Abstract
In applications, one of the basic problems is to solve the fixed point equationx=Tx withT a contractive mapping. Two theorems which can be implemented computationally to verify the existence of a solutionx * to the equation and to obtain a convergent approximate solution sequence {x n } are the classical Banach contraction mapping theorem and the newly established global convergence theorem of the ball algorithms in You, Xu and Liu [16]. These two theorems are compared on the basis of sensitivity, precision, computational complexity and efficiency. The comparison shows that except for computational complexity, the latter theorem is of far greater sensivity, precision and computational efficiency. This conclusion is supported by a number of numerical examples.
Zusammenfassung
Das Lösen von Fixpunktgleichungenx=Tx mit kontrahierendemT ist ein grundlegendes Problem in den Anwendungen. Zur Bestätigung der Existenz einer Lösungx und zur Bestimmung einer konvergenten Folge {x n } von Näherungslösungen kann man entweder den klassischen Banach'schen Fixpunktsatz oder einen neueren globalen Konvergenzsatz für Kugelalgorithmen in You, Xu und Lin [16] algorithmisch implementieren. Diese zwei Sätze werden hier in Bezug auf ihre Empfindlichkeit, Genauigkeit, rechnerische Komplexität und Effizienz verglichen. Dabei schneidet der zweitgenannte Satz in allen Aspekten außer der Komplexität deutlich besser ab. Dieses Ergebnis wird durch einige numerische Beispiele bestätigt.
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References
Birkhoff, G., Lynch, R. E.: Numerical solution of elliptic problems. SIAM Philadelphia (1984).
Collatz, L.: Functional analysis and numerical mathematics. New York: Academic Press 1966.
Neumaier, A.: A better estimate for fixed points of contractions. ZAMM62, 627 (1982).
Nickel, K.: A globally convergent ball Netwon method. SIAM J. Numer. Anal.18, 988–1003 (1981).
Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.
Pascali, D., Sburlan, S.: Nonlinear mapping of monotone type. Romania: Editure Academiei Bucuresti 1978.
Rall, L. B.: A comparison of the existence theorems of Kantorovich and Moore. SIAM J. Numer. Anal.17, 148–161 (1980).
Shi, X. Z.: Comparison about ball and point algorithms for nonlinear system of equations. Master's Thesis, Xi'an Jiaotong University, 1990.
Shi, X. Z., Xu, Z. B.: A further modification of ball Newton Algorithm (to appear).
Valentine, F. A.: A Lipschitz preserving extension for a vector function. Amer. J. Math.69, 83–93 (1945).
Williamson, T. E. Jr.: Geometric estimation of fixed point of Lipschitzian mapping II. J. Math. Anal. Appli.62, 600–609 (1978).
Xu, Z. B.: A computational ball test for the existence of solution to nonlinear operator equations. SIAM J. Numer. Anal.26, 239–248 (1989).
Xu, Z. B.:L p characeteristic inequalities and their applications. Acta Math. Sinica32, 209–218 (1989).
You, Z. Y., Xu, Z. B., Liu, K. K.: Generalized theory and some specializations of the region contraction algorithm I—Ball operation. In: Nickel, K. (ed.) Interval mathematic 1985. Berlin, Heidelberg, New York: Springer1986, 209–223 (Lecture Notes in Computer Science 212).
You, Z. Y., Xu, Z. B.: Ball algorithms for constructing solutions of nonlinear operator equations. J. Comput. Appl. Math.38, 457–471 (1991).
You, Z. Y., Xu, Z. B., Liu, K. K.: Generalize theory and some specializations of the region contraction algorithm II. In: Nickel, K. (ed.) Interval mathematics 1985. Berlin, Heidelberg, New York: Springer1986, 239–248 (Lecture Notes in Computer Science 212).
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Xu, Z., Shi, X. A comparison of point and ball iterations in the contractive mapping case. Computing 49, 75–85 (1992). https://doi.org/10.1007/BF02238651
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DOI: https://doi.org/10.1007/BF02238651