows from
{:[f(x)-L(c","a","b;f∣x)=(x-c)(x-a)(x-b)[c","a","b","x;f]],[f(x)-L(a","b","d;f∣x)=(x-a)(x-b)(x-d)[a","b","d","x;f]]:}\begin{aligned}
& f(x)-L(c, a, b ; f \mid x)=(x-c)(x-a)(x-b)[c, a, b, x ; f] \\
& f(x)-L(a, b, d ; f \mid x)=(x-a)(x-b)(x-d)[a, b, d, x ; f]
\end{aligned}
fora < x < ba<x<bthe relationship
L(a,b,d;f∣x) < f(x) < L(c,a,b;f∣x),L(a, b, d ; f \mid x)<f(x)<L(c, a, b ; f \mid x),
which generalizes the inequalities (2).(c < a < b < d)(c<a<b<d)
{:[L(a","b","d;f∣x)-L(a","b","b;f∣x)=(x-a)(x-b)(d-b)[a","b","b","d;f]],[L(a","a","b;f∣x)-L(c","a","b;f∣x)=(x-a)(x-b)(a-c)[c","a","a","b;f]]:}\begin{aligned}
& L(a, b, d ; f \mid x)-L(a, b, b ; f \mid x)=(x-a)(x-b)(d-b)[a, b, b, d ; f] \\
& L(a, a, b ; f \mid x)-L(c, a, b ; f \mid x)=(x-a)(x-b)(a-c)[c, a, a, b ; f]
\end{aligned}
It also follows that of all approximate valuesL(a,b,d;f∣x)L(a, b, d ; f \mid x),L(c,a,b;f∣x)L(c, a, b ; f \mid x)withc <= a < b <= d,quad L(a,b,b;f∣x)c \leq a<b \leq d, \quad L(a, b, b ; f \mid x)the best lower approximation off(x)f(x)andL(a,a,b;f∣x)L(a, a, b ; f \mid x)the best upper approximation off(x)f(x)fora < x < b quada<x<b \quadis.
A similar property can also be proven for concave functions of second order. Only in this case, the meaning of the inequalities is reversed.
The above considerations can be applied, for example, to the following functions:ln x\ln xwhich is convex on the set of positive real numbers of order 2,arctg x\operatorname{arctg} x, which is based on the interval[-(1)/(sqrt3),(1)/(sqrt3)]\left[-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right]concave of 2nd order and(1)/(2)ln(1+x^(2))\frac{1}{2} \ln \left(1+x^{2}\right), which is based on the interval[0,sqrt3][0, \sqrt{3}]is concave of order 2.
9. We assume in the following that the function on the intervalIIexplained functionffis sufficiently often differentiable so that all divided differences and interpolation polynomials that will occur exist.
To use the function represented by a tableffTo interpolate, we choose four valuesc,a,b,dc, a, b, dthe variable withc < a < b < dc<a<b<dand approximate the functionf(x)f(x), wherea < x < ba<x<bis divided by the arithmetic mean
P(c,a,b,d;f∣x)=(1)/(2){L(c,a,b;f∣x)+L(a,b,d;f∣x)}P(c, a, b, d ; f \mid x)=\frac{1}{2}\{L(c, a, b ; f \mid x)+L(a, b, d ; f \mid x)\}
of the polynomialsL(c,a,b;f∣x),quad L(a,b,d;f∣x).quadL(c, a, b ; f \mid x), \quad L(a, b, d ; f \mid x) . \quadThe error is then equal to
{:[(7)d(x)=f(x)-P(c","a","b","d;f∣x)=],[=(1)/(2)(x-a)(x-b){(x-c)(c-d)[c","a","x","b","d;f]+(2x-c-d)[a","x","b","d;f]}]:}\begin{align*}
d(x) & =f(x)-P(c, a, b, d ; f \mid x)= \tag{7}\\
& =\frac{1}{2}(x-a)(x-b)\{(x-c)(c-d)[c, a, x, b, d ; f]+(2 x-c-d)[a, x, b, d ; f]\}
\end{align*}
