The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.
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324
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The quadtree and related hierarchical data structures
– SAMET
- 1984
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267
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The Rapid Evaluation of Potential Fields in Particle Systems
– Greengard
- 1988
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143
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Parallel Computing: Theory and Practice
– Quinn
- 1993
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89
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Multiplication of multidigit numbers on automata
– Karatsuba, Ofman
- 1963
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79
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A Fully Parallel Algorithm for the Symmetric Eigenvalue Problem
– Sorensen
- 1987
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49
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A History of Mathematics
– Boyer
- 1991
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48
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Complexity of Bezout's Theorem I: Geometric Aspects
– Shub, Smale
- 1993
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47
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On the computational complexity and geometry of the first order theory of the reals
– Renegar
- 1992
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46
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Applied and
– Henrici
- 1974
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46
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On the e ciency of Algorithms of analysis
– Smale
- 1985
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44
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Newton’s method estimates from data at one point, in The Merging of Disciplines
– Smale
- 1986
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40
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Complexity of Bezout's Theorem II: Volumes and Probabilities in: Computational Algebraic Geometry
– Shub, Smale
- 1993
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40
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Complexity of Bezout's Theorem III: Condition Number and Packing
– Shub, Smale
- 1993
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39
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A Survey of Parallel Algorithms for Shared Memory Machines
– Karp, Ramachandran
- 1990
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38
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Solving systems of non-linear polynomial equations faster
– Canny, Kaltofen, et al.
- 1989
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37
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The Numerical Treatment of a Single Nonlinear Equation
– Householder
- 1970
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37
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Complexity of Bezout's theorem. IV. Probability of success; extensions
– Shub, Smale
- 1996
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34
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The fundamental theorem of algebra and complexity theory
– Smale
- 1981
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33
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Smale: Complexity of Bezout's theorem V: Polynomial time
– SHUB, SMALE
- 1994
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31
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On the worst-case arithmetic complexity of approximating zeros of polynomials
– Renegar
- 1987
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31
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Pseudozeros of polynomials and pseudospectra of companion matrices
– Toh, Trefethen
- 1994
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29
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A New O(n ) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem
– Dhillon
- 1997
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27
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The Pade table and its relation to certain algorithms of numerical analysis
– Gragg
- 1972
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27
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A bibliography on roots of polynomials
– McNamee
- 1993
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26
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On the Complexity of Sparse Elimination
– Emiris
- 1996
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23
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Elimination methods: an introduction
– Kapur, Lakshman
- 1992
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22
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Men of mathematics
– Bell
- 1937
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21
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A Principal Axis Transformation for NonHermitian Matrices
– Eckart, Young
- 1939
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20
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A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration
– Jenkins, Traub
- 1970
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20
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Optimal and nearly optimal algorithm for approximating complex polynomial zeros
– Pan
- 1996
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17
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The complexity of a scheme of functional elements realizing the multiplication of integers
– Toom
- 1963
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16
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Parallel complexity of tridiagonal symmetric eigenvalue problem
– Bini, Pan
- 1991
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16
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Simple algorithm for approximating all roots of a polynomial with real roots
– Ben-Or, Tiwari
- 1990
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15
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A Parallel Algorithm for the Nonsymmetric Eigenvalue Problem
– Dongarra, Sidani
- 1993
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14
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Optimal (up to Polylog Factors) Sequential and Parallel Algorithms for Approximating Complex Polynomial Zeros
– Pan
- 1995
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14
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On a New Method of Analysis and its Applications
– Turán
- 1984
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13
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Practical improvement of the divide-and-conquer eigenvalue algorithms
– Bini, Pan
- 1992
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13
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Sequential and parallel complexity of approximate evaluation of polynomial zeros
– Pan
- 1987
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13
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Randbemerkungen zu Hauptproblemen der Mathematik. II. Fundamentalsatz der Algebra und Grundlagen der
– Weyl
- 1924
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12
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Iteration methods for finding all zeros of a polynomial simultaneously
– Aberth
- 1973
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12
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A fast parallel algorithm for determining all roots of a polynomial with real roots
– Ben-Or, Feig, et al.
- 1988
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12
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A numerical method for locating the zeros of an analytic functions
– Delves, Lyness
- 1967
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11
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Algorithm 419: Zeros of a complex polynomial
– Jenkins, Traub
- 1972
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11
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On the Worst Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials
– Renegar
- 1989
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10
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On the complexity of polynomial zeros
– Bini, Gemignani
- 1992
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10
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Remark on algorithms to find roots of polynomials
– Goedecker
- 1994
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10
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Faster solution of the key equation for decoding BCH errorcorecting codes
– Pan
- 1997
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9
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Precision Polynomial Root Isolation is
– Neff
- 1994
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9
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and Efficient Parallel Evaluation of the Zeros of a Polynomial Having Only
– Pan
- 1989
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8
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On approximate zeros and rootfinding algorithms for a complex polynomial
– Kim
- 1988
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