## Univariate polynomials: Nearly optimal algorithms for numerical factorization and rootfinding (2001)

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Venue: | J. Symbolic Computation |

Citations: | 37 - 11 self |

### BibTeX

@ARTICLE{Pan01univariatepolynomials:,

author = {Victor Y. Pan},

title = {Univariate polynomials: Nearly optimal algorithms for numerical factorization and rootfinding},

journal = {J. Symbolic Computation},

year = {2001},

volume = {33},

pages = {2002}

}

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### Abstract

To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the n-th degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to recursive computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new rootfinder incorporates the earlier techniques of Schönhage and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the complexity of both complete factorization and rootfinding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, where the roots can be ill-conditioned, forming clusters. (The worst case bounds are supported by our previous algorithms as well.) All our algorithms allow processor efficient acceleration to achieve solution in polylogarithmic parallel time. Keywords Padé approximation, Graeffe’s lifting, univariate polynomials, rootfinding, numerical polynomial factorization, geometry of polynomial zeros, computational complexity

### Citations

1620 |
Time Series Analysis Forecasting and Control
- Box, Jenkins
- 1976
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Citation Context ...dependent interest due to their applications to time series analysis, Weiner filtering, noise variance estimation, covariance matrix computation, and the study of multi-channel systems (Wilson, 1969; =-=Box and Jenkins, 1976-=-; Barnett, 1983; Demeure and Mullis, 1989, 1990; Van Dooren, 1994). In the present work, we simplify the construction in Pan (1995a, 1996) where a higher precision of computing of the order of at leas... |

421 | The quadtree and related hierarchical data structures
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Citation Context ...973; Henrici, 1974); furthermore, Weyl’s quadtree root-finder alone (Weyl, 1924) has made an impact on computational geometry, image processing, template matching, and the n-body particle simulation (=-=Samet, 1984-=-; Greengard, 1988; Senoussi, 1994). Presently, polynomial root-finding is still a major research topic with highly important applications to computer algebra, in particular to the solution of polynomi... |

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Complexity and real computation
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Citation Context ...nomial root-finding is still a major research topic with highly important applications to computer algebra, in particular to the solution of polynomial systems of equations (Kapur and Lakshman, 1992; =-=Blum et al., 1997-=-; Pan, 1997; Mourrain and Pan, 1998, 2000) (cf. also Pan, 1998, 2001c; Pan and Chen, 1999 on the applications to the computation of approximate polynomial gcds and the algebraic eigenproblem). It was ... |

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The Rapid Evaluation of Potential Fields in Particle Systems
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Citation Context ... 1974); furthermore, Weyl’s quadtree root-finder alone (Weyl, 1924) has made an impact on computational geometry, image processing, template matching, and the n-body particle simulation (Samet, 1984; =-=Greengard, 1988-=-; Senoussi, 1994). Presently, polynomial root-finding is still a major research topic with highly important applications to computer algebra, in particular to the solution of polynomial systems of equ... |

164 | Multiplication of multidigit numbers on automata - Karatsuba, Ofman - 1963 |

158 |
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- 1966
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Citation Context ...er 4, No.163; Kirrinnis, 1992, Proof of Lemma 4.6) The first equation of Proposition 1.4.4 immediately follows from16 V. Y. Pan the Cauchy theorem on complex contour integrals of analytic functions (=-=Ahlfors, 1979-=-). Cauchy’s integral formula (Ahlfors, 1979) implies the second equation of Proposition 1.4.4 for every x equal to a zero of F (x). If F (x) has k distinct zeros, then the second equation is extended ... |

130 | Fast solution of Toeplitz systems of equations and computation of Padé approximations - Brent, Gustavson, et al. - 1980 |

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Aufgaben und Lehrsatze aus der Analysis
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- 1970
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Citation Context ...= {x, |x| ≤ 1}. Then, for any complex x, we have ∫ F (t) − F (x) R(t) dt = 0, t − x |x|=1 and if F (x) = 0 for |x| = 1, then f(x) = 1 2π √ −1 ∫ |x|=1 f(t) F (t) F (x) − F (t) · dt. x − t Proof. (Cf. =-=Pólya and Szegö, 1925-=-, III, Chapter 4, No.163; Kirrinnis, 1992, Proof of Lemma 4.6) The first equation of Proposition 1.4.4 immediately follows from16 V. Y. Pan the Cauchy theorem on complex contour integrals of analytic... |

