Error estimation in the numerical solving of operator equations

Ion Păvăloiu
(original), html, nonlinear equations in Banach spaces, Numerical Analysis, paper

Abstract

Let \(X\) be a Banach space and \(\varphi:X\rightarrow X\) a nonlinear operator. Assume the equation \(x=\varphi \left( x\right)\) has a solution \(x^{\ast}\), and the sequence \(x_{n}=\varphi \left( x_{n-1}\right) ,\ n\geq1\) converges with order at least \(k\geq2\).

We consider a setting useful for practical applications, where the sequence \(\left( x_{n}\right) _{n\geq0}\) is replaced by an approximate one \(\left( \xi_{n}\right) _{n\geq0\text{}}\) where \(\xi_{n}=\varphi^{\ast}\left( \xi_{n-1}\right) ,\ n\geq1\), where \(\left \Vert \varphi \left( x\right) -\varphi^{\ast}\left( x\right) \right \Vert \leq \varepsilon\) in a neighborhood of the solution.

We obtain a result regarding the evaluation of the errors \(\left \Vert \xi_{n}-x^{\ast}\right \Vert \), under the hypothesis that from a certain step, the sequence \(\left( \xi_{n}\right) _{n\geq0}\) is stationary.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Title

Original title (in Romanian)

Evaluarea erorilor în rezolvarea numerică a ecuaţiilor operatoriale

English translation of the title

Error estimation in the numerical solving of operator equations

Keywords

iterative methods; succesive approximations; approximate computations; convergence order; error estimation

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Cite this paper as:

I. Păvăloiu, Evaluarea erorilor în rezolvarea numerică a ecuaţiilor operatoriale, Studii şi cercetări matematice, 9 23 (1971), pp. 1459-1464 (in Romanian).

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Studii şi cercetări matematice

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Academia Republicii S.R.

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References

[1] Fujii, M., Remarks to Accelerated Iterative Processes for Numerical Solution of Equations. J. Sci. Hiroshima Univ. Ser. A-I, 27, 97-118. (1963).

[2]. Lankaster P., Error for the Newton – Raphson Method, Numerische Mathematik, 9, 1, 55-68 (1966).

[3] Ostrowski A’M., The Round- off stability  of iterations, Z,A.M.M,, 47, ,77-81 (1967).

[4]. Pavaloiu I., Asupra unor inegalitati recurente si aplicatii ale lor. St. cerc. mat. 8, 19, 1175-1179 (1967).

[5] I. Pavaloiu, Observatii asupra rezolvarii sistemelor de ecuatii cu ajutorul procedeelor iterative. Idem 9, 19, 1289-1298 (1967).

[6] Urabe, M.,  Convergence of Numerical lteratíon of Equations. J. Sci. Hiroshima IJniv. l Ser, A,. 19, 479-489 (1956).

[7] M. Urabe, Solution of Equation by Iteration Process. J. Sci. Hiroshima Univ. Ser. A-I, 26, 77-91 (1962).

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Error estimation in the numerical solving of operator equations

