Implicit QR for companion-like pencils

Boito, Paola; Eidelman, Yuli; Gemignani, Luca

doi:10.1090/mcom/3020

Cited by 11 publications

(19 citation statements)

References 20 publications

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“…Note that in Chebfun QR, and not QZ, is used by default for a polynomial scaled to be monic in the Chebyshev basis, and this may in some circumstances lead to a colleague matrix with a large norm C , which is undesirable in view of (1.4). See also [12,30] for a discussion for the monomial basis yielding the similar conclusion that QZ is preferred to QR when the leading coefficient is small. 2 The weaker notion of normwise stability for each root is (p + ∆pi)(xi) = 0 with…”

Section: Introductionmentioning

confidence: 95%

“…. , n − 1, the roots of T j+1 (x) are the eigenvalues of the (j + 1) × (j + 1) Jacobi matrix 12) where ij are the same as in (3.7). Let be as in (3.9).…”

Section: Roots Of Orthogonal Polynomials Are Insensitive To Perturbatmentioning

confidence: 99%

“…Some of these alternative methods are strong competitors of the linearization method, both for computational complexity and for stability (see, e.g., [7,10,11] for the EhrlichAberth method); on the other hand, special technologies, like subdivision [14] or preservation of structures such as quasiseparable [5,8,9,12,13], can be exploited in order to reduce the complexity of the linearization method to O(n 2 ). Among the many rootfinding algorithms available, our goal here is to understand the stability of linearization-based methods, as they are often easy to implement (given a black-box eigensolver) and widely used.…”

Section: Introductionmentioning

confidence: 99%

“…When p(x) is expressed in the standard monomial basis, many studies exist on the stability, or lack thereof, of the linearization-based rootfinders [12,20,21,49]. However, little seems to be known for p(x) expressed in a nonmonomial basis such as Chebyshev and other orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

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On the stability of computing polynomial roots via confederate linearizations

Nakatsukasa

Noferini

2015

Math. Comp.

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A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Finally, we focus on the special case of Chebyshev basis, in particular the Chebfun rootfinder: we discuss its stability and describe an optional functionality, made available for improved stability, for computing the roots of a general continuous function f (x), implemented in the recently updated version 5. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary.

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Section: Introductionmentioning

confidence: 95%

“…. , n − 1, the roots of T j+1 (x) are the eigenvalues of the (j + 1) × (j + 1) Jacobi matrix 12) where ij are the same as in (3.7). Let be as in (3.9).…”

Section: Roots Of Orthogonal Polynomials Are Insensitive To Perturbatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the stability of computing polynomial roots via confederate linearizations

Nakatsukasa

Noferini

2015

Math. Comp.

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“…The sequentially semiseparable representation used by Eidelman and Gohberg (1999); Vandebril et al (2005Vandebril et al ( , 2007; Eidelman et al (2005); Boito et al (2016) for a matrix M, consists of (n − 1) pairs of vectors p(i), q(i) of size r L , (n − 1) pairs of vectors g(i), h(i) of size r U , n − 1 matrices a(i) of https://cs.uwaterloo.ca/~astorjoh/ (Arne Storjohann) dimension r L × r L , and n − 1 matrices b(i) of dimension r U × r U such that…”

Section: Introductionmentioning

confidence: 99%

Time and space efficient generators for quasiseparable matrices

Pernet

Storjohann

2018

Journal of Symbolic Computation

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The class of quasiseparable matrices is defined by the property that any submatrix entirely below or above the main diagonal has small rank, namely below a bound called the order of quasiseparability. These matrices arise naturally in solving PDE's for particle interaction with the Fast Multi-pole Method (FMM), or computing generalized eigenvalues. From these application fields, structured representations and algorithms have been designed in numerical linear algebra to compute with these matrices in time linear in the matrix dimension and either quadratic or cubic in the quasiseparability order. Motivated by the design of the general purpose exact linear algebra library LinBox, and by algorithmic applications in algebraic computing, we adapt existing techniques introduce novel ones to use quasiseparable matrices in exact linear algebra, where sub-cubic matrix arithmetic is available. In particular, we will show, the connection between the notion of quasiseparability and the rank profile matrix invariant, that we have introduced in 2015. It results in two new structured representations, one being a simpler variation on the hierarchically semiseparable storage, and the second one exploiting the generalized Bruhat decomposition. As a consequence, most basic operations, such as computing the quasiseparability orders, applying a vector, a block vector, multiplying two quasiseparable matrices together, inverting a quasiseparable matrix, can be at least as fast and often faster than previous existing algorithms.

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A real QZ algorithm for structured companion pencils

Boito

Eidelman

Gemignani

2017

Calcolo

Self Cite

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We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an N × N structured matrix pencil using O(N ) flops per iteration and O(N ) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

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Implicit QR for companion-like pencils

Cited by 11 publications

References 20 publications

On the stability of computing polynomial roots via confederate linearizations

On the stability of computing polynomial roots via confederate linearizations

Time and space efficient generators for quasiseparable matrices

A real QZ algorithm for structured companion pencils

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