96 | Merzbach, A History of Mathematics - Boyer, C - 1991 |

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Citation Context ... is still a major research topic with highly important applications to computer algebra, in particular to the solution of polynomial systems of equations (Kapur and Lakshman, 1992; Blum et al., 1997; =-=Pan, 1997-=-; Mourrain and Pan, 1998, 2000) (cf. also Pan, 1998, 2001c; Pan and Chen, 1999 on the applications to the computation of approximate polynomial gcds and the algebraic eigenproblem). It was already und... |

68 | The fundamental theorem of algebra in terms of computational complexity - Schönhage - 1982 |

67 | On the efficiency of algorithms of analysis - Smale - 1985 |

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Stability of methods for solving Toeplitz systems of equations
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- 1985
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Citation Context ...tmead/Anderson (cf. Bini and Pan, 1994, Chapter 2, Theorem 13.1; Pan, 2001a, Chapter 5 or Pan and Wang, 2002). The symmetrization ensures positive definiteness and therefore weak numerical stability (=-=Bunch, 1985-=-). O(n log 3 n) ops are sufficient in the u steps of Stage 3. Summarizing, we arrive at the arithmetic cost estimates of Theorem 1.1.3. We perform all computations by Algorithm 1.2.1 with the precisio... |

58 | The fundamental theorem of algebra and complexity theory - Smale - 1981 |

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Polynomial and Linear Control Systems
- Barnett
- 1983
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Citation Context ...to their applications to time series analysis, Weiner filtering, noise variance estimation, covariance matrix computation, and the study of multi-channel systems (Wilson, 1969; Box and Jenkins, 1976; =-=Barnett, 1983-=-; Demeure and Mullis, 1989, 1990; Van Dooren, 1994). In the present work, we simplify the construction in Pan (1995a, 1996) where a higher precision of computing of the order of at least bn bits is ma... |

52 |
The Numerical Treatment of a Single Nonlinear Equation
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- 1970
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Citation Context ...lished by Elsevier Science Ltd.2 V. Y. Pan influenced by the polynomial root-finding problem in the 19th and 20th centuries included meromorphic functions, algebraic curves, and structured matrices (=-=Householder, 1970-=-; Gauss, 1973; Henrici, 1974); furthermore, Weyl’s quadtree root-finder alone (Weyl, 1924) has made an impact on computational geometry, image processing, template matching, and the n-body particle si... |

48 | Multivariate Polynomials, Duality and Structured Matrices - Mourrain, Pan - 2000 |

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An inequality about factors of polynomials
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- 1974
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Citation Context ... 1.3.4 in the case where assumption (1.3.6) is replaced by the inequality We need some auxiliary estimates. deg f < k. (1.4.1)Univariate Polynomials: Nearly Optimal Algorithms 15 Proposition 1.4.1. (=-=Mignotte, 1974-=-) If p = p(x) = ∏ l i=1 fi, deg p ≤ n, and all fi are polynomials, then ∏ l i=1 |fi| ≤ 2 n max |x=1| |p(x)| ≤ 2 n |p|2. The next two results extend the ones of Schönhage (1982b). Proposition 1.4.2. Fo... |

44 | Men of mathematics - Bell - 1937 |

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- 1987
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Citation Context ...ected complex point. These techniques have low arithmetic and Boolean cost and are customary for reducing the study to the canonical cases, say where all roots lie in the unit disc {x : |x| ≤ 1} (see =-=Renegar, 1987-=-; Pan, 1996; Kirrinnis, 1998). On the other hand, using these techniques requires precision of computation of the order of n or n log n bits, which creates an implementation problem for larger n. Alth... |

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- 2000
(Show Context)
Citation Context ...ich use no splitting (Pan, 1987, 2000). Our algorithms are quite involved, and their implementation would require a nontrivial work, incorporating numerous known implementation techniques and tricks (=-=Bini and Fiorentino, 2000-=-; Fortune, 2001; Bini and Pan, to appear). We do not touch this vast domain here and just briefly comment on the precision of computing. Our algorithms involve the shifts of the variable (or equivalen... |

40 |
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Citation Context ...ound (1.3.9) holds for a sufficiently large constant C ∗ . Due to Corollary 1.5.1, it remains to prove (1.4.1) under (1.3.8) in order to complete the proof of Theorem 1.3.4. By the Frobenius theorem (=-=Gragg, 1972-=-, Theorem 3.1), there exists a unique rational function F/G satisfying (1.3.4) for any given polynomial M and any pair of integers k and n such that 0 ≤ k ≤ n, deg F ≤ k, deg G ≤ n − k. Assuming furth... |

