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A new family of generalized quadrature methods for solving nonlinear equations. (English) Zbl 07539555

Summary: S. Weerakoon and T. G. I. Fernando [Appl. Math. Lett. 13, No. 8, 87–93 (2000; Zbl 0973.65037)] were resorted on a trapezoidal quadrature rule to derive an arithmetic mean Newton method with third-order convergence of the iterative scheme to solve nonlinear equations. Different quadrature methods have been developed, which form a special class of third-order iterative schemes requiring three evaluations of functions on \([f( x_n), f^\prime( x_n), f^\prime( y_n)]\) per iteration, where \(y_n\) is generated from the first Newton step. As an extension of these methods, we derive a new family of iterative schemes by using a new weight function \(H\) to generalize the quadrature methods, of which \(( x_{n + 1}- x_n) f^\prime( x_n)/H\) signifies an approximate area under the curve \(f^\prime(x)\) between \(x_n\) and \(x_{n + 1} \). Then, a generalization of the midpoint Newton method is obtained by using another weight function, which is based on three evaluations of functions on \([f( x_n), f^\prime( x_n), f^\prime(( x_n+ y_n)/2)]\) per iteration. The sufficient conditions of these two weight functions are derived, which guarantee that the convergence order of the proposed iterative schemes is three.

MSC:

65-XX Numerical analysis
41A25 Rate of convergence, degree of approximation
65D99 Numerical approximation and computational geometry (primarily algorithms)
65H05 Numerical computation of solutions to single equations

Citations:

Zbl 0973.65037
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Full Text: DOI

References:

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