An efficient algorithm for the complex roots problem

Journal of Complexity

Volumn 12, Issue 2, 1996, Pages 81-115

  Neff, Andrew C ^a   Reif, John H ^b,c  

^a Stratasys Inc   (United States)

^b Duke University   (United States)

^c Carnegie Mellon University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030163624     PISSN: 0885064X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcom.1996.0008     Document Type: Article

References (44)

1. [AHU 74] AHO, A. V., HOPCROFT, J. E., AND ULLMAN, J. D. (1974), "The Design and Analysis of Computer Algorithms," Addison-Wesley, Reading, MA.
2. [BFKT 86] BEN-OR, M., FEIG, E., KOZEN, D., AND TIWARI, P. (1986), A Fast parallel algorithm for determining all roots of a polynomial with Real Roots, SIAM J. Comput.
3. [BT 90] BEN-OR, M., AND TIWARI, P. (1990), Simple algorithms for approximating all roots of a polynomial with real roots, J. Complexity 6, 417-442.
4. [B 92] BINI, D. (1992), Divide and conquer techniques for the polynomial root-finding problem, in "Proceedings of the 1st World (International) Congress of Nonlinear Analysts, Tampa, August 1992."
5. [BP 86] BINI, D., AND PAN, V. (1986), Polynomial division and its computational complexity, J. Complexity 2, 179-203.
6. [BA 80] BITMEAD, R. R., AND ANDERSON, D. O. (1980), Asymptotically fast solution of Toeplitz and related systems of linear equations, Linear Algebra Appl. 34, 103-116.
7. [BM 75] BORODIN, A., AND MUNRO, I. (1975), "The Computational Complexity of Algebraic and Numeric Problems," American Elsevier, New York.
8. [BGY 80] BRENT, R. P., GUSTAVSON, F. G., AND YUN, D. Y. Y. (1980), Fast solution of Toeplitz systems of equations and computation of Padé approximants, J. Algorithms 1, 259-295.
9. [BT 71] BROWN, W. S. AND TRAUB, J. F. (1971), On Euclid's algorithm and the theory of subresultants, J. Assoc. Comput. Mach. 18, 505-514.
10. [BGH 82] BORODIN, A., VON ZUR GATHEN, J., AND HOPCROFT, J. (1982), Fast parallel matrix and GCD computations, Inform. and Control 52, 241-256.
11. [CR 93] CHERIYAN, J., AND REIF, J. H. (1993), Parallel and output sensitive algorithms for combinatoral and linear algebra problems, in "Proc. of Symposium on Parallel Algorithms and Architectures '93."
12. [C 66] COLLINS, G. E. (1966), Polynomial remainder sequences and determinants, Amer. Math. Monthly 73, 708-712.
13. [CL 82] COLLINS, G. E., AND LOOS, R. (1982), Real zeros of polynomials, in "Computing (1982)," Springer-Verlag, New York/Berlin.
14. [CN 94] COPPERSMITH D., AND NEFF, C. A. (1994), Roots of a polynomial and its derivatives, in "Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '94), Proceedings," pp. 271-279.
15. [E 87] EDELSBRUNNER, H. (1987), "Algorithms in Computational Geometry," Springer-Verlag, New York/Berlin.
16. [GM 77] GAL, S., AND MIRANKER, W. (1977), Optimal sequential and parallel search for finding a root, J. Combinatoral Theory 23.
17. [GH 72] GARGANTINI, I., AND HENRICI, P. Circular arithmetic and the determination of polynomial zeros, Numes. Math. 18, 305-320.
18. [H 74] HENRICI, P. (1974), "Applied and Computational Complex Analysis," Vols. 1-3,Wiley, New York.
19. [H 70] HOUSEHOLDER, A. S. (1970), "The Numerical Treatment of a Single Nonlinear Equation," McGraw-Hill, New York.
20. [J 92] JÁJÁ, J. (1992), "An Introduction to Parallel Algorithms," Addison-Wesley, Reading, MA.
