Rank-structured QR for Chebyshev rootfinding

Casulli, Angelo; Robol, Leonardo

doi:10.48550/arxiv.2010.11416

Cited by 1 publication

(5 citation statements)

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“…The relationship between componentwise perturbations to the colleague matrix and the backward error in the coefficients was described completely in 2019 by Noferini, Robol, and Vandebril in [27]. Recently (see [32] and [14]), it was observed that certain O(n 2 ) structured QR algorithms for colleague matrices are surprisingly stable, attaining the bound (2) in many cases, an observation that mirrors the discovery in [3] for the case of companion matrices. However, unlike in [3], all previously proposed O(n 2 ) structured QR algorithms for colleague matrices have polynomials for which the worst-case bound (4) is attained.…”

Section: Introductionmentioning

confidence: 91%

“…Lemma 2.5. Suppose that Q ∈ SU( 2) is a complex rotation matrix, and let Q be a floating point approximation to Q satisfying (14). Suppose further that…”

Section: Multiplication By Complex Plane Rotationsmentioning

confidence: 99%

“…Following [14], we construct polynomials p rand (x) by sampling Chebyshev expansion coefficients a i independently from a standard normal distribution, so that a i ∼ N (0, 1), for i = 0, 1, . .…”

Section: P Rand (X): Polynomials With Random Coefficientsmentioning

confidence: 99%

“…A Pathological Example from [14] Here we consider the order-n Chebyshev expansion of the smooth function…”

Section: F Cas (X)mentioning

confidence: 99%

“…described in [14]. The first 1430 Chebyshev expansion coefficients of f cas (x) are shown in Figure 8.…”

Section: F Cas (X)mentioning

confidence: 99%

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A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices

Serkh,

Rokhlin

2021

Preprint

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The roots of a monic polynomial expressed in a Chebyshev basis are known to be the eigenvalues of the so-called colleague matrix, which is a Hessenberg matrix that is the sum of a symmetric tridiagonal matrix and a rank-1 matrix. The rootfinding problem is thus reformulated as an eigenproblem, making the computation of the eigenvalues of such matrices a subject of significant practical importance. In this manuscript, we describe an O(n 2 ) explicit structured QR algorithm for colleague matrices and prove that it is componentwise backward stable, in the sense that the backward error in the colleague matrix can be represented as relative perturbations to its components. A recent result of Noferini, Robol, and Vandebril shows that componentwise backward stability implies that the backward error δc in the vector c of Chebyshev expansion coefficients of the polynomial has the bound δc c u, where u is machine precision. Thus, the algorithm we describe has both the optimal backward error in the coefficients and the optimal cost O(n 2 ). We illustrate the performance of the algorithm with several numerical examples.

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Section: Introductionmentioning

confidence: 91%

“…Lemma 2.5. Suppose that Q ∈ SU( 2) is a complex rotation matrix, and let Q be a floating point approximation to Q satisfying (14). Suppose further that…”

Section: Multiplication By Complex Plane Rotationsmentioning

confidence: 99%

Section: P Rand (X): Polynomials With Random Coefficientsmentioning

confidence: 99%

“…A Pathological Example from [14] Here we consider the order-n Chebyshev expansion of the smooth function…”

Section: F Cas (X)mentioning

confidence: 99%

“…described in [14]. The first 1430 Chebyshev expansion coefficients of f cas (x) are shown in Figure 8.…”

Section: F Cas (X)mentioning

confidence: 99%

See 3 more Smart Citations

A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices

Serkh,

Rokhlin

2021

Preprint

View full text Add to dashboard Cite

show abstract

Rank-structured QR for Chebyshev rootfinding

Abstract: We consider the computation of roots of polynomials expressed in the Chebyshev basis. We extend the QR iteration presented in [

Cited by 1 publication

References 28 publications

A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices

A Provably Componentwise Backward Stable $O(n^2)$ QR Algorithm for the Diagonalization of Colleague Matrices

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