Victor Y. Pan
ABSTRACT
In this paper we seek all real roots of a polynomial with real coefficients and real and nonreal roots. Somewhat paradoxically, one of the most effective solutions is by approximating these real roots semi-numerically together with all nonreal roots. Alternative methods are symbolic, based on Descartes' rule of signs (which can be combined with the continued fraction approximation algorithm) or the Sturm (or Sturm-Habicht) sequences. We combine various old and new techniques to devise semi-numerical algorithms that are effective where the real roots do not lie near the nonreal ones.
- A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974. Google Scholar
Digital Library
- R. P. Brent, Algorithms for Minimization Without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.Google Scholar
- D. A. Bini, G. Fiorentino, Design, Analysis, and Implementation of a Multiprecision Polynomial Rootfinder, Numerical Algorithms, 23, 127--173, 2000.Google ScholarCross Ref
- D. A. Bini, L. Gemignani, B. Meini, Computations with Infinite Toeplitz Matrices and Polynomials, Linear Algebra and Its Applications, 343--344, 21--61, 2002.Google Scholar
- D. A. Bini, L. Gemignani, V. Y. Pan, Inverse Power and Durand/Kerner Iteration for Univariate Polynomial Root-finding, Computers and Mathematics (with Applications), 47, 2/3, 447--459, 2004. (Also Technical Report TR 2002 020, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2002.)Google Scholar
- D. A. Bini, L. Gemignani, V. Y. Pan, Fast and Stable QR Eigenvalue Algorithms for Generalized Companion Matrices and Secular Equation, Numerische Math., 3, 373--408, 2005. (Also Technical Report 1470, Department of Math, University of Pisa, Pisa, Italy, July 2003.) Google ScholarDigital Library
- D. A. Bini, L. Gemignani, V. Y. Pan, Improved Initialization of the Accelerated and Robust QR-like Polynomial Root-finding, Electronic Transactions on Numerical Analysis, 17, 195--205, 2004.Google Scholar
- M. Ben-Or, P. Tiwari, Simple Algorithm for Approximating All Roots of a Polynomial with Real Roots, J. of Complexity, 6, 417--442, 1990. Google ScholarDigital Library
- D. Bini, V. Y. Pan, Computing Matrix Eigenvalues and Polynomial Zeros Where the Output Is Real, SIAM Journal on Computing, 27, 4, 1099--1115, 1998. (Proceedings version in Proc. ACM/SIAM Symposium on Discrete Algorithms (SODA'91), ACM Press, New York, and SIAM Publications, Philadelphia, January 1991.) Google ScholarDigital Library
- C. Carstensen, Linear Construction of Companion Matrices, Linear Algebra and Its Applications, 149, 191--214, 1991.Google ScholarCross Ref
- G. Collins, A. Akritas, Polynomial Real Root Isolation Using Descartes' Rule of Signs, SYMSAC '76, 272--275, ACM Press, New York, 1976. Google ScholarDigital Library
- F. Cazals, J.-C. Faugère, M. Pouget, F. Rouillier, The Implicit Structure of Ridges of a Smooth Parametric Surface, Comput. Aided Geom. Des., 23, 7, 582--598, 2006. Also Technical Report 5608, INRIA, France, 2005. Isolation by Differentiation, Proc. of Symposium on Symbolic and Algebraic Computation (SYMSAC 1976), 15--25, ACM Press, New York, 1976. Google ScholarDigital Library
- G. Collins, R. Loos, Real Zeros of Polynomials, in Computer Algebra: Symbolic and Algebraic Computation, (B. Buchberger, G. Collins, and R. Loos, editors), 83--94, Springer-Verlag, Vienna, 1982 (2nd edition). Google ScholarDigital Library