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- 1993
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Citation Context ... raised the question of devising optimal numerical polynomial root-finders, which would use smaller computational time, and proposed some effective algorithms. In the vast literature on root-finding (=-=McNamee, 1993-=-, 1997), including thousands of publications, not many items are devoted to this important problem, however see Bini and Pan (to appear), Kim and Sutherland (1994), Kirrinnis (1998), Neff and Reif (19... |

39 |
Structured Matrices and Polynomials: Unified Superfast Algorithms
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- 2001
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Citation Context .... To solve the u Toeplitz linear systems (where u = O(log n)), we first symmetrize them and then apply the MBA algorithm of Morf and Bitmead/Anderson (cf. Bini and Pan, 1994, Chapter 2, Theorem 13.1; =-=Pan, 2001-=-a, Chapter 5 or Pan and Wang, 2002). The symmetrization ensures positive definiteness and therefore weak numerical stability (Bunch, 1985). O(n log 3 n) ops are sufficient in the u steps of Stage 3. S... |

31 |
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(Show Context)
Citation Context ...5a, 1996) where a higher precision of computing of the order of at least bn bits is maintained throughout. This is indeed required for approximating the ill-conditioned zeros of p(x) (see Fact 1.1 in =-=Pan, 1996-=-) and its higher order derivatives p (l) (x) unless n−l = o(n). Now we observe that the factorization of p(x) and p (l) (x) does not require such a high computational precision (theUnivariate Polynom... |

25 |
Factorization of the covariance generating function of a pure moving average process
- Wilson
- 1969
(Show Context)
Citation Context ...lems are of independent interest due to their applications to time series analysis, Weiner filtering, noise variance estimation, covariance matrix computation, and the study of multi-channel systems (=-=Wilson, 1969-=-; Box and Jenkins, 1976; Barnett, 1983; Demeure and Mullis, 1989, 1990; Van Dooren, 1994). In the present work, we simplify the construction in Pan (1995a, 1996) where a higher precision of computing ... |

24 | An Efficient Algorithm for the Complex Roots Problem
- Neff, Reif
- 1996
(Show Context)
Citation Context ...umption of the isolation (by reversing Graeffe’s lifting process with using the Padé approximation) (Pan, 1995a, 1996) and to achieving balanced splitting (Gel’fond, 1958; Coppersmith and Neff, 1994; =-=Neff and Reif, 1994-=-; Pan, 1995a, 1996). The techniques in the papers Kirrinnis (1998), Neff and Reif (1994) and Schönhage (1982b) are more important and should last longer than the computational complexity estimates. Ki... |

21 |
Optimal (up to Polylog Factors) Sequential and Parallel Algorithms for Approximating Complex Polynomial Zeros
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- 1995
(Show Context)
Citation Context ...Boolean cost estimates. Further improvements by the order of magnitude were due to relaxing the assumption of the isolation (by reversing Graeffe’s lifting process with using the Padé approximation) (=-=Pan, 1995-=-a, 1996) and to achieving balanced splitting (Gel’fond, 1958; Coppersmith and Neff, 1994; Neff and Reif, 1994; Pan, 1995a, 1996). The techniques in the papers Kirrinnis (1998), Neff and Reif (1994) an... |

20 |
Randbemerkungen zu Hauptproblemen der Mathematik, II, Fundamentalsatz der Algebra and Grundlagen der
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Citation Context ... the 19th and 20th centuries included meromorphic functions, algebraic curves, and structured matrices (Householder, 1970; Gauss, 1973; Henrici, 1974); furthermore, Weyl’s quadtree root-finder alone (=-=Weyl, 1924-=-) has made an impact on computational geometry, image processing, template matching, and the n-body particle simulation (Samet, 1984; Greengard, 1988; Senoussi, 1994). Presently, polynomial root-findi... |

18 | Sequential and parallel complexity of approximate evaluation of polynomial zeros
- Pan
- 1996
(Show Context)
Citation Context ...imating a single root or a few roots of a polynomial, but in these cases the computational cost is slightly lower and the implementation is simpler in our distinct approaches, which use no splitting (=-=Pan, 1987-=-, 2000). Our algorithms are quite involved, and their implementation would require a nontrivial work, incorporating numerous known implementation techniques and tricks (Bini and Fiorentino, 2000; Fort... |