OF

I. PAVALOIU(Cluj)

In this paper we will study the evaluation of errors that appear in the approximate solution of operational equations, when their solution is done with rapidly converging iterative procedures. We will give an evaluation of the errors in question using a delimitation of the solutions of some nonlinear recurrent inequalities. In the papers [1], [2], [3], [4], [6] and [7] the authors studied this problem for iterative procedures that have the convergence order 1 and then applied the results found to the study of the errors of procedures with convergence order 2.
We consider the equation
(1) x − f ( x ) = 0 (1) x − f ( x ) = 0 {:(1)x-varphi(x)=0:}\begin{equation*} x-\varphi(x)=0 \tag{1} \end{equation*}(1)x−f(x)=0
and for its solution, the iterative procedure
(2) f ( x n ) = f ( x n − 1 ) , n = 1 , 2 , … , x 0 ∈ X (2) f x n = f x n − 1 , n = 1 , 2 , … , x 0 ∈ X {:(2)varphi(x_(n))=varphi(x_(n-1))","n=1","2","dots","x_(0)in X:}\begin{equation*} \varphi\left(x_{n}\right)=\varphi\left(x_{n-1}\right), n=1,2, \ldots, x_{0} \in X \tag{2} \end{equation*}(2)f(xn)=f(xn−1),n=1,2,…,x0∈X
where f f varphi\varphifis an operator defined on the space X X XXXand with values ​​in X X XXX. Throughout what follows, we will assume that the iterative procedure (2) satisfies the following conditions:
a) There is at least one element x ¯ ∈ X x ¯ ∈ X bar(x)in X\bar{x} \in Xx¯∈Xfor which x ¯ = f ( x ¯ ) x ¯ = f ( x ¯ ) bar(x)=varphi( bar(x))\bar{x}=\varphi(\bar{x})x¯=f(x¯).
b) lim n → ∞ ‖ x n − x ¯ ‖ = 0 lim n → ∞   x n − x ¯ = 0 lim_(n rarr oo)||x_(n)-( bar(x))||=0\lim _{n \rightarrow \infty}\left\|x_{n}-\bar{x}\right\|=0limn→∞‖xn−x¯‖=0, where ( x n ) n = 0 ∞ x n n = 0 ∞ (x_(n))_(n=0)^(yes)\left(x_{n}\right)_{n=0}^{\infty}(xn)n=0∞is the string generated by procedure (2).
e) The operator f f varphi\varphifadmits derivatives (in the Fréchet sense) up to the order k k kkkincluding ( k ≧ 2 k ≧ 2 k >= 2k \geqq 2k≧2- natural number), for which the following conditions are met:
φ ( i ) ( x ¯ ) = θ i , i = 1 , 2 , … , k − 1 ; sup x ∈ S ¯ ‖ φ ( k ) ( x ) ‖ ≦ M < + ∞ , φ ( i ) ( x ¯ ) = θ i , i = 1 , 2 , … , k − 1 ; sup x ∈ S ¯   φ ( k ) ( x ) ≦ M < + ∞ , varphi^((i))( bar(x))=theta_(i),i=1,2,dots,k-1;s u p_(x in bar(S))||varphi^((k))(x)|| <= M < +oo,\varphi^{(i)}(\bar{x})=\theta_{i}, i=1,2, \ldots, k-1 ; \sup _{x \in \bar{S}}\left\|\varphi^{(k)}(x)\right\| \leqq M<+\infty,f(i)(x¯)=ii,i=1,2,…,k−1;supx∈S¯‖f(k)(x)‖≦M<+∞,
where S = { x ∈ X : ‖ x − x ¯ ‖ ≦∣ x ¯ − x 0 ‖ } S = x ∈ X : ‖ x − x ¯ ‖ ≦∣ x ¯ − x 0 ‖ S={x in X:||x-( bar(x))||≦∣( bar(x))-x_(0)||}S=\left\{x \in X:\|x-\bar{x}\| \leqq \mid \bar{x}-x_{0} \|\right\}S={x∈X:‖x−x¯‖≦∣x¯−x0‖}and θ i θ i theta_(i)\theta_{i}iiis the null operation i − i − i-i-i−linear.
If we note with
