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Бини, Дарио Андреа; Fiorentino, Giuseppe

doi:10.1023/a:1019199917103

Cited by 153 publications

(52 citation statements)

References 33 publications

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“…In practice, we have used a modified version of the program newgrqd.f described in reference [158] to obtain the differential approximants. Our program uses Mathematica to obtain the polynomials Q j exactly (i.e., with exact rational coefficients), and then MPSolve [159,160] to compute the N K zeros of Q K to arbitrarily high precision (100 digits in our case). We have computed all the differential approximants of first and second order (i.e., K = 1, 2) satisfying |N i − N j | ≤ 1 and using at least 36 coefficients of the corresponding series.…”

Section: A Series Analysismentioning

confidence: 99%

“…We wish to thank Dario Bini for supplying us the MPSolve 2.1.1 package [159,160] for computing roots of polynomials, and for many discussions about its use; Ian Enting, Tony Guttmann and Iwan Jensen for correspondence concerning the finite-lattice method; Yaoban Chan and Andrew Rechnitzer for providing us their C++ code to compute differential approximants, and for assistance in its use; and Hubert Saleur and Robert Shrock for many helpful conversations throughout the course of this work. We also wish to thank two referees for thoughtful comments on the first version of this paper.…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Spanning Forests and the q-State Potts Model in the Limit q →0

Jacobsen

Salas²,

Sokal³

2005

J Stat Phys

108

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We study the q-state Potts model with nearest-neighbor coupling v = e βJ − 1 in the limit q, v → 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q, v) = (0, 0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 ≤ L ≤ 10, as well as the limiting curves B of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w = w 0 , where w 0 = −1/4 (resp. w 0 = −0.1753 ± 0.0002) for the square (resp. triangular) lattice. For w > w 0 we find a non-critical disordered phase that is compatible with the predicted asymptotic freedom as w → +∞. For w < w 0 our results are compatible with a massless Berker-Kadanoff phase with central charge c = −2 and leading thermal scaling dimension x T,1 = 2 (marginally irrelevant operator). At w = w 0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w 0 , while the correlation length diverges as w ↓ w 0 (and is infinite at w = w 0 ). The critical behavior at w = w 0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the central charge is c = −1, the leading thermal scaling dimension is x T,1 = 0, and the critical exponents are ν = 1/d = 1/2 and α = 1.

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Section: A Series Analysismentioning

confidence: 99%

Section: Acknowledgmentsmentioning

confidence: 99%

Spanning Forests and the q-State Potts Model in the Limit q →0

Jacobsen

Salas²,

Sokal³

2005

J Stat Phys

108

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“…By applying the following theorem (cf. [34], [19], [61,Remark 6.4], [12,13,and 14], [15, Sections 3.1 and 3.2]) for i = 1 we obtain the upper bound p(c) p (c) d on the smallest distance r d (c, p) from a complex point c to a closest root of a black box polynomial p. Theorem 6. For a polynomial p of (1) and a complex point c, write…”

Section: Root Radii Estimationmentioning

confidence: 99%

“…Its formal support is weaker than by Theorem 2, but since 2000 it has been serving quite efficiently in initializing Ehrlich's iterations of MPSolve (cf. [12], [15]). 10 9 Throughout the paper we count m times a root of multiplicity m. 10 As we point out in Sections 3.2 and 6 (in its beginning), we can substantially simplify both real and complex root-finding if we narrow the search for the roots to the sets of suspect segments and suspect annuli, respectively.…”

Section: Root Radii Estimationmentioning

confidence: 99%

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Old and New Nearly Optimal Polynomial Root-Finders

Pan

2019

Computer Algebra in Scientific Computing

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Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By incorporating some known and novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.

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The application of the Ehrlich‐Aberth method to structured polynomial eigenvalue problems

Noferini

2011

Proc Appl Math and Mech

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An algorithm for the application of the Ehrlich-Aberth method to polynomial eigenvalue problems (PEPs) is proposed. The computational complexity of the algorithm is only quadratic with respect to the degree of the polynomial, where customary matrix methods have cubic complexity. For the case of some structured PEPs a strategy is given in order to exploit the structure and obtain the induced coupling in the spectrum. Numerical experiments are provided for the particular case of even/odd PEPs. This communication is based on joint work with Dario A. Bini and Luca Gemignani.

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Untitled

Cited by 153 publications

References 33 publications

Spanning Forests and the q-State Potts Model in the Limit q →0

Spanning Forests and the q-State Potts Model in the Limit q →0

Old and New Nearly Optimal Polynomial Root-Finders

The application of the Ehrlich‐Aberth method to structured polynomial eigenvalue problems

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