On the Bounding of the Error in Approximating the Roots of an Equation by Linear or Quadratic Interpolation

Tiberiu Popoviciu
(original), Approximation Theory, divided differences, Inequalities, interpolation, Newton method, Numerical Analysis, paper, secant/chord method, solving F(x)=0 iteratively

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Tiberiu Popoviciu
Institutul de Calcul

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 T. Popoviciu, Sur la délimitation de l’erreur dans l’approximation des racines d’une équation par interpolation linéaire ou quadratique, Rev. Roumaine Math. Pures Appl., 13 (1968), pp. 75-78 (in French)

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1968 b -Popoviciu- Rev. Roum. Math. Pures Appl. – Sur la delimitation de l’erreur dans l’approximation des racines d’une equation par interpolation lineaire ou quadratique

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Revue Roumaine de Mathématiques Pures et Appliquées

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1968 b -Popoviciu- Rev. Roum. Math. Pures Appl. - On the delimitation of the error in the approximation
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ON THE DELIMITATION OF THE ERROR IN THE APPROXIMATION OF THE ROOTS OF AN EQUATION BY LINEAR OR QUADRATIC INTERPOLATION
by
TIBERIU POPOVICIU

We first specify the delimitation of the error given by AM OSTROWSKI
[1] in the approximation of the roots of an equation by linear interpolation.
We also give an analogous result in the case of quadratic interpolation.
  1. Consider a real function f = f ( x ) f = f ( x ) f=f(x)f=f(x)f=f(x), defined and continuous on an interval I I IIIof non-zero length. We will designate by z a root of the equation
f ( x ) = 0 f ( x ) = 0 f(x)=0f(x)=0f(x)=0
We will designate by [ x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f ] x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f [x_(1)^('),x_(2)^('),dots,x_(n+1)^(');f]\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+1}^{\prime}; f\right][x1′,x2′,…,xn+1′;f]the divided difference (of order n n nnn) And p p pppar L ( x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f ∣ x ) = [ x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f ] x n + … L x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f ∣ x = x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f x n + … L(x_(1)^('),x_(2)^('),dots,x_(n+1)^(');f∣x)=[x_(1)^('),x_(2)^('),dots,x_(n+1)^(');f]x^(n)+dotsL\left(x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+1}^{\prime}; f \mid x\right)=\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+1}^{\prime} ; f\right] x^{n}+\ldotsL(x1′,x2′,…,xn+1′;f∣x)=[x1′,x2′,…,xn+1′;f]xn+…; the Lagrange-Hermite interpolation polynomial on the knots x 1 ′ x 1 ′ x_(1)^(')x_{1}^{\prime}x1′, x 2 ′ , … , x n ′ x 2 ′ , … , x n ′ x_(2)^('),dots,x_(n)^(')x_{2}^{\prime}, \ldots, x_{n}^{\prime}x2′,…,xn′which are not necessarily distinct. When the nodes are not distinct, the successive derivatives of the function also intervene in the divided difference and in the corresponding interpolation polynomial f f fff, according to well-known rules.
2. Let x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2points of I I IIIAnd y y yyythe root of the Lagrange-Hermite polynomial L ( x 1 , x 2 ; f ∣ x ) L x 1 , x 2 ; f ∣ x L(x_(1),x_(2);f∣x)L\left(x_{1}, x_{2}; f \mid x\right)L(x1,x2;f∣x)Let us propose to delimit the error z − y z − y zyzyz−yof approximation y y yyythus calculated from z z zzz, by doing on the function f f fffsome suitable assumptions.