ω n = ( M k ! ) 1 k − 1 ⋅ ‖ x n − x ¯ ‖ , n = 0 , 1 , 2 , … ω n = M k ! 1 k − 1 ⋅ x n − x ¯ , n = 0 , 1 , 2 , … omega_(n)=((M)/(k!))^((1)/(k-1))*||x_(n)-( bar(x))||,quad n=0,1,2,dots\omega_{n}=\left(\frac{M}{k!}\right)^{\frac{1}{k-1}} \cdot\left\|x_{n}-\bar{x}\right\|, \quad n=0,1,2, \ldotsohn=(Mk!)1k−1⋅‖xn−x¯‖,n=0,1,2,…
and we assume that ω 0 < 1 ω 0 < 1 omega_(0) < 1\omega_{0}<1oh0<1, taking into account hypotheses a) and c) we will show that the equality holds
lim n → ∞ ω n = 0 lim n → ∞   ω n = 0 lim_(n rarr oo)omega_(n)=0\lim _{n \rightarrow \infty} \omega_{n}=0limn→∞ohn=0
that is, the string ( x n ) n = 0 ∞ x n n = 0 ∞ (x_(n))_(n=0)^(oo)\left(x_{n}\right)_{n=0}^{\infty}(xn)n=0∞is convergent.
Indeed, applying the generalized Taylor formula we deduce the inequalities
ω n ≦ ω 0 k n , n = 1 , 2 , … ω n ≦ ω 0 k n , n = 1 , 2 , … omega_(n) <= omega_(0)^(k^(n)),quad n=1,2,dots\omega_{n} \leqq \omega_{0}^{k^{n}}, \quad n=1,2, \ldotsohn≦oh0kn,n=1,2,…
from which we deduce the required equality.
We first prove the following lemma:
Lemma 1. If δ δ delta\deltadis a non-negative number, ( ρ n ) n = 0 ∞ ρ n n = 0 ∞ (rho_(n))_(n=0)^(oo)\left(\rho_{n}\right)_{n=0}^{\infty}(rn)n=0∞is a sequence of non-negative numbers that satisfies the recurrence relation
(3) ρ n ≦ ρ n − 1 k + δ , k ≧ 2 , n = 1 , 2 , … (3) ρ n ≦ ρ n − 1 k + δ , k ≧ 2 , n = 1 , 2 , … {:(3)rho_(n) <= rho_(n-1)^(k)+delta","k >= 2","quad n=1","2","dots:}\begin{equation*} \rho_{n} \leqq \rho_{n-1}^{k}+\delta, k \geqq 2, \quad n=1,2, \ldots \tag{3} \end{equation*}(3)rn≦rn−1k+d,k≧2,n=1,2,…
(4) ρ 0 < 1 (4) ρ 0 < 1 {:(4)rho_(0) < 1:}\begin{equation*} \rho_{0}<1 \tag{4} \end{equation*}(4)r0<1
and ā is the smallest positive solution (if it exists) of the equation
(5) α = ( α + ρ 0 ) k − ρ 0 k + δ (5) α = α + ρ 0 k − ρ 0 k + δ {:(5)alpha=(alpha+rho_(0))^(k)-rho_(0)^(k)+delta:}\begin{equation*} \alpha=\left(\alpha+\rho_{0}\right)^{k}-\rho_{0}^{k}+\delta \tag{5} \end{equation*}(5)a=(a+r0)k−r0k+d
Then for anything n = 1 , 2 , … n = 1 , 2 , … n=1,2,dotsn=1,2, \ldotsn=1,2,…the inequalities occur:
(6) ρ n ≦ ρ 0 k n + α ¯ (6) ρ n ≦ ρ 0 k n + α ¯ {:(6)rho_(n) <= rho_(0)^(kn)+ bar(alpha):}\begin{equation*} \rho_{n} \leqq \rho_{0}^{k n}+\bar{\alpha} \tag{6} \end{equation*}(6)rn≦r0kn+a¯
Proof. We will prove inequalities (6) by induction. For n = 1 n = 1 n=1n=1n=1we obtain the inequality
ρ 1 ≦ ( ρ 0 + α ¯ ) k + δ ρ 1 ≦ ρ 0 + α ¯ k + δ rho_(1) <= (rho_(0)+( bar(alpha)))^(k)+delta\rho_{1} \leqq\left(\rho_{0}+\bar{\alpha}\right)^{k}+\deltar1≦(r0+a¯)k+d
and if α ¯ α ¯ bar(alpha)\bar{\alpha}a¯is a solution of equation (5), then obviously the inequality holds
ρ 1 ≦ ρ 0 k + α ¯ ρ 1 ≦ ρ 0 k + α ¯ rho_(1) <= rho_(0)^(k)+ bar(alpha)\rho_{1} \leqq \rho_{0}^{k}+\bar{\alpha}r1≦r0k+a¯
