Error estimation in the numerical solving of operator equations

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.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

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

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

Scanned paper.

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

Google Scholar Profile

[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).