Fast approximate computations with Cauchy matrices and polynomials

Pan, Victor Y.

doi:10.1090/mcom/3204

Cited by 17 publications

(19 citation statements)

References 37 publications

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“…Multipoint polynomial evaluation (MPE) and FMM. In context of polynomial root-finding we further recall that the algorithm of [64] and [66] evaluates a polynomial of degree d at O(d) points at the same asymptotic cost of performing FMM, that is, by using O(d log 2 (d)) arithmetic operations with the precision of order O(log(d)).…”

Section: E Fast Multipole Methods (Fmm) and Fast Multipoint Polynomialmentioning

confidence: 99%

“…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. Such applications of Theorem 2 have been explored in the papers [74] and [66].…”

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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Section: E Fast Multipole Methods (Fmm) and Fast Multipoint Polynomialmentioning

confidence: 99%

Section: Root Radii Estimationmentioning

confidence: 99%

Old and New Nearly Optimal Polynomial Root-Finders

Pan

2019

Computer Algebra in Scientific Computing

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“…We consider in this section the numerical solution of Cauchy‐like matrices. It is known that Cauchy‐like matrices are related to other types of structured matrices including Toeplitz matrices, Vandermonde matrices, Hankel matrices, and their variants . Consider the kernel

κ false(x, y false) = 1 false/ false(x - y false), x \neq y \in double-struckC

.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…It is known that Cauchy-like matrices are related to other types of structured matrices including Toeplitz matrices, Vandermonde matrices, Hankel matrices, and their variants. [55][56][57][58][59][60] Consider the kernel (x, ) = 1∕(x − ), x ≠ ∈ C. Let x i , y j (i, j = 1 ∶ n) be 2n pair-wise distinct points in C. The Cauchy matrix is then given by  = [ (x i , )] i, =1∶n , which is known to be invertible. 61 Given two matrices w, v ∈ C n×p , the (i, j) entry of a Cauchy-like matrix A associated with generators w and v is defined by 62…”

Section: Cauchy-like Matricesmentioning

confidence: 99%

SMASH: Structured matrix approximation by separation and hierarchy

Cai

Chow

Erlandson

et al. 2018

Numerical Linear Algebra App

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Summary This paper presents an efficient method to perform structured matrix approximation by separation and hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given the points in a domain, a tree structure is first constructed based on an adaptive partition of the computational domain to facilitate subsequent approximation procedures. In contrast to existing schemes based on either analytic or purely algebraic approximations, SMASH takes advantage of both approaches and greatly improves efficiency. The algorithm follows a bottom‐up traversal of the tree and is able to perform the operations associated with each node on the same level in parallel. A strong rank‐revealing factorization is applied to the initial analytic approximation in the separation regime so that a special structure is incorporated into the final nested bases. As a consequence, the storage is significantly reduced on one hand and a hierarchy of the original grid is constructed on the other hand. Due to this hierarchy, nested bases at upper levels can be computed in a similar way as the leaf level operations but on coarser grids. The main advantages of SMASH include its simplicity of implementation, its flexibility to construct various hierarchical rank structures, and a low storage cost. The efficiency and robustness of SMASH are demonstrated through various test problems arising from integral equations, structured matrices, etc.

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“…The Moenck-Borodin algorithm uses nearly linear arithmetic time, and [13] proved that this algorithm supports multipoint polynomial evaluation at a low Boolean cost as well (see also J. van der Hoeven (2008) [31], Pan and Tsigaridas (2013a,b) [26], [27], Kobel and Sagraloff (2013) [14], [24], and Pan (2015a) [25]). This immediately implies extension of our algorithm that support Corollary 6 to refining all simple isolated roots of a polynomial at a nearly optimal Boolean cost, but actually such an extension can be also obtained directly by using classical polynomial evaluation algorithm.…”

Section: Compute and Output The Values σmentioning

confidence: 99%

Accelerated approximation of the complex roots and factors of a univariate polynomial

Pan

Tsigaridas

2017

Theoretical Computer Science

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The algorithms of Pan (1995) [20] and Pan (2002) [22]) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require a precision of computing that exceeds the degree of the polynomial. This causes numerical stability problems when the degree is large. We observe, however, that such a difficulty disappears at the initial stage of the algorithms, and in our present paper we extend this stage to root-finding within a nearly optimal arithmetic and Boolean complexity bounds provided that some mild initial isolation of the roots of the input polynomial has been ensured. Furthermore our algorithm is nearly optimal for the approximation of the roots isolated in a fixed disc, square or another region on the complex plane rather than all complex roots of a polynomial. Moreover the algorithm can be applied to a polynomial given by a black box for its evaluation (even if its coefficients are not known); it promises to be of practical value for polynomial root-finding and factorization, the latter task being of interest on its own right. We also provide a new support for a winding number algorithm, which enables extension of our progress to obtaining mild initial approximations to the roots. We conclude with summarizing our algorithms and their extension to the approximation of isolated multiple roots and root clusters.1

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Fast approximate computations with Cauchy matrices and polynomials

Cited by 17 publications

References 37 publications

Old and New Nearly Optimal Polynomial Root-Finders

Old and New Nearly Optimal Polynomial Root-Finders

SMASH: Structured matrix approximation by separation and hierarchy

Accelerated approximation of the complex roots and factors of a univariate polynomial

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