which proves the statement of the lemma for the case n = 1 n = 1 n=1n=1n=1From (4) and (5) we deduce that for n = 1 , 2 , … n = 1 , 2 , … n=1,2,dotsn=1,2, \ldotsn=1,2,…inequalities occur
(7) C k 1 ⋅ ρ 0 k n ( k − 1 ) α ¯ + … + C k k − 1 ρ 0 k n α ¯ k − 1 + α ¯ k + δ ≦ α ¯ . (7) C k 1 ⋅ ρ 0 k n ( k − 1 ) α ¯ + … + C k k − 1 ρ 0 k n α ¯ k − 1 + α ¯ k + δ ≦ α ¯ . {:(7)C_(k)^(1)*rho_(0)^(k^(n)(k-1)) bar(alpha)+dots+C_(k)^(k-1)rho_(0)^(k^(n)) bar(alpha)^(k-1)+ bar(alpha)^(k)+delta <= bar(alpha).:}\begin{equation*} C_{k}^{1} \cdot \rho_{0}^{k^{n}(k-1)} \bar{\alpha}+\ldots+C_{k}^{k-1} \rho_{0}^{k^{n}} \bar{\alpha}^{k-1}+\bar{\alpha}^{k}+\delta \leqq \bar{\alpha} . \tag{7} \end{equation*}(7)Ck1⋅r0kn(k−1)a¯+…+Ckk−1r0kna¯k−1+a¯k+d≦a¯.
We assume by induction that the inequality holds:
ρ n − 1 ≦ ρ 0 k n − 1 + α ¯ ρ n − 1 ≦ ρ 0 k n − 1 + α ¯ rho_(n-1) <= rho_(0)^(k^(n-1))+ bar(alpha)\rho_{n-1} \leqq \rho_{0}^{k^{n-1}}+\bar{\alpha}rn−1≦r0kn−1+a¯
which by substituting it in (3) we obtain
ρ n ≦ ( ρ 0 k n − 1 + α ¯ ) k + δ ρ n ≦ ρ 0 k n − 1 + α ¯ k + δ rho_(n) <= (rho_(0)^(k^(n-1))+( bar(alpha)))^(k)+delta\rho_{n} \leqq\left(\rho_{0}^{k^{n-1}}+\bar{\alpha}\right)^{k}+\deltarn≦(r0kn−1+a¯)k+d
from which, expanding according to Newton's binomial formula and taking into account inequalities (7.), we deduce
ρ n = ρ 0 k n + α ¯ ρ n = ρ 0 k n + α ¯ rho_(n)=rho_(0)^(k^(n))+ bar(alpha)\rho_{n}=\rho_{0}^{k^{n}}+\bar{\alpha}rn=r0kn+a¯
which proves the statement of the lemma.
For the study of errors we will consider, alongside the iterative procedure (2), the following approximate procedure,
(8) ξ n = φ ∗ ( ξ n − 1 ) , unde ξ 0 = x 0 , n = 1 , 2 , … (8) ξ n = φ ∗ ξ n − 1 ,  unde  ξ 0 = x 0 , n = 1 , 2 , … {:(8)xi_(n)=varphi^(**)(xi_(n-1))","" unde "xi_(0)=x_(0)","n=1","2","dots:}\begin{equation*} \xi_{n}=\varphi^{*}\left(\xi_{n-1}\right), \text { unde } \xi_{0}=x_{0}, n=1,2, \ldots \tag{8} \end{equation*}(8)xn=f∗(xn−1), where x0=x0,n=1,2,…
where the operator φ ∗ φ ∗ varphi^(**)\varphi^{*}f∗is defined on the space X X XXXand with values ​​in X X XXX.
Theorem 1. Given a negative real number e. We assume that relative to e there is at least one positive number M 1 M 1 M_(1)M_{1}M1so that the conditions are met:
a) Equation (5) where
ρ 0 = ( M 1 k ! ) 1 k − 1 ⋅ ‖ x ¯ − x 0 ‖ , δ = ε ( M 1 k ! ) 1 k − 1 ρ 0 = M 1 k ! 1 k − 1 ⋅ x ¯ − x 0 , δ = ε M 1 k ! 1 k − 1 rho_(0)=((M_(1))/(k!))^((1)/(k-1))*||( bar(x))-x_(0)||,quad delta=epsi((M_(1))/(k!))^((1)/(k-1))\rho_{0}=\left(\frac{M_{1}}{k!}\right)^{\frac{1}{k-1}} \cdot\left\|\bar{x}-x_{0}\right\|, \quad \delta=\varepsilon\left(\frac{M_{1}}{k!}\right)^{\frac{1}{k-1}}r0=(M1k!)1k−1⋅‖x¯−x0‖,d=e(M1k!)1k−1
has at least one positive real solution.
b) Fréchet derivative φ ( k ) φ ( k ) varphi^((k))\varphi^{(k)}f(k)check the inequality
