A generalization of the Jenkins-Traub method

Author:
J. A. Ford

Journal:
Math. Comp. **31** (1977), 193-203

MSC:
Primary 65H05

DOI:
https://doi.org/10.1090/S0025-5718-1977-0428703-9

MathSciNet review:
0428703

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Abstract: A class of methods for finding zeros of polynomials is derived which depends upon an arbitrary parameter . The Jenkins-Traub algorithm is a special case, corresponding to the choice . Global convergence is proved for large and small values of and a duality between pairs of members is exhibited. Finally, we show that many members of the class (including the Jenkins-Traub method) converge with *R*-order at least 2.618..., which improves upon the result obtained by Jenkins and Traub [3].

**[1]**C. G. Broyden and J. A. Ford,*A new method of polynomial deflation*, J. Inst. Math. Appl.**16**(1975), no. 3, 271–281. MR**0418443****[2]**J. A. FORD, "A generalization of the Jenkins-Traub method," Technical Report CSM-9, Univ. of Essex Computing Centre, June 1975.**[3]**M. A. Jenkins and J. F. Traub,*A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration*, Numer. Math.**14**(1969/1970), 252–263. MR**0258271**, https://doi.org/10.1007/BF02163334**[4]**M. A. Jenkins and J. F. Traub,*A three-stage algorithm for real polynomials using quadratic iteration.*, SIAM J. Numer. Anal.**7**(1970), 545–566. MR**0279995**, https://doi.org/10.1137/0707045**[5]**Morris Marden,*The Geometry of the Zeros of a Polynomial in a Complex Variable*, Mathematical Surveys, No. 3, American Mathematical Society, New York, N. Y., 1949. MR**0031114****[6]**J. M. Ortega and W. C. Rheinboldt,*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810****[7]**G. Peters and J. H. Wilkinson,*Practical problems arising in the solution of polynomial equations*, J. Inst. Math. Appl.**8**(1971), 16–35. MR**0298931**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0428703-9

Article copyright:
© Copyright 1977
American Mathematical Society