17 |
A numerical method for locating zeros of an analytic function
- Delves, Lyness
- 1967
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Citation Context ...nstants c > 0 and d. Splitting algorithms first compute a crude initial splitting and then refine it by means of Newton’s iterative process. They have been completely developed in several papers (see =-=Delves and Lyness, 1967-=-; Grau, 1971; Schönhage, 1982b; McNamee, 1993, 1997; Kirrinnis, 1998). Let us first state the basic splitting results of Schönhage and Kirrinnis and then our main splitting theorem. Theorem 1.1.1. (Ki... |

16 |
The complexity of the matrix eigenproblem
- Pan, Chen
- 1999
(Show Context)
Citation Context ...o computer algebra, in particular to the solution of polynomial systems of equations (Kapur and Lakshman, 1992; Blum et al., 1997; Pan, 1997; Mourrain and Pan, 1998, 2000) (cf. also Pan, 1998, 2001c; =-=Pan and Chen, 1999-=- on the applications to the computation of approximate polynomial gcds and the algebraic eigenproblem). It was already understood by the Greeks at about 500 B.C. that even the solution of the equation... |

16 |
Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients
- Schönhage
- 1982
(Show Context)
Citation Context ...factors. Apparently, our asymptotic bit-operation cost bounds can be improved by roughly the logarithmic factor if one applies fast integer arithmetic based on the binary segmentation techniques (cf. =-=Schönhage, 1982-=-a,b; Bini and Pan 1994, Section 3.3; Kirrinnis, 1998). Asymptotically, these techniques are slightly superior to the FFT-based arithmetic, on which we rely in our estimates. At the stage of splitting ... |

15 | Polynomial root finding using iterated eigenvalue computation
- Fortune
- 2001
(Show Context)
Citation Context ...1987, 2000). Our algorithms are quite involved, and their implementation would require a nontrivial work, incorporating numerous known implementation techniques and tricks (Bini and Fiorentino, 2000; =-=Fortune, 2001-=-; Bini and Pan, to appear). We do not touch this vast domain here and just briefly comment on the precision of computing. Our algorithms involve the shifts of the variable (or equivalently of the orig... |

14 | Approximating Complex Polynomial Zeros: Modified Quadtree (Weyl’s) Construction and Improved Newton’s Iteration
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(Show Context)
Citation Context ...e polynomial q(x) = p(x + X), obtained from p(x) when the origin is shifted into a complex point X. We use the following auxiliary result.24 V. Y. Pan Proposition 2.2.3. (Schönhage, 1982b) (Cf. also =-=Pan, 2000-=-) O(n log 2 n) ops performed with O(n)-bit precision are sufficient to approximate within the relative error bound c/n d (for any fixed pair of c > 0 and d ≥ 0) all root radii rj of a polynomial p(x),... |

14 |
Computation of approximate polynomial GCDs and an extension
- Pan
(Show Context)
Citation Context .... To solve the u Toeplitz linear systems (where u = O(log n)), we first symmetrize them and then apply the MBA algorithm of Morf and Bitmead/Anderson (cf. Bini and Pan, 1994, Chapter 2, Theorem 13.1; =-=Pan, 2001-=-a, Chapter 5 or Pan and Wang, 2002). The symmetrization ensures positive definiteness and therefore weak numerical stability (Bunch, 1985). O(n log 3 n) ops are sufficient in the u steps of Stage 3. S... |

13 |
Specified precision polynomial root isolation is NC
- Neff
- 1994
(Show Context)
Citation Context ...xing the assumption of the isolation (by reversing Graeffe’s lifting process with using the Padé approximation) (Pan, 1995a, 1996) and to achieving balanced splitting (Gel’fond, 1958; Coppersmith and =-=Neff, 1994-=-; Neff and Reif, 1994; Pan, 1995a, 1996). The techniques in the papers Kirrinnis (1998), Neff and Reif (1994) and Schönhage (1982b) are more important and should last longer than the computational com... |

12 | A 2000 Updated Supplementary Bibliography on Roots of Polynomials - McNamee |

11 | Numerical univariate polynomial GCD
- Emiris, Galligo, et al.
- 1996
(Show Context)
Citation Context ...d then in Neff and Reif (1996). In this case, however, each descending step (1.2.4) is replaced by a recursive Euclidean process, prone to severe problems of numerical stability (cf. Schönhage, 1985; =-=Emiris et al., 1996-=-, 1997) and to blowing up the precision and the Boolean cost of the computations. In Bini and Pan (to appear) we give some critical comments on an attempt to avoid this problem made in Neff and Reif (... |

