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Pawe l J. Expansions of one density via polynomials orthogonal with respect to the other
 J. Math. Anal. Appl
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Polynomial perturbations of bilinear functionals and Hessenberg matrices
"... This paper deals with symmetric and nonsymmetric polynomial perturbations of symmetric quaside nite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit a ..."
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This paper deals with symmetric and nonsymmetric polynomial perturbations of symmetric quaside nite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals.
Darboux transformations of Jacobi matrices and Pade
 approximation, Linear Algebra Appl. 435
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Stability and sensitivity of tridiagonal LU factorization without pivoting
"... In this paper the accuracy of LU factorization of tridiagonal matrices without pivoting is considered. Two types of componentwise condition numbers for the L and U factors of tridiadonal matrices are presented and compared. One type is a condition number with respect to small relative perturbations ..."
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In this paper the accuracy of LU factorization of tridiagonal matrices without pivoting is considered. Two types of componentwise condition numbers for the L and U factors of tridiadonal matrices are presented and compared. One type is a condition number with respect to small relative perturbations of each entry of the matrix. The other type is a condition number with respect to small componentwise perturbations of the kind appearing in the backward error analysis of the usual algorithm for the LU factorization. We show that both condition numbers are of similar magnitude. This means that the algorithm is componentwise forward stable, i.e., the forward errors are of similar magnitude to those produced by a componentwise backward stable method. Moreover the presented condition numbers can be computed in O(n) flops, which allows to estimate with low cost the forward errors.
A note on the Geronimus transformation and Sobolev orthogonal polynomials
 Numer. Algorithms, DOI
, 2013
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THE LINEAR PENCIL APPROACH TO RATIONAL INTERPOLATION
, 908
"... ABSTRACT. It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils zB − A of two tridiagonal matrices A, B, fol ..."
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ABSTRACT. It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils zB − A of two tridiagonal matrices A, B, following Spiridonov and Zhedanov. In the present paper, beside revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Padé approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to ∞, we compute explicitly the spectrum and the numerical range of the underlying linear pencil. 1.
Verblunsky Parameters and Linear Spectral Transformations
 Methods and Applications of Analysis 16 (2009
"... In this paper we analyze the behavior of Verblunsky parameters for hermitian linear functionals deduced from canonical linear spectral transformations of a quasidefinite hermitian linear functional. Some illustrative examples are studied. Key words: Quasidefinite hermitian linear functionals, ort ..."
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In this paper we analyze the behavior of Verblunsky parameters for hermitian linear functionals deduced from canonical linear spectral transformations of a quasidefinite hermitian linear functional. Some illustrative examples are studied. Key words: Quasidefinite hermitian linear functionals, orthogonal polynomials,
On the relation between Darboux transformations and polynomial mappings
 Journal of Approximation Theory
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