Of L ( x 1 , x 2 ; f ∣ y ) = f ( z ) = 0 L x 1 , x 2 ; f ∣ y = f ( z ) = 0 L(x_(1),x_(2);f∣y)=f(z)=0L\left(x_{1}, x_{2} ; f \mid y\right)=f(z)=0L(x1,x2;f∣y)=f(z)=0and from the well-known expression of the rest of the Lagrange-Hermite interpolation formula, we deduce that
(2)
L ( x 1 , x 2 ; f ∣ y ) − L ( x 1 , x 2 ; f ∣ z ) = = f ( z ) − L ( x 1 , x 2 ; f ∣ z ) = [ x 1 , x 2 , z ; f ] ( z − x 1 ) ( z − x 2 ) L x 1 , x 2 ; f ∣ y − L x 1 , x 2 ; f ∣ z = = f ( z ) − L x 1 , x 2 ; f ∣ z = x 1 , x 2 , z ; f z − x 1 z − x 2 {:[L(x_(1),x_(2);f∣y)-L(x_(1),x_(2);f∣z)=],[=f(z)-L(x_(1),x_(2);f∣z)=[x_(1),x_(2),z;f](z-x_(1))(z-x_(2))]:}\begin{gathered} L\left(x_{1}, x_{2} ; f \mid y\right)-L\left(x_{1}, x_{2} ; f \mid z\right)= \\ =f(z)-L\left(x_{1}, x_{2} ; f \mid z\right)=\left[x_{1}, x_{2}, z ; f\right]\left(z-x_{1}\right)\left(z-x_{2}\right) \end{gathered}L(x1,x2;f∣y)−L(x1,x2;f∣z)==f(z)−L(x1,x2;f∣z)=[x1,x2,z;f](z−x1)(z−x2)
REV. ROUM. MATH. PURE AND APPL., VOLUME XIII, NO 1. p. 75-78, BUCHAREST, 1968
But
L ( x 1 , x 2 ; f ∣ y ) − L ( x 1 , x 2 ; f ∣ z ) = [ x 1 , x 2 ; f ] ( y − z ) L x 1 , x 2 ; f ∣ y − L x 1 , x 2 ; f ∣ z = x 1 , x 2 ; f ( y − z ) L(x_(1),x_(2);f∣y)-L(x_(1),x_(2);f∣z)=[x_(1),x_(2);f](yz)L\left(x_{1}, x_{2}; f \mid y\right)-L\left(x_{1}, x_{2}; f \mid z\right)=\left[x_{1}, x_{2}; f\right](yz)L(x1,x2;f∣y)−L(x1,x2;f∣z)=[x1,x2;f](y−z)
and it follows that
(3) [ x 1 , x 2 ; f ] ( z − y ) = − [ x 1 , x 2 , z ; f ] ( z − x 1 ) ( z − x 2 ) (3) x 1 , x 2 ; f ( z − y ) = − x 1 , x 2 , z ; f z − x 1 z − x 2 {:(3)[x_(1),x_(2);f](zy)=-[x_(1),x_(2),z;f](z-x_(1))(z-x_(2)):}\begin{equation*} \left[x_{1}, x_{2}; f\right](zy)=-\left[x_{1}, x_{2}, z; f\right]\left(z-x_{1}\right)\left(z-x_{2}\right) \tag{3} \end{equation*}(3)[x1,x2;f](z−y)=−[x1,x2,z;f](z−x1)(z−x2)
If we assume that [ x 1 , x 2 ; f ] ≠ 0 x 1 , x 2 ; f ≠ 0 [x_(1),x_(2);f]!=0\left[x_{1}, x_{2}; f\right] \neq 0[x1,x2;f]≠0, we deduce that
(4) z − y = − [ x 1 , x 2 , z ; f ] [ x 1 , x 2 ; f ] ( z − x 1 ) ( z − x 2 ) (4) z − y = − x 1 , x 2 , z ; f x 1 , x 2 ; f z − x 1 z − x 2 {:(4)zy=-([x_(1),x_(2),z;f])/([x_(1),x_(2);f])(z-x_(1))(z-x_(2)):}\begin{equation*} zy=-\frac{\left[x_{1}, x_{2}, z; f\right]}{\left[x_{1}, x_{2}; f\right]}\left(z-x_{1}\right)\left(z-x_{2}\right) \tag{4} \end{equation*}(4)z−y=−[x1,x2,z;f][x1,x2;f](z−x1)(z−x2)
This is also an immediate consequence of the application of the "regula falsi" method and in particular that of Newton if x 1 = x 2 x 1 = x 2 x_(1)=x_(2)x_{1}=x_{2}x1=x2.
If we assume that f f fffhas a second derivative on I I III, We have.
