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On a class of quadratically convergent iteration formulae. (English) Zbl 1078.65036

Almost all iterative techniques for solving a nonlinear equation require having one or more initial guesses for the desired root. If the interval containing the root is known, several classical iterative procedures to refine it (for example Newton’s method, regula falsi, Wu method and various modifications of these methods) are available. The problem is, that the method may fail to convergence in case the initial guess is far from the zero point or the derivative of the right-side function is small in the vicinity of the required root.
The authors try to eliminate this defect by a modification of the iteration process. Using the Wu and Wu theorems two classes of iterative algorithms for transcendent equation solving are proposed. The developed algorithms do not require the derivative of the right-side function and have quadratic speed of convergence. The suggested algorithms are compared with the original Newton’s method on several numerical experiments.

MSC:

65H05 Numerical computation of solutions to single equations
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References:

[1] Dowell, M.; Jarrat, P., A modified regula-falsi method for computing the root of an equation, Bit, 11, 168-174, (1971) · Zbl 0236.65036
[2] Wu, X.; Wu, H.W., On a class of quadratic convergence iteration formulae without derivatives, Applied mathematics and computation, 10, 7, 77-80, (2000) · Zbl 1023.65042
[3] Wu, X.; Fu, D., New high-order convergence iteration methods without employing derivatives for solving nonlinear equations, Computers and mathematics with applications, 41, 489-495, (2001) · Zbl 0985.65047
[4] Ostrowski, A.M., Solution of equations in Euclidean and Banach space, (1973), Academic Press New York · Zbl 0304.65002
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