The condition of Vandermonde-like matrices involving orthogonal polynomials☆
TLDR
In this article, the condition number of the n X n matrix p = [Pi-ltxj)lY,Y,j-l is investigated, where p,((.) = p, (.; dX) are orthogonal polynomials with respect to some weight distribution dX, and xi are pairwise distinct real numbers.About:
This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 70 citations till now. The article focuses on the topics: Orthogonal polynomials & Condition number.read more
Citations
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Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study
TL;DR: In this paper, the authors show that the convergence and absolute convergence of orthogonal polynomials on infinite intervals and on the untt crrcle can be explained by the convergence of Christoffel functions.
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Fast algorithms for discrete polynomial transforms
TL;DR: A new fast algorithm for the computation of the matrix-vector product Pa in O(N log 2 N) arithmetical operations is presented, which divides into a fast transform which replaces Pa with C N+1 ā and a subsequent fast cosine transform.
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Stability analysis of algorithms for solving confluent Vandermonde-like systems
TL;DR: A method is derived for computing a “stable” ordering of the points $\alpha _i $; it mimics the interchanges performed by Gaussian elimination with partial pivoting, using only $O(n^2)$ operations.
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A Korovkin-type theory for finite Toeplitz operators via matrix algebras
TL;DR: In this article, it was shown that the Frobenius-optimal preconditioners can be used to define new and just known linear positive operators uniformly approximating the function f. The convergence features of these PCG have been naturally studied by means of the Weierstrass-Jackson Theorem, owing to the profound relationship between the spectral features of the matrices generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself.
Fast Solution of Yandermonde-L ike Systems Involving Orthogonal Polynomials
TL;DR: The authors propose des algorithmes rapides de resolution des systemes primal et dual Px=b and dual P T a=f ou P est une matrice du type de Vandermonde, in which les elements polynomiaux satisfont a relation de recurrence a trois termes.
References
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Remark on “Algorithm 500: Minimization of Unconstrained Multivariate Functions [E4]”
David F. Shanno,K. H. Phua +1 more
TL;DR: The subroutine incorporates two nonlinear optimization methods, a conjugate gradient algorithm and a variable metric algorithm, with the choice of method left to the user, with two differences.