‖ φ ( k ) ( x ) ‖ ≦ M 1 < + ∞ φ ( k ) ( x ) ≦ M 1 < + ∞ ||varphi^((k))(x)|| <= M_(1) < +oo\left\|\varphi^{(k)}(x)\right\| \leqq M_{1}<+\infty‖f(k)(x)‖≦M1<+∞
for anything w ∈ S ′ w ∈ S ′ w inS^(')w \in S^{\prime}In∈S′where S ′ = { x ∈ X : ‖ x − x ¯ ‖ ≤ ‖ x ¯ − x 0 ‖ + α ¯ ( k ! M 1 ) 1 k − 1 } S ′ = x ∈ X : ‖ x − x ¯ ‖ ≤ x ¯ − x 0 + α ¯ k ! M 1 1 k − 1 S^(')={x in X:||x-( bar(x))|| <= ||( bar(x))-x_(0)||+( bar(alpha))((k!)/(M_(1)))^((1)/(k-1))}S^{\prime}=\left\{x \in X:\|x-\bar{x}\| \leq\left\|\bar{x}-x_{0}\right\|+\bar{\alpha}\left(\frac{k!}{M_{1}}\right)^{\frac{1}{k-1}}\right\}S′={x∈X:‖x−x¯‖≤‖x¯−x0‖+a¯(k!M1)1k−1}and ā is the smallest positive solution of equation (5).
e) The number M 1 M 1 M_(1)M_{1}M1and the element x 0 x 0 x_(0)x_{0}x0are such that the inequality holds ρ 0 < 1 ρ 0 < 1 rho_(0) < 1\rho_{0}<1r0<1d )
The operator φ ∗ φ ∗ varphi^(**)\varphi^{*}f∗is chosen so that the inequality is verified
‖ φ ( x ) − φ ∗ ( x ) ‖ ≦ ε , pentru orice x ∈ S ′ . φ ( x ) − φ ∗ ( x ) ≦ ε ,  pentru orice  x ∈ S ′ . ||varphi(x)-varphi^(**)(x)|| <= epsi," pentru orice "x inS^(').\left\|\varphi(x)-\varphi^{*}(x)\right\| \leqq \varepsilon, \text { pentru orice } x \in S^{\prime} .‖f(x)−f∗(x)‖≦e, for anything x∈S′.
Under these conditions all successive iterations ( ξ n ) n = 0 ∞ ξ n n = 0 ∞ (xi_(n))_(n=0)^(oo)\left(\xi_{n}\right)_{n=0}^{\infty}(xn)n=0∞belong to the sphere S ′ S ′ S^(')S^{\prime}S′and the delimitation takes place
(9) ‖ x ¯ − ξ n ‖ ≦ ( k ! M 1 ) 1 k − 1 ⋅ ( ρ 0 k n + α ¯ ) , n = 1 , 2 , … (9) x ¯ − ξ n ≦ k ! M 1 1 k − 1 ⋅ ρ 0 k n + α ¯ , n = 1 , 2 , … {:(9)||( bar(x))-xi_(n)|| <= ((k!)/(M_(1)))^((1)/(k-1))*(rho_(0)^(k^(n))+( bar(alpha)))","quad n=1","2","dots:}\begin{equation*} \left\|\bar{x}-\xi_{n}\right\| \leqq\left(\frac{k!}{M_{1}}\right)^{\frac{1}{k-1}} \cdot\left(\rho_{0}^{k^{n}}+\bar{\alpha}\right), \quad n=1,2, \ldots \tag{9} \end{equation*}(9)‖x¯−xn‖≦(k!M1)1k−1⋅(r0kn+a¯),n=1,2,…
Demonstration. Indeed, because ξ 0 = x 0 ∈ S ′ ξ 0 = x 0 ∈ S ′ xi_(0)=x_(0)inS^(')\xi_{0}=x_{0} \in S^{\prime}x0=x0∈S′we will have
‖ x ¯ − ξ 1 ‖ ≦ ‖ φ ( x ¯ ) − φ ( ξ 0 ) ‖ + ‖ φ ( ξ 0 ) − φ ∗ ( ξ 0 ) ‖ ≦ M 1 k ! ‖ x ¯ − ξ 0 ‖ k + ε x ¯ − ξ 1 ≦ φ ( x ¯ ) − φ ξ 0 + φ ξ 0 − φ ∗ ξ 0 ≦ M 1 k ! x ¯ − ξ 0 k + ε ||( bar(x))-xi_(1)|| <= ||varphi(( bar(x)))-varphi(xi_(0))||+||varphi(xi_(0))-varphi^(**)(xi_(0))|| <= (M_(1))/(k!)||( bar(x))-xi_(0)||^(k)+epsi\left\|\bar{x}-\xi_{1}\right\| \leqq\left\|\varphi(\bar{x})-\varphi\left(\xi_{0}\right)\right\|+\left\|\varphi\left(\xi_{0}\right)-\varphi^{*}\left(\xi_{0}\right)\right\| \leqq \frac{M_{1}}{k!}\left\|\bar{x}-\xi_{0}\right\|^{k}+\varepsilon‖x¯−x1‖≦‖f(x¯)−f(x0)‖+‖f(x0)−f∗(x0)‖≦M1k!‖x¯−x0‖k+eand noting
