Design, analysis, and implementation of a multiprecision polynomial rootfinder

Numerical Algorithms

Volumn 23, Issue 2-3, 2000, Pages 127-173

  Bini, Dario Andrea ^a   Fiorentino, Giuseppe ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

Adaptive algorithms; Clustered roots; Inclusion theorems; Mathematical software; Polynomial roots; Root neighborhoods; Simultaneous iterations

Indexed keywords

EID: 0034396488     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1019199917103     Document Type: Article

References (49)

1. O. Aberth, Iteration methods for finding all zeros of a polynomial simultaneously, Math. Comp. 27(122) (1973) 339-344.
2. D.A. Adams, A stopping criterion for polynomial root finding, Comm. ACM 10 (1967) 655-658.
3. D.H. Bailey, A Portable High Performance Multiprecision Package, RNR Technical Report RNR-90-022, NAS Applied Rsearch Branch, NASA Ames Research center, Moffet Field, CA.
4. D. Bini and G. Fiorentino, Adaptive multiprecision algorithm for univariate polynomial zeros, in: Proc. of the 1st Internat. MATHEMATICA Sympos. (1995) pp. 53-60.
5. D. Bini and G. Fiorentino, A multiprecision implementation of a poly-algorithm for univariate polynomial zeros, in: Proc. of the POSSO Workshop on Software (1995).
6. D.A. Bini and G. Fiorentino, Mpsolve: Numerical computation of polynomial roots v2.0, FRISCO Technical Report (1998).
7. D.A. Bini and G. Fiorentino, On the parallel evaluation of a sparse polynomial at a point, Numer. Algorithms 20(4) (1999) 323-329.
8. D.A. Bini, Numerical computation of polynomial zeros by means of Aberth's method, Numer. Algorithms 13 (1996) 179-200.
9. D. Bini and V. Pan, Polynomial and Matrix Computations, Vol. 1: Fundamental Algorithms, Progress in Theoretical Computer Science (Birkhäuser, Boston, 1994).
10. W. Börsch-Soupan, A-posteriori error bounds for the zeros of polynomials, Numer. Math. 5 (1963) 380-398.
11. C. Carstensen, Inclusion of the roots of a polynomial based on Gerschgorin's theorem, Numer. Math. 59 (1991) 349-360.
12. D. Coppersmith and C.A. Neff, Roots of a polynomial and its derivatives, in: Proc. of the 5th Annual ACM-SIAM Sympos. on Discrete Algorithms, Arlington, VA, USA (January 23-25, 1994) (ACM, New York, 1994) pp. 271-279.
13. E. Durand, Solutions Numériques des Equations Algébriques, Tome 1: Equations du Type F(X) = 0; Racines d'un Polynôme (Masson, Paris, 1960).
14. L.W. Ehrlich, A modified Newton method for polynomials, Comm. ACM 10(2) (1967) 107-108.
15. L. Elsener, A remark on simultaneous inclusions of the zeros of a polynomial by Gerschgorin's theorem, Numer. Math. 21 (1973) 425-427.
16. W. Gautschi, Questions of numerical condition related to polynomials, in: Recent Advances in Numerical Analysis, eds. C. de Boor and G.H. Golub (Academic Press, New York, 1978) pp. 45-72.
17. G.H. Golub and C.F. Van Loan, Matrix Computations (Johns Hopkins Univ. Press, Baltimore, MD, 1989).
18. X. Gourdon, Algorithmique du théorème fondamental de l'algèbre, Rapport de Recherche No. 1852, INRIA (1993).
19. T. Granlund, GNU MP: The GNU multiple precision arithmetic library, edition 2.0 (April 1996).
20. H. Guggenheimer, Initial approximations in Durand-Kerner's root finding method, BIT 26 (1986) 537-539.
21. P. Henrici, Applied and Computational Complex Analysis, Vol. 1 (Wiley, New York, 1974).
22. M. Igarashi, A termination criterion for iterative methods used to find the zeros of polynomials, Math. Comp. 42 (1984) 165-171.