The errord_(1)(x)d_{1}(x), which is used in the approximation off(x)f(x)by the arithmetic mean of the polynomialsL(a,a,b;f∣x),quad L(a,b,b;f∣x),quadL(a, a, b ; f \mid x), \quad L(a, b, b ; f \mid x), \quadso throughP(a,a,b,b;f∣x)P(a, a, b, b ; f \mid x), can be obtained by substituting in (7)c,dc, dthrougha,ba, bA simple calculation then gives
{:[(8)d(x)-d_(1)(x)=(1)/(2)(x-a)(x-b){(x-c)(c-a)[c","a","a","x","b;f]+],[+(x-d)(d-b)[a","x","b","b","d;f]+(b-a)(b-d)[a","a","x","b","b;f]}]:}\begin{align*}
d(x)-d_{1}(x) & =\frac{1}{2}(x-a)(x-b)\{(x-c)(c-a)[c, a, a, x, b ; f]+ \tag{8}\\
& +(x-d)(d-b)[a, x, b, b, d ; f]+(b-a)(b-d)[a, a, x, b, b ; f]\}
\end{align*}
We now assume that the pointsc,a,b,dc, a, b, dare equidistant, which is the case with most interpolation tables, and want to find the value offfat the center of the interval[a,b][a, b]estimate, a case that also occurs very frequently. We then setx=mu=(a+b)/(2)(=(c+d)/(2))x=\mu=\frac{a+b}{2}\left(=\frac{c+d}{2}\right), and from (7), (8) follows
{:[d(mu)=9((b-a)/(2))^(4)[c","a","mu","b","d;f]","quadd_(1)(mu)=((b-a)/(2))^(4)[a","a","mu","b","b;f]],[(9)d(mu)-d_(1)(mu)=((b-a)/(2))^(4){3[c","a","a","mu","b;f]+3[a","mu","b","b","d;f]+2[a","a","mu","b","b;f]}.]:}\begin{align*}
& d(\mu)=9\left(\frac{b-a}{2}\right)^{4}[c, a, \mu, b, d ; f], \quad d_{1}(\mu)=\left(\frac{b-a}{2}\right)^{4}[a, a, \mu, b, b ; f] \\
& d(\mu)-d_{1}(\mu)=\left(\frac{b-a}{2}\right)^{4}\{3[c, a, a, \mu, b ; f]+3[a, \mu, b, b, d ; f]+2[a, a, \mu, b, b ; f]\} . \tag{9}
\end{align*}
If we now assume thatffconcave or convex of 3rd order, the result is|d(mu)| > |d_(1)(mu)||d(\mu)|>\left|d_{1}(\mu)\right|. It follows in this case thatP(a,a,b,b;f∣x)P(a, a, b, b ; f \mid x)at the center of the interval[a,b][a, b]a better approximation forffprovides as the polynomialP(c,a,b,d;f∣x)P(c, a, b, d ; f \mid x). By the way,ffapproached by both from the same side.
The above considerations can be applied, for example, to the functions already mentioned:ln x\ln x, which is concave of order 3 on the set of positive real numbers,arctg x\operatorname{arctg} x, which is based on the interval[0,1][0,1]is convex of 3rd order and(1)/(2)ln(1+x^(2))\frac{1}{2} \ln \left(1+x^{2}\right), which is based on the interval[-sqrt2+1,sqrt2-1][-\sqrt{2}+1, \sqrt{2}-1]is concave of 3rd order.
In practical execution of the calculations, one needs to calculate the polynomialP(a,a,b,b;f∣x)P(a, a, b, b ; f \mid x)except the values of the functionff, which can be seen in the table, also the valuesf^(')(a),f^(')(b)f^{\prime}(a), f^{\prime}(b)the derivative at the pointsa,ba, b. The calculation of these values is usually made easier by the fact that the derivative of the functionffis a rational function, such asln x\ln x,arctg x,(1)/(2)ln(1+x^(2))\operatorname{arctg} x, \frac{1}{2} \ln \left(1+x^{2}\right)10.