ρ 1 = ( M 1 k ! ) 1 k − 1 ‖ x ¯ − ξ 1 ‖ şi ρ 0 = ( M 1 k ! ) 1 k − 1 ‖ x ¯ − ξ 0 ‖ ρ 1 = M 1 k ! 1 k − 1 x ¯ − ξ 1  şi  ρ 0 = M 1 k ! 1 k − 1 x ¯ − ξ 0 rho_(1)=((M_(1))/(k!))^((1)/(k-1))||( bar(x))-xi_(1)||" şi "rho_(0)=((M_(1))/(k!))^((1)/(k-1))||( bar(x))-xi_(0)||\rho_{1}=\left(\frac{M_{1}}{k!}\right)^{\frac{1}{k-1}}\left\|\bar{x}-\xi_{1}\right\| \text { şi } \rho_{0}=\left(\frac{M_{1}}{k!}\right)^{\frac{1}{k-1}}\left\|\bar{x}-\xi_{0}\right\|r1=(M1k!)1k−1‖x¯−x1‖ and r0=(M1k!)1k−1‖x¯−x0‖
we deduce the inequality
ρ 1 ≦ ρ 0 k + δ . ρ 1 ≦ ρ 0 k + δ . rho_(1) <= rho_(0)^(k)+delta.\rho_{1} \leqq \rho_{0}^{k}+\delta .r1≦r0k+d.
Thus the conditions of Lemma 1 are satisfied for ρ 0 ρ 0 rho_(0)\rho_{0}r0and ρ 1 ρ 1 rho_(1)\rho_{1}r1.
Either α ¯ α ¯ bar(alpha)\bar{\alpha}a¯the smallest positive solution of equation (5). Taking into account lemma 1 and condition c) we have
ρ 1 ≦ ρ 0 k + α ¯ ρ 1 ≦ ρ 0 k + α ¯ rho_(1) <= rho_(0)^(k)+ bar(alpha)\rho_{1} \leqq \rho_{0}^{k}+\bar{\alpha}r1≦r0k+a¯
from which we easily deduce that ξ 1 ∈ S ′ ξ 1 ∈ S ′ xi_(1)inS^(')\xi_{1} \in S^{\prime}x1∈S′.
We now assume by induction that ξ n − 1 ∈ S ′ ξ n − 1 ∈ S ′ xi_(n-1)inS^(')\xi_{n-1} \in S^{\prime}xn−1∈S′and we demonstrate that ξ n ∈ S ′ ξ n ∈ S ′ xi_(n)inS^(')\xi_{n} \in S^{\prime}xn∈S′Indeed, we have:
‖ x ¯ − ξ n ‖ ≦ ‖ φ ( x ¯ ) − φ ( ξ n − 1 ) ‖ + ‖ φ ( ξ n − 1 ) − φ ∗ ( ξ n − 1 ) ‖ ≦ ≦ M 1 k ! ‖ x → − ξ n − 1 ‖ π + ε x ¯ − ξ n ≦ φ ( x ¯ ) − φ ξ n − 1 + φ ξ n − 1 − φ ∗ ξ n − 1 ≦ ≦ M 1 k ! x → − ξ n − 1 π + ε {:[||( bar(x))-xi_(n)|| <= ||varphi(( bar(x)))-varphi(xi_(n-1))||+||varphi(xi_(n-1))-varphi^(**)(xi_(n-1))|| <= ],[ <= (M_(1))/(k!)||( vec(x))-xi_(n-1)||^(pi)+epsi]:}\begin{gathered} \left\|\bar{x}-\xi_{n}\right\| \leqq\left\|\varphi(\bar{x})-\varphi\left(\xi_{n-1}\right)\right\|+\left\|\varphi\left(\xi_{n-1}\right)-\varphi^{*}\left(\xi_{n-1}\right)\right\| \leqq \\ \leqq \frac{M_{1}}{k!}\left\|\vec{x}-\xi_{n-1}\right\|^{\pi}+\varepsilon \end{gathered}‖x¯−xn‖≦‖f(x¯)−f(xn−1)‖+‖f(xn−1)−f∗(xn−1)‖≦≦M1k!‖x→−xn−1‖p+e
where, noting with ρ n = ( M 1 k ! ) 1 k − 1 ‖ x ¯ − ξ n ‖ , n = 2 , 3 , … ρ n = M 1 k ! 1 k − 1 x ¯ − ξ n , n = 2 , 3 , … rho_(n)=((M_(1))/(k!))^((1)/(k-1))||( bar(x))-xi_(n)||,n=2,3,dots\rho_{n}=\left(\frac{M_{1}}{k!}\right)^{\frac{1}{k-1}}\left\|\bar{x}-\xi_{n}\right\|, n=2,3, \ldotsrn=(M1k!)1k−1‖x¯−xn‖,n=2,3,…, we have
ρ n ≦ ρ n − 1 k + δ ρ n ≦ ρ n − 1 k + δ rho_(n) <= rho_(n-1)^(k)+delta\rho_{n} \leqq \rho_{n-1}^{k}+\deltarn≦rn−1k+d
Applying lemma 1 we deduce inequality (9) from which it follows that ξ n ∈ S ′ ξ n ∈ S ′ xi_(n)inS^(')\xi_{n} \in S^{\prime}xn∈S′.
Note. It may happen that during the calculations, starting with a certain step N N NNN, let's have ξ N = ξ N + 1 = … ξ N = ξ N + 1 = … xi_(N)=xi_(N+1)=dots\xi_{N}=\xi_{N+1}=\ldotsxN=xN+1=…In this case for any n == N , N + 1 , … n == N , N + 1 , … n==N,N+1,dotsn= =N, N+1, \ldotsn==N,N+1,…inequality occurs