(5) [ x 1 , x 2 ; f ] = f ′ ( ξ ) , [ x 1 , x 2 , z ; f ] = 1 2 f ′ ′ ( ξ 1 ) (5) x 1 , x 2 ; f = f ′ ( ξ ) , x 1 , x 2 , z ; f = 1 2 f ′ ′ ξ 1 {:(5)[x_(1),x_(2);f]=f^(')(xi)","quad[x_(1),x_(2),z;f]=(1)/(2)f^('')(xi_(1)):}\begin{equation*} \left[x_{1}, x_{2} ; f\right]=f^{\prime}(\xi), \quad\left[x_{1}, x_{2}, z ; f\right]=\frac{1}{2} f^{\prime \prime}\left(\xi_{1}\right) \tag{5} \end{equation*}(5)[x1,x2;f]=f′(ξ),[x1,x2,z;f]=12f′′(ξ1)
Or ξ ξ xi\xiξrespectively ξ 1 ξ 1 xi_(1)\xi_{1}ξ1, is (when x 1 ≠ x 2 x 1 ≠ x 2 x_(1)!=x_(2)x_{1} \neq x_{2}x1≠x2) inside the smallest interval containing the points x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2respectively the points x 1 , x 2 , z x 1 , x 2 , z x_(1),x_(2),zx_{1}, x_{2}, zx1,x2,z. If therefore the derivative f ′ f ′ f^(')f^{\prime}f′of f f fffdoes not cancel on I I III(more generally within the smallest interval containing the points x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2, or on x 1 x 1 x_(1)x_{1}x1if x 1 = x 2 x 1 = x 2 x_(1)=x_(2)x_{1}=x_{2}x1=x2), we deduce from this
z − y = − f ′ ′ ( ξ 1 ) 2 f ′ ( ξ ) ( z − x 1 ) ( z − x 2 ) z − y = − f ′ ′ ξ 1 2 f ′ ( ξ ) z − x 1 z − x 2 z-y=-(f^('')(xi_(1)))/(2f^(')(xi))(z-x_(1))(z-x_(2))z-y=-\frac{f^{\prime \prime}\left(\xi_{1}\right)}{2 f^{\prime}(\xi)}\left(z-x_{1}\right)\left(z-x_{2}\right)z−y=−f′′(ξ1)2f′(ξ)(z−x1)(z−x2)
  1. The preceding formulas make sense only if the root z exists and are of interest only if y y yyydoes not go out of the interval I I III.
Let's suppose that
0 < m 1 ≦ | [ x 1 ′ , x 2 ′ ; f ] | ≦ M 1 < + ∞ 0 < m 2 ≦ | [ x 1 ′ , x 2 ′ , x 3 ′ ; f ] | ≦ M 2 < + ∞ 0 < m 1 ≦ x 1 ′ , x 2 ′ ; f ≦ M 1 < + ∞ 0 < m 2 ≦ x 1 ′ , x 2 ′ , x 3 ′ ; f ≦ M 2 < + ∞ {:[0 < m_(1) <= |[x_(1)^('),x_(2)^(');f]| <= M_(1) < +oo],[0 < m_(2) <= |[x_(1)^('),x_(2)^('),x_(3)^(');f]| <= M_(2) < +oo]:}\begin{aligned} & 0<m_{1} \leqq\left|\left[x_{1}^{\prime}, x_{2}^{\prime} ; f\right]\right| \leqq M_{1}<+\infty \\ & 0<m_{2} \leqq\left|\left[x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime} ; f\right]\right| \leqq M_{2}<+\infty \end{aligned}0<m1≦|[x1′,x2′;f]|≦M1<+∞0<m2≦|[x1′,x2′,x3′;f]|≦M2<+∞
for all groups of 3 distinct points x 1 ′ , x 2 ′ , x 3 ′ x 1 ′ , x 2 ′ , x 3 ′ x_(1)^('),x_(2)^('),x_(3)^(')x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}x1′,x2′,x3′of I I III. The function f f fffthen has a continuous derivative, is strictly monotone and is convex or concave on I I III. So if the function changes sign, the root z z zzzexists and is unique. The point y belongs to the smallest interval containing the points x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2, z if f ( x 1 ) f ( x 2 ) < 0 f x 1 f x 2 < 0 f(x_(1))f(x_(2)) < 0f\left(x_{1}\right) f\left(x_{2}\right)<0f(x1)f(x2)<0, or if the point x 1 = x 2 x 1 = x 2 x_(1)=x_(2)x_{1}=x_{2}x1=x2is on a suitable side of z z zzz.