23. B. Jackson, A zero-free interval for chromatic polynomials of graphs, Combin. Probab. Comput. 2(3) (1993) 325-336.
24. 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 (1970) 252-263.
25. I.O. Kerner, Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen, Numer. Math. 8 (1966) 290-294.
26. P. Kirrinnis, Partial fraction decomposition in C(z) and simultaneous Newton iteration for factorization in C[z], J. Complexity 14 (1998) 378-444.
27. P. Kirrinnis, Fast computation of contour integrals of rational functions, J. Complexity, to appear.
28. W.S. Luk and T.C. Chen, Parallelizing a class of root finding methods by means of pseudo-deflation, Report, Department of Computer Science, The Chinese University of Hong Kong, Shatin, Hong Kong (1994).
29. K. Madsen and K. Reid, Fortran subroutines for finding polynomial zeros, Technical Report HL 75/1172(C.13), Computer Science and Systems Divisions, A.E.R.E. Harwell, Oxford (1975).
30. V.H. Maehly, Zur iterativen auflösing algebraischer gleichungen, Z. Angew. Math. Phys. 5 (1954) 260-263.
31. J.M. McNamee, A bibliography on roots of polynomials, J. Comput. Appl. Math. 47 (1993) 391-394.
32. J.M. McNamee, A comparison of methods for terminating polynomial iterations, J. Comput. Appl. Math. 21 (1998) 239-244.
33. R.G. Mosier, Root neighborhoods of a polynomial, Math. Comp. 47 (1986) 265-273.
34. A. Ostrowski, On a theorem by J.L. Walsh concerning the moduli of roots of algebraic equations, Bull. A.M.S. 47 (1941) 742-746.
35. V.Y. Pan, On approximating complex polynomial zeros: Modified quadtree (Weyl's) construction and improved Newton's iteration, in: 5th Annual ACM-SIAM Sympos. on Discrete Algorithms (ACM Press, New York, and SIAM, Philadelphia, PA, 1994).
36. V.Y. Pan, Sequential and parallel complexity of approximate evaluation of polynomial zeros, Comput. Math. Appl. 14(8) (1987) 591-622.
37. V.Y. Pan, Solving a polynomial equation: Some history and recent progress, SIAM Rev. 39(2) (1997) 187-220.
38. V.Y. Pan, Optimal and nearly optimal algorithms for approximating complex polynomial zeros, Comput. Math. Appl. 31 (1996) 97-138.
39. N. Psycharis, Analysis of the Lee-Yang zeros in lattice compact QED with scalars and fermions in 3D and in lattice noncompact QED in 4D, Ph.D. thesis, University of Glasgow (December 1998).
40. J. Renegar, On the computational complexity of approximating solutions for real algebraic formulas, SIAM J. Comput. 21(6) (1992) 1008-1025.
41. J. Renegar, On the worst case arithmetic complexity of approximating zeros of polynomials, J. Complexity 3 (1987) 90-113.
42. F. Rouillier, Solving zero-dimensional systems through the rational univariate representation, Technical Report, Loria (to appear in Journal of AAECC).
43. A. Schönhage, The fundamental theorem of algebra in terms of computational complexity, Technical Report, Mathematische Institut der Universität Tübingen (1982).
44. B.T. Smith, Error bounds for the zeros of a polynomial based upon Gerschgorin's theorem, J. ACM 17 (1970) 661-674.
45. A. Sokal, personal communication (1999).
46. H.J. Stetter, Analysis of zero cluster in multivariate polynomial systems, in: Proc. of ISSAC'96, ed. R. Caviness (1996) pp. 127-136.
47. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1980).
48. P. Tilli, Convergence conditions of some methods for the simultaneous computation of polynomial zeros, Calcolo 35(1) (1998) 3-15.
49. J.H. Wilkinson, Practical problems arising in the solution of polynomial equations, J. Inst. Math. Appl. 8 (1971) 16-35.