Fast Computation of the Zeros of a Polynomial via Factorization of the Companion Matrix

Aurentz, Jared L.; Vandebril, Raf; Watkins, David S.

doi:10.1137/120865392

Cited by 20 publications

(26 citation statements)

References 10 publications

(14 reference statements)

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“…For avoiding over-and underflow and for easily computing the eigenvectors we refer to [2]. Figures 1 and 2 illustrate that the accuracy of our proposed algorithm AMVW is comparable to that of LAPACK, significantly better than BEGG, and slightly better than CGXZ.…”

Section: Polynomials With Random Coefficientsmentioning

confidence: 94%

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Fast and Backward Stable Computation of Roots of Polynomials

Aurentz¹,

Mach²,

Vandebril³

et al. 2015

SIAM J. Matrix Anal. & Appl.

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A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly shifted QR algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rankone, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of 3n − 1 Givens rotators plus the rank-one part, so only O(n) storage space is required. In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly. Francis's algorithm implemented on this representation requires only O(n) flops per iteration and thus O(n 2 ) flops overall. The algorithm is described, normwise backward stability is proved, and an extensive set of numerical experiments is presented. The algorithm is shown to be about as accurate as the (slow) Francis QR algorithm applied to the companion matrix without exploiting the structure. It is faster than other fast methods that have been proposed, and its accuracy is comparable or better.

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Section: Polynomials With Random Coefficientsmentioning

confidence: 94%

“…In some of our previous work [2,3] we have made use of core transformations that are not unitary, but in this paper we will use only unitary ones. Thus, in this paper, the term core transformation will mean unitary core transformation.…”

mentioning

confidence: 99%

Fast and Backward Stable Computation of Roots of Polynomials

Aurentz¹,

Mach²,

Vandebril³

et al. 2015

SIAM J. Matrix Anal. & Appl.

Self Cite

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“…Then we obtain β 3 as the minimum value of f over all stationary points in [−1, 1] along with the endpoints ±1. The dominant cost for this bound is computing the stationary points, which are the real eigenvalues of a companion matrix of order 2n − 3 in [−1, 1]; these can be computed in O(n 2 ) flops [2]. To summarize, we can separate the new upper bounds into three main categories.…”

Section: Computing the Boundsmentioning

confidence: 99%

Bounds for the Distance to the Nearest Correlation Matrix

Higham

Strabić²

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. In a wide range of practical problems correlation matrices are formed in such a way that, while symmetry and a unit diagonal are ensured, they may lack semidefiniteness. We derive a variety of new upper bounds for the distance from an arbitrary symmetric matrix to the nearest correlation matrix. The bounds are of two main classes: those based on the eigensystem and those based on a modified Cholesky factorization. Bounds from both classes have a computational cost of O(n 3 ) flops for a matrix of order n but are much less expensive to evaluate than the nearest correlation matrix itself. For unit diagonal A with |a ij | ≤ 1 for all i = j the eigensystem bounds are shown to overestimate the distance by a factor at most 1 + n √ n. We show that for a collection of matrices from the literature and from practical applications the eigensystem-based bounds are often good order of magnitude estimates of the actual distance; indeed the best upper bound is never more than a factor 5 larger than a related lower bound. The modified Cholesky bounds are less sharp but also less expensive, and they provide an efficient way to test for definiteness of the putative correlation matrix. Both classes of bounds enable a user to identify an invalid correlation matrix relatively cheaply and to decide whether to revisit its construction or to compute a replacement, such as the nearest correlation matrix.

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“…Though this may not be the best way to address the polynomial root-finding problem, from the point of view of efficiency and storage (see, for instance, Moler (1991)), it has been extensively used because of the advantages of the QR algorithm (robustness and backward stability). Nonetheless, to overcome the mentioned drawbacks on the efficiency (measured in number of operations) and storage, several fast variants of the QR method have been proposed, which take advantage of the structure of the companion matrix (see, for instance, Aurentz et al (2013); Bini et al (2004Bini et al ( , 2005Bini et al ( , 2010; Calvetti et al (2002); Chandrasekaran et al (2008); Gemignani (2007); Van Barel et al (2010)), but none of them has been proved to be stable. In a different line of research, also variants of C 1 ,C 2 have been proposed, devoted to improve the accuracy in the case of multiple roots, where the standard companion matrix gives less accurate results than for simple roots (see Brugnano & Trigiante (1995); Niu & Sakurai (2003)).…”

Section: Introductionmentioning

confidence: 99%

Backward stability of polynomial root-finding using Fiedler companion matrices

2015

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Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices. In other words, given a polynomial p(z), the question is to determine whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm for the standard eigenvalue problem, are the set of roots of a nearby polynomial or not. We show that, if the coefficients of p(z) are bounded in absolute value by a moderate number, then algorithms for polynomial root-finding using Fiedler matrices are backward stable, and Fiedler matrices are as good as the Frobenius companion matrices. This allows us to use Fiedler companion matrices with favorable structures in the polynomial root-finding problem. However, when some of the coefficients of the polynomial are large, Fiedler companion matrices may produce larger backward errors than Frobenius companion matrices, although in this case neither Frobenius nor Fiedler matrices lead to backward stable computations. To prove this we obtain explicit expressions for the change, to first order, of the characteristic polynomial coefficients of Fielder matrices under small perturbations. We show that, for all Fiedler matrices except the Frobenius ones, this change involves quadratic terms in the coefficients of the characteristic polynomial of the original matrix, while for the Frobenius matrices it only involves linear terms. We present extensive numerical experiments that support these theoretical results. The effect of balancing these matrices is also investigated.

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Fast Computation of the Zeros of a Polynomial via Factorization of the Companion Matrix

Cited by 20 publications

References 10 publications

Fast and Backward Stable Computation of Roots of Polynomials

Fast and Backward Stable Computation of Roots of Polynomials

Bounds for the Distance to the Nearest Correlation Matrix

Backward stability of polynomial root-finding using Fiedler companion matrices

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