From (4) we then deduce the following delimitations of the error z − y z − y z-yz-yz−yof approximation y y yyyof z z zzz,
m 2 M 1 | z − x 1 | | z − x 2 | ≦ | z − y | ≦ M 2 m 1 | z − x 1 | | z − x 2 | m 2 M 1 z − x 1 z − x 2 ≦ | z − y | ≦ M 2 m 1 z − x 1 z − x 2 (m_(2))/(M_(1))|z-x_(1)||z-x_(2)| <= |z-y| <= (M_(2))/(m_(1))|z-x_(1)||z-x_(2)|\frac{m_{2}}{M_{1}}\left|z-x_{1}\right|\left|z-x_{2}\right| \leqq|z-y| \leqq \frac{M_{2}}{m_{1}}\left|z-x_{1}\right|\left|z-x_{2}\right|m2M1|z−x1||z−x2|≦|z−y|≦M2m1|z−x1||z−x2|
The coefficients m 2 M 1 , M 2 m 1 m 2 M 1 , M 2 m 1 (m_(2))/(M_(1)),(M_(2))/(m_(1))\frac{m_{2}}{M_{1}}, \frac{M_{2}}{m_{1}}m2M1,M2m1of these delimitations are, in general, better than m 2 m 1 2 M 1 3 , M 2 M 1 2 m 1 3 m 2 m 1 2 M 1 3 , M 2 M 1 2 m 1 3 (m_(2)m_(1)^(2))/(M_(1)^(3)),(M_(2)M_(1)^(2))/(m_(1)^(3))\frac{m_{2} m_{1}^{2}}{M_{1}^{3}}, \frac{M_{2} M_{1}^{2}}{m_{1}^{3}}m2m12M13,M2M12m13found by AM Ostrowski [1].
4. We propose to obtain an analogous result by considering an interpolation polynomial on 3 nodes.
Suppose that the function f f fffbe continuous, strictly monotonic and have a root z z zzzwithin the interval I I III. Let us consider three points x 1 , x 2 , x 3 ∈ I x 1 , x 2 , x 3 ∈ I x_(1),x_(2),x_(3)in Ix_{1}, x_{2}, x_{3} \in Ix1,x2,x3∈I, not all combined, such as x 1 ≦ x 2 ≦ x 3 , x 1 < z < x 3 x 1 ≦ x 2 ≦ x 3 , x 1 < z < x 3 x_(1) <= x_(2) <= x_(3),x_(1) < z < x_(3)x_{1} \leqq x_{2} \leqq x_{3}, x_{1}<z<x_{3}x1≦x2≦x3,x1<z<x3. Then the Lagrange-Hermite polynomial L ( x 1 , x 2 , x 2 ⋅ f + x ) L x 1 , x 2 , x 2 ⋅ f + x L(x_(1),x_(2),x^(2)*f+x)L\left(x_{1}, x_{2}, x^{2} \cdot f+x\right)L(x1,x2,x2⋅f+x)has y ′ y ′ y^(')y^{\prime}y′(and only one) on the interval ( x 1 , x 3 x 1 , x 3 x_(1),x_(3)x_{1}, x_{3}x1,x3). If we please wander L ( x 1 , x 2 , x 3 ; f ∣ x ) = L ( x ) L x 1 , x 2 , x 3 ; f ∣ x = L ( x ) L(x_(1),x_(2),x_(3);f∣x)=L(x)L\left(x_{1}, x_{2}, x_{3} ; f \mid x\right)=L(x)L(x1,x2,x3;f∣x)=L(x), the formula, analysis and moment L ( x 1 , x 2 , x 3 , j ∣ x ) = L x 1 , x 2 , x 3 , j ∣ x = L(x_(1),x_(2),x_(3),j∣x)=L\left(x_{1}, x_{2}, x_{3}, j \mid x\right)=L(x1,x2,x3,I∣x)=
(6) L ( y ′ ) − L ( z ) = [ x 1 , x 2 , x 3 , z ; f ] ( z − x 1 ) ( z − x 2 ) ( z − x 3 ) (6) L y ′ − L ( z ) = x 1 , x 2 , x 3 , z ; f z − x 1 z − x 2 z − x 3 {:(6)L(y^('))-L(z)=[x_(1),x_(2),x_(3),z;f](z-x_(1))(z-x_(2))(z-x_(3)):}\begin{equation*} L\left(y^{\prime}\right)-L(z)=\left[x_{1}, x_{2}, x_{3}, z ; f\right]\left(z-x_{1}\right)\left(z-x_{2}\right)\left(z-x_{3}\right) \tag{6} \end{equation*}(6)L(y′)−L(z)=[x1,x2,x3,z;f](z−x1)(z−x2)(z−x3)
will be used for error delimitation z − y ′ z − y ′ z-y^(')z-y^{\prime}z−y′of approximation y ′ y ′ y^(')y^{\prime}y′of z z zzz.