(10) ‖ ξ n − x ¯ ‖ ≦ ( k ! M 1 ) 1 k − 1 α ¯ (10) ξ n − x ¯ ≦ k ! M 1 1 k − 1 α ¯ {:(10)||xi_(n)-( bar(x))|| <= ((k!)/(M_(1)))^((1)/(k-1)) bar(alpha):}\begin{equation*} \left\|\xi_{n}-\bar{x}\right\| \leqq\left(\frac{k!}{M_{1}}\right)^{\frac{1}{k-1}} \bar{\alpha} \tag{10} \end{equation*}(10)‖xn−x¯‖≦(k!M1)1k−1a¯
Example. Be φ ( x ) = 1 2 ( x + 0 , 2 x ) φ ( x ) = 1 2 x + 0 , 2 x varphi(x)=(1)/(2)(x+(0,2)/(x))\varphi(x)=\frac{1}{2}\left(x+\frac{0,2}{x}\right)f(x)=12(x+0,2x)which generates the following iterative process
x n = 1 2 ( x n − 1 + 0 , 2 x n − 1 ) , x 0 ≠ 0 , n = 1 , 2 , … x n = 1 2 x n − 1 + 0 , 2 x n − 1 , x 0 ≠ 0 , n = 1 , 2 , … x_(n)=(1)/(2)(x_(n-1)+(0,2)/(x_(n-1))),x_(0)!=0,n=1,2,dotsx_{n}=\frac{1}{2}\left(x_{n-1}+\frac{0,2}{x_{n-1}}\right), x_{0} \neq 0, n=1,2, \ldotsxn=12(xn−1+0,2xn−1),x0≠0,n=1,2,…
which is used to calculate the positive root of the equation
x 2 − 0 , 2 = 0 x 2 − 0 , 2 = 0 x^(2)-0,2=0x^{2}-0,2=0x2−0,2=0
Assuming that ε = 10 − 7 ε = 10 − 7 epsi=10^(-7)\varepsilon=10^{-7}e=10−7and x 0 = 0 , 44 x 0 = 0 , 44 x_(0)=0,44x_{0}=0,44x0=0,44, we find ‖ x ¯ − x 0 ‖ ≦ 0 , 01 , M 1 = 4 x ¯ − x 0 ≦ 0 , 01 , M 1 = 4 ||( bar(x))-x_(0)|| <= 0,01,M_(1)=4\left\|\bar{x}-x_{0}\right\| \leqq 0,01, M_{1}=4‖x¯−x0‖≦0,01,M1=4, ρ 0 = 0 , 02 , δ = 2 ⋅ 10 − 7 , α ¯ ≦ 3 ⋅ 10 − 7 ρ 0 = 0 , 02 , δ = 2 ⋅ 10 − 7 , α ¯ ≦ 3 ⋅ 10 − 7 rho_(0)=0,02,delta=2*10^(-7), bar(alpha) <= 3*10^(-7)\rho_{0}=0,02, \delta=2 \cdot 10^{-7}, \bar{\alpha} \leqq 3 \cdot 10^{-7}r0=0,02,d=2⋅10−7,a¯≦3⋅10−7The successive interactions and therefore the values ​​of the function φ ∗ φ ∗ varphi^(**)\varphi^{*}f∗are given in the table.
Table
No. crt. ξ n ξ n xi_(n)\xi_{n}xn φ ⋆ ( ξ n ) φ ⋆ ξ n varphi^(***)(xi_(n))\varphi^{\star}\left(\xi_{n}\right)f⋆(xn)
0 0,44 0,44727272
1 0,44727272 0,44721359
2 0,44721359 0,44721359
3 0,44721359 -
Nr. crt. xi_(n) varphi^(***)(xi_(n)) 0 0,44 0,44727272 1 0,44727272 0,44721359 2 0,44721359 0,44721359 3 0,44721359 -| Nr. crt. | $\xi_{n}$ | $\varphi^{\star}\left(\xi_{n}\right)$ | | :---: | :---: | :---: | | 0 | 0,44 | 0,44727272 | | 1 | 0,44727272 | 0,44721359 | | 2 | 0,44721359 | 0,44721359 | | 3 | 0,44721359 | - |
The table shows that ξ 2 = ξ 3 = … = 0 , 44721359 ξ 2 = ξ 3 = … = 0 , 44721359 xi_(2)=xi_(3)=dots=0,44721359\xi_{2}=\xi_{3}=\ldots=0,44721359x2=x3=…=0,44721359, from which, taking into account the above observation and inequality (10), we obtain
| 0 , 2 − 0 , 44721359 | ≦ 1 , 5 ⋅ 10 − 7 | 0 , 2 − 0 , 44721359 | ≦ 1 , 5 ⋅ 10 − 7 |sqrt(0,2)-0,44721359| <= 1,5*10^(-7)|\sqrt{0,2}-0,44721359| \leqq 1,5 \cdot 10^{-7}|0,2−0,44721359|≦1,5⋅10−7
Received by the editorial office on October 10, 1970.