- Q. Du, M. Jin, T. Y. Li, Z. Zeng, Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems, SIAM J. Sci. Computing, 17, 6, 1347--1368, 1996. Google ScholarDigital Library
- Q. Du, M. Jin, T. Y. Li, Z. Zeng, The Quasi-Laguerre Iteration, Math. of Computation, 66, 217, 345--361, 1997. Google ScholarDigital Library
- Z. Du, V. Sharma, C. K. Yap, Amortized Bound for Root Isolation via Sturm Sequences, Chapter 8, pages 105--121 in Symbolic Numeric Computing (D. Wang and L. Zhi, editors), Birkhäuser, Basel/Boston, 2007.Google Scholar
- I. Z. Emiris, A. Kakargias, M. Teillaud, E. Tsigaridas, S. Pion, Towards an Open Curved Kernel, Proceedings of Annual ACM Symp. on Computational Geometry, 438--446, ACM Press, New York, 2004. Google ScholarDigital Library
- I. Z. Emiris, B. Mourrain, E. P. Tsigaridas, Real Algebraic Numbers: Compexity Analysis and Experimentation, in Reliable Implementations of Real Number Algorithms: Theory and Practice, P. Hertling, C. Hoffmann, W. Luther, N. Revol editors, LNCS, Springer (to appear), also available in www.inria.fr/rrrt/rr-5897.html, 2007. Google ScholarDigital Library
- S. Fortune, An Iterated Eigenvalue Algorithm for Approximating Roots of Univariate Polynomials, J. of Symbolic Computation, 33, 5, 627--646, 2002. (Proc. version in Proc. Intern. Symp. on Symbolic and Algebraic Computation (ISSAC'01), 121--128, ACM Press, New York, 2001.) Google ScholarDigital Library
- M. Fiedler, Expressing a Polynomial As the Characteristic Polynomial of a Symmetric Matrix, Linear Algebra and Its Applications, 141, 265--270, 1990.Google ScholarCross Ref
- L. V. Foster, Generalization of Laguerre's Method: Higher Order Methods, SIAM J. on Numerical Analysis, 18, 6, 1004--1018, 1981.Google ScholarCross Ref
- G. H. Golub, C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 1996 (third addition).Google Scholar
- P. Henrici, Applied and Computational Complex Analysis, 1, John Wiley, New York, 1974.Google Scholar
- H. Hong, Bounds for Absolute Positiveness of Multivariate Polynomials, Journal of Symbolic Computation, 25, 5, 571--585, 1998. Google ScholarDigital Library
- E. Hansen, M. Patrick, J. Rusnack, Some Modification of Laguerre's Method, BIT, 17, 409--417, 1977.Google ScholarCross Ref
- J. R. Johnson, Algorithms for Polynomial Real Root Isolation, Quantifier Elimination and Cylindrical Algebraic Decomposition, (B. Caviness and J. R. Johnson, editors), 269--299, Springer, 1998.Google Scholar
- W. Krandick, Isolierung reeller Nullstellen von Polynomen, In Wissenschaftliches Rechnen (J. Herzberger, editor), 105--154, Akademie Verlag, Berlin, 1995.Google Scholar
- P. Kirrinnis, Polynomial Factorization and Partial Fraction Decomposition by Simultaneous Newton's Iteration, J. of Complexity, 14, 378--444, 1998. Google ScholarDigital Library
- B. Mourrain, J. Técourt, M. Teillaud, On the Computation of an Arrangement of Quadrics in 3d, Comput. Geom. Theory Appl., 30, 2, 145--164, 2005. Google ScholarDigital Library
- J. M. McNamee, Bibliography on Roots of Polynomials, J. Computational and Applied Mathematics, 47, 391--394, 1993. Google ScholarDigital Library
- J. M. McNamee, A Supplementary Bibliography on Roots of Polynomials, J. Computational and Applied Mathematics, 78, 1, 1997.Google Scholar
- J. M. McNamee, An Updated Supplementary Bibliography on Roots of Polynomials, J. Computational and Applied Mathematics, 110, 305--306, 1999. Google ScholarDigital Library