If L ( y ′ ) ≠ L ( z ) L y ′ ≠ L ( z ) L(y^('))!=L(z)L\left(y^{\prime}\right) \neq L(z)L(y′)≠L(z)from (6) it follows that
(7) z − y ′ = − [ x 1 , x 2 , x 3 , z ; f ] [ y ′ , z ; L ] ( z − x 1 ) ( z − x 2 ) ( z − x 3 ) (7) z − y ′ = − x 1 , x 2 , x 3 , z ; f y ′ , z ; L z − x 1 z − x 2 z − x 3 {:(7)z-y^(')=-([x_(1),x_(2),x_(3),z;f])/([y^('),z;L])(z-x_(1))(z-x_(2))(z-x_(3)):}\begin{equation*} z-y^{\prime}=-\frac{\left[x_{1}, x_{2}, x_{3}, z ; f\right]}{\left[y^{\prime}, z ; L\right]}\left(z-x_{1}\right)\left(z-x_{2}\right)\left(z-x_{3}\right) \tag{7} \end{equation*}(7)z−y′=−[x1,x2,x3,z;f][y′,z;L](z−x1)(z−x2)(z−x3)
  1. To go further, let us note that the derivative L ′ ( x ) L ′ ( x ) L^(')(x)L^{\prime}(x)L′(x)being of degree 1 and y ′ , z ∈ ( x 1 , x 3 ) y ′ , z ∈ x 1 , x 3 y^('),z in(x_(1),x_(3))y^{\prime}, z \in\left(x_{1}, x_{3}\right)y′,z∈(x1,x3)the divided difference [ y ′ , z ; L ] y ′ , z ; L [y^('),z;L]\left[y^{\prime}, z ; L\right][y′,z;L]is between L ′ ( x 1 ) L ′ x 1 L^(')(x_(1))L^{\prime}\left(x_{1}\right)L′(x1)And L ′ ( x 3 ) L ′ x 3 L^(')(x_(3))L^{\prime}\left(x_{3}\right)L′(x3).
Doing the calculations, we find
L ′ ( x 1 ) = x 2 − x 1 x 3 − x 1 { 2 [ x 1 , x 2 ; f ] − [ x 2 , x 3 ; f ] } + x 3 − x 2 x 3 − x 1 [ x 1 , x 2 ; f ] L ′ ( x 3 ) = x 3 − x 2 x 3 − x 1 { 2 [ x 2 , x 3 ; f ] − [ x 1 , x 2 ; f ] } + x 2 − x 1 x 3 − x 1 [ x 2 , x 3 ; f ] L ′ x 1 = x 2 − x 1 x 3 − x 1 2 x 1 , x 2 ; f − x 2 , x 3 ; f + x 3 − x 2 x 3 − x 1 x 1 , x 2 ; f L ′ x 3 = x 3 − x 2 x 3 − x 1 2 x 2 , x 3 ; f − x 1 , x 2 ; f + x 2 − x 1 x 3 − x 1 x 2 , x 3 ; f {:[L^(')(x_(1))=(x_(2)-x_(1))/(x_(3)-x_(1)){2[x_(1),x_(2);f]-[x_(2),x_(3);f]}+(x_(3)-x_(2))/(x_(3)-x_(1))[x_(1),x_(2);f]],[L^(')(x_(3))=(x_(3)-x_(2))/(x_(3)-x_(1)){2[x_(2),x_(3);f]-[x_(1),x_(2);f]}+(x_(2)-x_(1))/(x_(3)-x_(1))[x_(2),x_(3);f]]:}\begin{aligned} L^{\prime}\left(x_{1}\right) & =\frac{x_{2}-x_{1}}{x_{3}-x_{1}}\left\{2\left[x_{1}, x_{2} ; f\right]-\left[x_{2}, x_{3} ; f\right]\right\}+\frac{x_{3}-x_{2}}{x_{3}-x_{1}}\left[x_{1}, x_{2} ; f\right] \\ L^{\prime}\left(x_{3}\right) & =\frac{x_{3}-x_{2}}{x_{3}-x_{1}}\left\{2\left[x_{2}, x_{3} ; f\right]-\left[x_{1}, x_{2} ; f\right]\right\}+\frac{x_{2}-x_{1}}{x_{3}-x_{1}}\left[x_{2}, x_{3} ; f\right] \end{aligned}L′(x1)=x2−x1x3−x1{2[x1,x2;f]−[x2,x3;f]}+x3−x2x3−x1[x1,x2;f]L′(x3)=x3−x2x3−x1{2[x2,x3;f]−[x1,x2;f]}+x2−x1x3−x1[x2,x3;f]
