Rate of convergence of the Pólya algorithm from polyhedral sets

Huotari, Robert; Marano, M.; Navas, J.; Quesada, José M.

doi:10.1016/j.jat.2005.03.007

Cited by 1 publication

(2 citation statements)

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“…where x (p) is the l p -solution of Gx = b and x (st) is the strict Chebyshev solution. We formulate this property in the form (3.1) to underline that Gx (p) is the l p -approximation to b by vectors from a linear subspace V. The properties of the Pólya algorithm and the strict Chebyshev approximations are the subject of a series of papers by Huotari and his coauthors (see, for example, [28,29,30,46,47,48]). In [28] Egger and Huotari give two important examples for the approximation over closed convex sets in R m .…”

Section: The Chebyshev and L P -Solutions Of An Overdetermined Systemmentioning

confidence: 99%

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From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Ziętak

2017

Banach Center Publ.

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The main aim of this review paper is approximation of a complex rectangular matrix, with respect to the unitarily invariant norms, by matrices from a linear subspace or from a convex closed subset of matrices. In particular, we focus on the properties and characterizations of the strict spectral approximant which is in some sense the best among all approximants to a given matrix with respect to the spectral norm. We formulate a conjecture that the strict spectral approximant to a matrix by matrices from a linear subspace is the limit of approximants with respect to the cp norms of Schatten when p tends to infinity. This conjecture corresponds to the conjecture stated by Rogers and Ward on approximation by positive semidefinite matrices and to the known analogous property of the strict Chebyshev approximation of a vector, proved by Descloux. We describe some special cases of an approximation of a matrix that confirm our conjecture and we discuss an attempt to prove the conjecture. Additionally, we focus on orthogonality of matrices in the sense of Birkhoff and James and approximation by matrices whose spectrum is in a strip. For the last case we recall a conjecture on characterization of the best approximation of a matrix in the spectral norm by matrices whose spectrum is in a strip. This kind of approximation is a generalization of the famous concept of Halmos of approximation by positive operators.

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Section: The Chebyshev and L P -Solutions Of An Overdetermined Systemmentioning

confidence: 99%

“…Some characterizations of sets on which the Pólya algorithm converges to the strict approximants are given in [46]. Future results, presented in the series of papers by Marano and his coauthors (see [48,69,77]), concern a rate of convergence of the Pólya algorithm.…”

Section: The Chebyshev and L P -Solutions Of An Overdetermined Systemmentioning

confidence: 99%

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Ziętak

2017

Banach Center Publ.

View full text Add to dashboard Cite

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Rate of convergence of the Pólya algorithm from polyhedral sets

Cited by 1 publication

References 11 publications

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

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