Abstract
In 1984, Wang and Zheng (J. Comput. Math. 1984, 1, 70–76) introduced a new fourth order iterative method for the simultaneous computation of all zeros of a polynomial. In this paper, we present new local and semilocal convergence theorems with error estimates for Wang–Zheng’s method. Our results improve the earlier ones due to Wang and Wu (Computing 1987, 38, 75–87) and Petković, Petković, and Rančić (J. Comput. Appl. Math. 2007, 205, 32–52).
Highlights
Over the last few decades, we have observed rapid development of the theory of iterative methods for simultaneously finding all roots of a polynomial.The present paper deals with a thorough local and semilocal convergence analysis of a well-known iterative method, which was introduced in [34].In the paper, (K, | · |) stands for a valued field with absolute value | · | and K[z] stands for the ring of polynomials over K
We present two local convergence theorems (Theorem 5 and Theorem 6) and a semilocal convergence theorem (Theorem 9)
We provide two semilocal convergence theorems for Wang–Zheng’s method (1), which improve the result of Petković, Petković, and Rančić (Theorem 2)
Summary
Over the last few decades, we have observed rapid development of the theory of iterative methods for simultaneously finding all roots of a polynomial (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] and the references given therein).The present paper deals with a thorough local and semilocal convergence analysis of a well-known iterative method, which was introduced in [34].In the paper, (K, | · |) stands for a valued field with absolute value | · | and K[z] stands for the ring of polynomials over K. Over the last few decades, we have observed rapid development of the theory of iterative methods for simultaneously finding all roots of a polynomial (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] and the references given therein).
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