If λ ≦ [ x 1 ′ , x 2 ′ ; f ] ≦ μ λ ≦ x 1 ′ , x 2 ′ ; f ≦ μ lambda <= [x_(1)^('),x_(2)^(');f] <= mu\lambda \leqq\left[x_{1}^{\prime}, x_{2}^{\prime} ; f\right] \leqq \muλ≦[x1′,x2′;f]≦μfor everything x 1 ′ , x 2 ′ ∈ I x 1 ′ , x 2 ′ ∈ I x_(1)^('),x_(2)^(')in Ix_{1}^{\prime}, x_{2}^{\prime} \in Ix1′,x2′∈I, we find that
(8) min ( λ , 2 λ − μ ) ≦ L ′ ( x 1 ) , L ′ ( x 3 ) ≦ max ( μ , 2 μ − λ ) (8) min ( λ , 2 λ − μ ) ≦ L ′ x 1 , L ′ x 3 ≦ max ( μ , 2 μ − λ ) {:(8)min(lambda","2lambda-mu) <= L^(')(x_(1))","L^(')(x_(3)) <= max(mu","2mu-lambda):}\begin{equation*} \min (\lambda, 2 \lambda-\mu) \leqq L^{\prime}\left(x_{1}\right), L^{\prime}\left(x_{3}\right) \leqq \max (\mu, 2 \mu-\lambda) \tag{8} \end{equation*}(8)min(λ,2λ−μ)≦L′(x1),L′(x3)≦max(μ,2μ−λ)
Now suppose that the function f f fffcheck the conditions
(9) { 0 < m 1 ≦ | [ x 1 ′ , x 2 ′ ; f ] | ≦ M 1 < + ∞ 0 < m 3 ≦ | [ x 1 ′ , x 2 ′ , x 3 ′ , x 4 ′ ; f ] | ≦ M 3 < + ∞ (9) 0 < m 1 ≦ x 1 ′ , x 2 ′ ; f ≦ M 1 < + ∞ 0 < m 3 ≦ x 1 ′ , x 2 ′ , x 3 ′ , x 4 ′ ; f ≦ M 3 < + ∞ {:(9){[0 < m_(1) <= |[x_(1)^('),x_(2)^(');f]| <= M_(1) < +oo],[0 < m_(3) <= |[x_(1)^('),x_(2)^('),x_(3)^('),x_(4)^(');f]| <= M_(3) < +oo]:}:}\left\{\begin{array}{l} 0<m_{1} \leqq\left|\left[x_{1}^{\prime}, x_{2}^{\prime} ; f\right]\right| \leqq M_{1}<+\infty \tag{9}\\ 0<m_{3} \leqq\left|\left[x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}, x_{4}^{\prime} ; f\right]\right| \leqq M_{3}<+\infty \end{array}\right.(9){0<m1≦|[x1′,x2′;f]|≦M1<+∞0<m3≦|[x1′,x2′,x3′,x4′;f]|≦M3<+∞
for any group of 4 distinct points x 1 ′ , x 2 ′ , x 3 ′ , x 4 ′ x 1 ′ , x 2 ′ , x 3 ′ , x 4 ′ x_(1)^('),x_(2)^('),x_(3)^('),x_(4)^(')x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}, x_{4}^{\prime}x1′,x2′,x3′,x4′of I I IIIand that M 1 < 2 m 1 M 1 < 2 m 1 M_(1) < 2m_(1)M_{1}<2 m_{1}M1<2m1.