EVALUATION OF ERRORS IN THE NUMERICAL SOLUTION OF OPERATIONAL EQUATIONS

(RÉSUMÉ)

In the work we give an evaluation of errors for the case where a given operational equation is solved with the iterative method having an order of convergence k ( k ≧ 2 ) k ( k ≧ 2 ) k(k >= 2)k(k \geqq 2)k(k≧2). Thus we generalize the results of works [1], [2], [3], [4], [5] and [7].

BIBLIOGRAPHY

  1. Fujit, M., Remarks to Accelerated Iterative Processes for Numerical Solution of Equations. J. Sci. Hiroshima Univ. Ser. A-I, 27, 97-118 (1963).
  2. Lankaster, P., Error for the Newton - Raphson Method. Numerische Mathematik, 9, 1, 55-68 (1966).
    .3. Ostrowski, A.M., The Round - off Stability of iterations. Z.A.M.M., 47, 77-81 (1967).
  3. Păvăloiv, I., On some recurrent inequalities and their applications. St. cerc. mat. 8, 19, 11751179 (1967).
    • Observations on the solution of systems of equations using iterative procedures. Idem, 9, 19, 1289-1298 (1967).
  5. Jrabe, M., Convergence of Numerical Iteration of Equations. J. Sci. Hiroshima Univ. 1 Ser. A, 19, 479-489 (1956).
    • Solution of Equations by Iteration Process. J. Sci. Hiroshima Univ. Ser. A-I, 26, 77-91 (1962).
1971

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