- J. M. McNamee, A 2000 Updated Supplementary Bibliography on Roots of Polynomials, J. Computational and Applied Mathematics, 142, 433--434, 2002. Google ScholarDigital Library
- F. Malek, R. Vaillancourt, Polynomial Zerofinding Iterative Matrix Algorithms, Computers and Math. with Applications, 29,1, 1--13, 1995.Google ScholarCross Ref
- F. Malek, R. Vaillancourt, A Composite Polynomial Zerofinding Matrix Algorithm, Computers and Math. with Applications, 30,2, 37--47, 1995.Google ScholarCross Ref
- V. Y. Pan, An Algebraic Approach to Approximate Evaluation of a Polynomial on a Set of Real Points, Advances in Computational Mathematics, 3, 41--58, 1995.Google ScholarCross Ref
- V. Y. Pan, Solving a Polynomial Equation: Some History and Recent Progress, SIAM Review, 39, 2, 187--220, 1997. Google ScholarDigital Library
- V. Y. Pan, Approximating Complex Polynomial Zeros: Modified Quadtree (Weyl's) Construction and Improved Newton's Iteration, J. of Complexity, 16, 1, 213--264, 2000. Google ScholarDigital Library
- V. Y. Pan, Structured Matrices and Polynomials: Unified Superfast Algorithms, Birkhäuser/Springer, Boston/New York, 2001. Google ScholarDigital Library
- V. Y. Pan, Univariate Polynomials: Nearly Optimal Algorithms for Factorization and Rootfinding, Journal of Symbolic Computation, 33, 5, 701--733, 2002. Google ScholarDigital Library
- B. Parlett, Laguerre's Method Applied to the Matrix Eigenvalue Problem, Math. of Computation, 18, 464--485, 1964.Google ScholarCross Ref
- B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, New Jersey, 1980, and SIAM, Philadelphia, 1998. Google ScholarDigital Library
- V. Y. Pan, M. Abu Tabanjeh, Z. Q. Chen, E. Landowne, A. Sadikou, New Transformations of Cauchy Matrices and Trummer's Problem, Computers & Mathematics (with Applications), 35, 12, 1--5, 1998.Google Scholar
- V. Y. Pan, D. Ivolgin, B. Murphy, R. E. Rosholt, Y. Tang, X. Wang, X. Yan, Root-finding with Eigen-solving, Chapter 14, pages 219--245 in Symbolic-Numeric Computation, (Dongming Wang and Li-Hong Zhi, editors), Birkhäuser, Basel/Boston, 2007.Google Scholar
- V. Y. Pan, M.-h. Kim, A. Sadikou, X. Huang, A. Zheng, On Isolation of Real and Nearly Real Zeros of a Univariate Polynomial and Its Splitting Into Factors, J. of Complexity, 12, 5720--594, 1996. Google ScholarDigital Library
- V. Y. Pan, A. Zheng, X. Huang, Y. Yu, Fast Multipoint Polynomial Evaluation and Interpolation via Computation with Structured Matrices, Annals of Numerical Math., 4, 483--510, 1997.Google Scholar
- J. Renegar, On the Worst-Case Arithmetic Complexity of Approximating Zeros of Polynomials, J. of Complexity, 3, 90--113, 1987. Google ScholarDigital Library
- F. Rouillier, Z. Zimmermann, Efficient Isolation of Polynomial's Real Roots, J. of Computational and Applied Mathematics, 162, 1, 33--50, 2004. Google ScholarDigital Library
- A. Schönhage, The Fundamental Theorem of Algebra in Terms of Computational Complexity, Mathematics Department, University of Tübingen, Germany, 1982.Google Scholar
- G. W. Stewart, Matrix Algorithms, Vol II: Eigensystems, SIAM, Philadelphia, 2001. Google ScholarDigital Library
- A. Terui and T. Sasaki, Durand-Kerner Method for the Real Roots, Japan J. Indust. Appl. Math., 19, 1, 19--38, 2002.Google ScholarCross Ref
- X. Zou, Analysis of the Quasi-Laguerre Method, Numerische Math., 82, 491--519, 1999.Google ScholarCross Ref
Index Terms
- Real root-finding
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