21. [JT 70] JENKINS, M. A., AND TRAUB, J. F. (1970), A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration, Numer. Math. 14, 252-263.
22. [Mar 66] MARDEN, M. (1966), "Geometry of Polynomials," Amer. Math. Soc., Providence, RI.
23. [M 92] MIGNOTTE, M. (1992), "Mathematics for Computer Algebra," Springer-Verlag, New York/Berlin.
24. [N 90] NEFF, C. A. (1990), Specified precision polynomial root isolation is in NC, in "Proc. 24th Annual IEEE Symposium on Foundations of Computer Science," pp. 138-145.
25. [N 94] NEFF, C. A. (1994), Specified precision polynomial root isolation is in NC, J. Comput. System Sci. 48, 429-463.
26. 1+ε log b) algorithm for the complex roots problem, in "Proc. 35th Annual IEEE FOCS, Santa Fe, NM.
27. [Pn 87] PAN, V. Y. (1987), Algebraic complexity of computing polynomial zeros, Comput. Math. Appl. 14, 285-304.
28. [P 87] PAN, V. Y., (1987), Sequential and parallel complexity of approximate evaluation of polynomial zeros, Comput. Math. Appl. 14, 591-622.
29. [P 89] PAN, V. Y., (1989), Fast and efficient parallel evaluation of the zeros of a polynomial having only real zeros, Comput. Math. Appl. 17, 1475-1480.
30. [P 94] PAN, V. Y. (1994), Deterministic improvement of complex polynomial factorization based on the properties of the associated resultant, unpublished manuscript.
31. [P 95] PAN, V. Y. (1995), Improved algorithms for approximating complex polynomial zeros, in "Proc. of the 27th ACM Symposium on Theory of Computing (STOC), Las Vegas, NV, May 1995.
32. [PS 85] PREPARATA, F. P. AND SHAMOS, M. I. "Computational Geometry: An Introduction," Springer-Verlag, New York/Berlin, 1985.
33. [R 93a] REIF, J. H. (Ed.), (1993), "Synthesis of Parallel Algorithms," Morgan Kaufmann, Los Altos, CA.
34. 3 n) algorithm for the real root problem, in "34th Annual IEEE Conference on Foundations of Computer Science (FOCS '93), Palo Alto, CA, Proceedings.
35. [R 95] REIF, J. H. (1995), Work efficient parallel solution of Toeplitz systems and polynomial GCD, in "27th Annual ACM Symposium of Theory of Computing (STOC95), Las Vegas, NV, May 1995."
36. [R 85] RENEGAR, J. (1985), On the Cost of approximating all roots of a complex polynomial, Math. Programming 32, 319-336.
37. [R 87] RENEGAR, J. (1987), On the worst-case arithmetic complexity of approximating zeros of polynomials, in "Proc. 2nd Symposium on the Complexity of Approx. Solved Problems, 1987."
38. [S 82] SCHöNHAGE, A. (1982), "The Fundamental Theorem of Algebra in Terms of Computational Complexity," unpublished manuscript, Univ. of Tübingen, Tü̈bingen, Germany.
39. [SS 85] SHUB, M., AND SMALE, S. (1985), Computational Complexity: On the geometry of polynomials and a theory of cost, Part I, Ann. Sci. Ecole Norm. Sup. 4, 107-142.
40. [SS 86] SHUB, M., AND SMALE, S. (1986), Computational Complexity: On the geometry of polynomials and a theory of cost, II, SIAM J. Comput. 15, 145-161.
41. [S 81] SMALE, S. (1981), The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. 4, 1-36.
42. [T 68] TURAN, P. (1968), On the approximate solution of algebraic equations, Comm. Math. Phys. Class Hungar. Acad. 18, 223-236.
43. [T 75] TURAN, P. (1975), Power sum method and approximative solution of algebraic equations, Math. Comput. 29, 311-318.
44. [T 84] TURAN, P. (1984), "On a New Method of Analysis and Its Applications," Wiley, New York.