In the following, we give estimates for the error resulting from the approximations considered previously. We will only consider one case, since the others are similar, namely the approximation of the functionffin the spotlightmu\muof the interval[a,b][a, b]by the polynomialP(a,a,b,b;f∣x)P(a, a, b, b ; f \mid x).
If you set
M=s u p|[x_(1),x_(2),x_(3),x_(4),x_(5);f]|,M=\sup \left|\left[x_{1}, x_{2}, x_{3}, x_{4}, x_{5} ; f\right]\right|,
where the supremum extends over all groups of 5 different pointsx_(1),x_(2),x_(3),x_(4),x_(5)x_{1}, x_{2}, x_{3}, x_{4}, x_{5}of the interval[a,b][a, b]then from (9) we get the estimate
|d_(1)(mu)| <= ((b-a)/(2))^(4)M.\left|d_{1}(\mu)\right| \leq\left(\frac{b-a}{2}\right)^{4} M .
Now ownsffa 4th order derivative, then
M=(1)/(24)s u p_(x in[a,b])|f^((4))(x)|M=\frac{1}{24} \sup _{x \in[a, b]}\left|f^{(4)}(x)\right|
Of course, this estimate is only of interest ifffis bounded.
In the case of a non-concave or non-convex function of 4th order, this estimate can be further refined. Indeed, note that
(d)/(dx)[a,a,x,b,b;f]=[a,a,x,x,b,b;f]\frac{d}{d x}[a, a, x, b, b ; f]=[a, a, x, x, b, b ; f]
it follows that the function[a,a,x,b,b;f][a, a, x, b, b ; f]is monotonic. Therefore,d_(1)(mu)d_{1}(\mu)in this case always between
{:(11)((b-a)/(2))^(4)[a","a","a","b","b;f]quad" und "quad((b-a)/(2))^(4)[a","a","b","b","b;f].:}\begin{equation*}
\left(\frac{b-a}{2}\right)^{4}[a, a, a, b, b ; f] \quad \text { und } \quad\left(\frac{b-a}{2}\right)^{4}[a, a, b, b, b ; f] . \tag{11}
\end{equation*}
Numerical example. If the values of the functionf(x)=ln xf(x)=\ln xfor the points 1,2,3,4, then the polynomial
P(2,2,3,3;f∣x)=f(2)+(x-2)[f(3)-f(2)]+(1)/(2)(x-2)(x-3)[f^(')(3)-f^(')(2)]P(2,2,3,3 ; f \mid x)=f(2)+(x-2)[f(3)-f(2)]+\frac{1}{2}(x-2)(x-3)\left[f^{\prime}(3)-f^{\prime}(2)\right]
at the center of the interval [ 2,3 ] a better approximation than the polynomial
P(1,2,3,4;f∣x)=f(2)+(x-2)[f(3)-f(2)]+(1)/(4)(x-2)(x-3)[f(4)-f(3)-f(2)+f(1)]P(1,2,3,4 ; f \mid x)=f(2)+(x-2)[f(3)-f(2)]+\frac{1}{4}(x-2)(x-3)[f(4)-f(3)-f(2)+f(1)]
According to formula (10), the amount of error in the first approximation is<= (1)/(1024)\leq \frac{1}{1024}. However, if one considers thatln x\ln xis concave of 3rd order and convex of 4th order, then the absolute value of the first number in (11) gives the better bound
If the 11th decimal place is ignored during the calculation, the result is
{:[P(2","2","3","3;f∣2","5)~~0","8958797346+0","0208333333=0","916713 o 679],[P(1","2","3","4;f∣2","5)~~0","8958797346+o","o 253415692=o","9212213 o 38.]:}\begin{aligned}
& P(2,2,3,3 ; f \mid 2,5) \approx 0,8958797346+0,0208333333=0,916713 o 679 \\
& P(1,2,3,4 ; f \mid 2,5) \approx 0,8958797346+o, o 253415692=o, 9212213 o 38 .
\end{aligned}
In the same table, the value ofln 2,5\ln 2,5as 0.9162907319. So 3 decimal places of our approximate value agree with the value ofln 2,5\ln 2,5agree.