From the first condition (9) it follows that the divided difference [ x 1 ′ , x 2 ′ ; f x 1 ′ , x 2 ′ ; f x_(1)^('),x_(2)^(');fx_{1}^{\prime}, x_{2}^{\prime} ; fx1′,x2′;f] is of invariable sign (otherwise it would also cancel itself out, which is impossible). We therefore have, either 0 < m 1 ≦ [ x 1 ′ , x 2 ′ ; f ] ≦ M 1 0 < m 1 ≦ x 1 ′ , x 2 ′ ; f ≦ M 1 0 < m_(1) <= [x_(1)^('),x_(2)^(');f] <= M_(1)0<m_{1} \leqq\left[x_{1}^{\prime}, x_{2}^{\prime} ; f\right] \leqq M_{1}0<m1≦[x1′,x2′;f]≦M1for everything x 1 ′ , x 2 ′ ∈ I x 1 ′ , x 2 ′ ∈ I x_(1)^('),x_(2)^(')in Ix_{1}^{\prime}, x_{2}^{\prime} \in Ix1′,x2′∈Ior else − M 1 ≦ [ x 1 ′ , x 2 ′ ; f ] ≦ − m 1 < 0 − M 1 ≦ x 1 ′ , x 2 ′ ; f ≦ − m 1 < 0 -M_(1) <= [x_(1)^('),x_(2)^(');f] <= -m_(1) < 0-M_{1} \leqq\left[x_{1}^{\prime}, x_{2}^{\prime} ; f\right] \leqq-m_{1}<0−M1≦[x1′,x2′;f]≦−m1<0for everything x 1 ′ x 1 ′ x_(1)^(')x_{1}^{\prime}x1′, x 2 ′ ∈ I x 2 ′ ∈ I x_(2)^(')in Ix_{2}^{\prime} \in Ix2′∈I.
From (8) it follows that 0 < 2 m 1 − M 1 = min ( m 1 , 2 m 1 − M 1 ) ≦≦ | L ′ ( x 1 ) | , | L ′ ( x 3 ) | ≦ max ( M 1 , 2 M 1 − m 1 ) ≦ 2 M 1 − m 1 0 < 2 m 1 − M 1 = min m 1 , 2 m 1 − M 1 ≦≦ L ′ x 1 , L ′ x 3 ≦ max M 1 , 2 M 1 − m 1 ≦ 2 M 1 − m 1 0 < 2m_(1)-M_(1)=min(m_(1),2m_(1)-M_(1))≦≦|L^(')(x_(1))|,|L^(')(x_(3))| <= max(M_(1),2M_(1)-m_(1)) <= 2M_(1)-m_(1)0<2 m_{1}-M_{1}=\min \left(m_{1}, 2 m_{1}-M_{1}\right) \leqq \leqq\left|L^{\prime}\left(x_{1}\right)\right|,\left|L^{\prime}\left(x_{3}\right)\right| \leqq \max \left(M_{1}, 2 M_{1}-m_{1}\right) \leqq 2 M_{1}-m_{1}0<2m1−M1=min(m1,2m1−M1)≦≦|L′(x1)|,|L′(x3)|≦max(M1,2M1−m1)≦2M1−m1and the formula.
(7) gives us the delimitation
m 3 2 M 1 − m 1 | z − x 1 | | z − x 2 | | z − x 3 | ≦ | z − y ′ | ≦ ≦ M 3 2 m 1 − M 1 | z − x 1 | | z − x 2 | | z − x 3 | m 3 2 M 1 − m 1 z − x 1 z − x 2 z − x 3 ≦ z − y ′ ≦ ≦ M 3 2 m 1 − M 1 z − x 1 z − x 2 z − x 3 {:[(m_(3))/(2M_(1)-m_(1))|z-x_(1)||z-x_(2)||z-x_(3)| <= |z-y^(')| <= ],[ <= (M_(3))/(2m_(1)-M_(1))|z-x_(1)||z-x_(2)||z-x_(3)|]:}\begin{gathered} \frac{m_{3}}{2 M_{1}-m_{1}}\left|z-x_{1}\right|\left|z-x_{2}\right|\left|z-x_{3}\right| \leqq\left|z-y^{\prime}\right| \leqq \\ \leqq \frac{M_{3}}{2 m_{1}-M_{1}}\left|z-x_{1}\right|\left|z-x_{2}\right|\left|z-x_{3}\right| \end{gathered}m32M1−m1|z−x1||z−x2||z−x3|≦|z−y′|≦≦M32m1−M1|z−x1||z−x2||z−x3|
Received May 3, 1967

Institute of ComputingAcademy of the Socialist Republicfrom Romania, Cluj Branch

BIBLIOGRAPHY

  1. Ostrowski, AM, Solution of Equations and Systems of Equations, 1966.
1968

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