11 |
The Euclid Algorithm and the Fast Computation of Cross-covariance and Autocovariance Sequences
- Demeure, Mullis
- 1989
(Show Context)
Citation Context ...ations to time series analysis, Weiner filtering, noise variance estimation, covariance matrix computation, and the study of multi-channel systems (Wilson, 1969; Box and Jenkins, 1976; Barnett, 1983; =-=Demeure and Mullis, 1989-=-, 1990; Van Dooren, 1994). In the present work, we simplify the construction in Pan (1995a, 1996) where a higher precision of computing of the order of at least bn bits is maintained throughout. This ... |

10 | Polynomial Root-finding Algorithms and Branched Covers - Kim, Sutherland - 1994 |

8 |
Polynomial Factorization and Partial Fraction Decomposition by Simultaneous Newton’s Iteration
- Kirrinnis
- 1998
(Show Context)
Citation Context ...st bounds can be improved by roughly the logarithmic factor if one applies fast integer arithmetic based on the binary segmentation techniques (cf. Schönhage, 1982a,b; Bini and Pan 1994, Section 3.3; =-=Kirrinnis, 1998-=-). Asymptotically, these techniques are slightly superior to the FFT-based arithmetic, on which we rely in our estimates. At the stage of splitting into two factors, an acceleration by the logarithmic... |

8 | Variations on computing reciprocals of power series - Schönhage |

7 |
Roots of a polynomial and its derivatives
- Coppersmith, Neff
- 1994
(Show Context)
Citation Context ...were due to relaxing the assumption of the isolation (by reversing Graeffe’s lifting process with using the Padé approximation) (Pan, 1995a, 1996) and to achieving balanced splitting (Gel’fond, 1958; =-=Coppersmith and Neff, 1994-=-; Neff and Reif, 1994; Pan, 1995a, 1996). The techniques in the papers Kirrinnis (1998), Neff and Reif (1994) and Schönhage (1982b) are more important and should last longer than the computational com... |

7 |
Deterministic improvement of complex polynomial factorization based on the properties of the associated resultant
- Pan
- 1995
(Show Context)
Citation Context ...Boolean cost estimates. Further improvements by the order of magnitude were due to relaxing the assumption of the isolation (by reversing Graeffe’s lifting process with using the Padé approximation) (=-=Pan, 1995-=-a, 1996) and to achieving balanced splitting (Gel’fond, 1958; Coppersmith and Neff, 1994; Neff and Reif, 1994; Pan, 1995a, 1996). The techniques in the papers Kirrinnis (1998), Neff and Reif (1994) an... |

7 | A Newton–Raphson method for moving-average spectral factorization using the Euclid algorithm - Demeure, Mullis - 1990 |

6 |
Approximate polynomial gcds, Padé approximation, polynomial zeros, and bipartite graphs
- Pan
- 1998
(Show Context)
Citation Context ...ch topic with highly important applications to computer algebra, in particular to the solution of polynomial systems of equations (Kapur and Lakshman, 1992; Blum et al., 1997; Pan, 1997; Mourrain and =-=Pan, 1998-=-, 2000) (cf. also Pan, 1998, 2001c; Pan and Chen, 1999 on the applications to the computation of approximate polynomial gcds and the algebraic eigenproblem). It was already understood by the Greeks at... |

6 |
Applied and Computational Complex Analysis 1
- Henrici
- 1974
(Show Context)
Citation Context ...2 V. Y. Pan influenced by the polynomial root-finding problem in the 19th and 20th centuries included meromorphic functions, algebraic curves, and structured matrices (Householder, 1970; Gauss, 1973; =-=Henrici, 1974-=-); furthermore, Weyl’s quadtree root-finder alone (Weyl, 1924) has made an impact on computational geometry, image processing, template matching, and the n-body particle simulation (Samet, 1984; Green... |

6 |
The Fundamental Theorem of Algebra
- Schönhage
- 1982
(Show Context)
Citation Context ...implies the upper bound κiδ on the error of approximation of root zi of p(x) where κi is the condition number of zi under the same norm ‖ · ‖1.) We begin with an auxiliary result. Proposition 1.7.1. (=-=Schönhage, 1982-=-b, Section 5) Let ∆k = |p − f1, . . . , fk| ≤ kɛ|p|/n, (1.7.2) ∆ = |f1 − fg| ≤ ɛk|f1|, (1.7.3) for some non-constant polynomials f1, . . . , fk, f and g and for ( k∏ ) ɛk ≤ ɛ|p|/ n |fi| . (1.7.4) Then... |