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The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more complicated variant for the special case of polynomials with real coefficients, commonly known as the "RPOLY" algorithm. The latter is "practically a standard in black-box polynomial root-finders".〔Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007), Numerical Recipes: The Art of Scientific Computing, 3rd ed., Cambridge University Press, page 470.〕 This article describes the complex variant. Given a polynomial ''P'', : with complex coefficients it computes approximations to the ''n'' zeros of ''P''(''z''), one at a time in roughly increasing order of magnitude. After each root is computed, its linear factor is removed from the polynomial. Using this ''deflation'' guarantees that each root is computed only once and that all roots are found. The real variant follows the same pattern, but computes two roots at a time, either two real roots or a pair of conjugate complex roots. By avoiding complex arithmetic, the real variant can be faster (by a factor of 4) than the complex variant. The Jenkins–Traub algorithm has stimulated considerable research on theory and software for methods of this type. ==Overview== The Jenkins–Traub algorithm calculates all of the roots of a polynomial with complex coefficients. The algorithm starts by checking the polynomial for the occurrence of very large or very small roots. If necessary, the coefficients are rescaled by a rescaling of the variable. In the algorithm proper, roots are found one by one and generally in increasing size. After each root is found, the polynomial is deflated by dividing off the corresponding linear factor. Indeed, the factorization of the polynomial into the linear factor and the remaining deflated polynomial is already a result of the root-finding procedure. The root-finding procedure has three stages that correspond to different variants of the inverse power iteration. See Jenkins and Traub.〔Jenkins, M. A. and Traub, J. F. (1970), (A Three-Stage Variables-Shift Iteration for Polynomial Zeros and Its Relation to Generalized Rayleigh Iteration ), Numer. Math. 14, 252–263.〕 A description can also be found in Ralston and Rabinowitz〔Ralston, A. and Rabinowitz, P. (1978), A First Course in Numerical Analysis, 2nd ed., McGraw-Hill, New York.〕 p. 383. The algorithm is similar in spirit to the two-stage algorithm studied by Traub.〔Traub, J. F. (1966), (A Class of Globally Convergent Iteration Functions for the Solution of Polynomial Equations ), Math. Comp., 20(93), 113–138.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Jenkins–Traub algorithm」の詳細全文を読む スポンサード リンク
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