An improvement of Gargantini’s simultaneous inclusion method for polynomial roots by Schröder’s correction. (English) Zbl 0793.65037
A class of new interval methods for the simultaneous inclusion of polynomial roots is presented. The methods are based on Gargantini’s method which is accelerated using Schröder’s modification of Newton’s corrections. The \(R\)-orders of convergence of the new methods are proved to be greater than 3.5. Further more numerical experiments verify a higher computational efficiency than a priori existing algorithms.
Reviewer: C.Bendtsen (Lyngby)
MSC:
| 65H05 | Numerical computation of solutions to single equations |
| 65Y20 | Complexity and performance of numerical algorithms |
| 65G30 | Interval and finite arithmetic |
| 26C10 | Real polynomials: location of zeros |
| 12Y05 | Computational aspects of field theory and polynomials (MSC2010) |
Keywords:
\(R\)-orders of convergence; interval methods; simultaneous inclusion of polynomial roots; Newton’s corrections; numerical experiments; computational efficiencySoftware:
PASCAL-XSCReferences:
| [1] | Aberth, O., Iteration methods for finding all zeros of a polynomial simultaneously, Math. Comp., 27, 339-344 (1973) · Zbl 0282.65037 |
| [2] | High accuracy arithmetic-extended scientific computation, ACRITH-XSC Language Reference (1990), IBM Corporation, SC33-6462-00 |
| [3] | Alefeld, G.; Herzberger, J., Introduction to Interval Computation (1983), Academic Press: Academic Press New York · Zbl 0552.65041 |
| [4] | Börsch-Supan, W., A posteriori error bounds for the zeros of polynomials, Numer. Math., 5, 380-398 (1963) · Zbl 0133.08401 |
| [5] | Burmeister, W.; Schmidt, J. W., On the R-order of coupled sequences arising in single-step type methods, Numer. Math., 53, 653-661 (1988) · Zbl 0655.65074 |
| [6] | Ehrlich, L. W., A modified Newton method for polynomials, Comm. ACM, 10, 107-108 (1967) · Zbl 0148.39004 |
| [7] | Gargantini, I., Further applications of circular arithmetic: Schroeder-like algorithms with error bounds for finding zeros of polynomials, SIAM J. Numer. Math., 15, 497-510 (1978) · Zbl 0384.65020 |
| [8] | Gargantini, I.; Henrici, P., Circular arithmetic and the determination of polynomial zeros, Numer. Math., 18, 305-320 (1972) · Zbl 0228.65038 |
| [9] | Henrici, P., Applied and Computational Complex Analysis, Vol. I (1974), Wiley: Wiley New York · Zbl 0313.30001 |
| [10] | Klatte, R.; Kulisch, U.; Neaga, M.; Ratz, D.; Ullrich, C., PASCAL-XSC, Sprachbeschreibung mit Beispielen (1991), Springer: Springer Berlin · Zbl 0757.68023 |
| [11] | Maehly, V. H., Zur iterativen Auflösung algebraischer Gleichungen, Z. Angew. Math. Phys., 5, 260-263 (1954) · Zbl 0055.11102 |
| [12] | Nourein, A. W.M., An improvement on two iteration methods for simultaneous determination of the zeros of a polynomial, Internat. J. Comput. Math., 6, 241-252 (1977) · Zbl 0403.65013 |
| [13] | Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046 |
| [14] | Petković, M. S., Iterative Methods for Simultaneous Inclusion of Polynomial Zeros (1989), Springer: Springer Berlin · Zbl 0689.65028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
