Fast inversion of Chebyshev-Vandermonde systems

Numerische Mathematik (Impact Factor: 1.61). 08/1995; 67(1). DOI: 10.1007/s002110050018
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This paper contains two fast algorithms for inversion of Chebyshev-- Vandermonde matrices of the first and second kind. They are based on special representations of the Bezoutians of Chebyshev polynomials of both kinds. The paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster, but also more stable than other algorithms available. It is also efficient to use the above two algorithms for solving Chebyshev--Vandermode systems of equations with preprocessing.

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    • "Chebyshev–V. Chebyshev Gohberg-Olshevsky [12] Three–Term V. Real orthogonal Calvetti-Reichel [9] Szegö–Vandermonde Szegö Olshevsky [19] (H, m)-semiseparable– (H, m)-semiseparable BEGOTZ [4] Vandermonde (new derivation in this paper) "
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    ABSTRACT: We use the language of signal flow graph representation of digital filter structures to solve three purely mathematical problems, including fast inversion of certain polynomial-Vandermonde matrices, deriving an analogue of the Horner and Clenshaw rules for polynomial evaluation in a (H,m)-quasiseparable basis, and computation of eigenvectors of (H,m)-quasiseparable classes of matrices. While algebraic derivations are possible, using elementary operations (specifically, flow reversal) on signal flow graphs provides a unified derivation, reveals connections with systems theory, etc.
    Linear Algebra and its Applications 04/2010; 432(8):2032-2051. DOI:10.1016/j.laa.2009.08.016 · 0.94 Impact Factor
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    • "Björck-Pereyra [2] Chebyshev-Vandermonde Chebyshev polynomials Gohberg-Olshevsky [14] Reichel-Opfer [26] Three-Term Vandermonde Real orthogonal polynomials Calvetti-Reichel [7] Higham [19] Szegö-Vandermonde Szegö polynomials Olshevsky [23] BEGKO [3] 1.2. "
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    ABSTRACT: Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomial-Vandermonde matrices involving real orthogonal polynomials, and the Szegö polynomials. In this paper we consider a new more general class of polynomials that we suggest to call Hessenberg order m quasisseparable polynomials, or (H, m)-quasiseparable polynomials. The new class is wide enough to include all of the above important special cases, e.g., monomials, real orthogonal polynomials and the Szcgö polynomials, as well as new subclasses. We derive a fast O(n 2) Traub-like algorithm to invert the associated (H, m)-quasisseparable-Vandermonde matrices. The class of quasiseparable matrices is garnering a lot of attention recently; it has been found to be useful in designing a number fo fast algorithms. The derivation of our new Traub-like algorithm is also based on exploiting quasiseparable structure of the corresponding Hessenberg matrices. Preliminary numerical experiments are presented comparing the algorithm to standard structure ignoring methods. This paper extends our recent results in [6] from the (H,0)-and (H,1)-quasiseparable cases to the more general (H, m)-quasiseparable case. Mathematics Subject Classification (2000)15A09–15–04–15B05
    Numerical Methods for Structured Matrices and Applications, 12/2009: pages 127-154;
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    • "For the discussions of Vandermonde-like matrix, there are a large number of papers in the literature. Usually, one makes sufficiently use of the special recurrence structure and orthogonality of Q k (x) to derive the fast inversions of Vandermonde-like matrices and fast algorithms for associated linear systems by direct computation; see, e.g., [1] [2] [4] [5] [7] [8] [17] and references therein. On the other hand, Vandermonde-like or polynomial Vandermonde matrix is an important class of displacement structure matrices. "
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    ABSTRACT: In the present paper, confluent polynomial Vandermonde-like matrices with general recurrence structure are introduced. Three kinds of displacement structure equations and two kinds of fast inversion formulas for this class of matrices are derived by using displacement structure matrix method. A relationship between confluent polynomial Vandermonde-like matrices and confluent Cauchy-like matrices is pointed out.
    Journal of Computational and Applied Mathematics 05/2005; 180(2):229-243. DOI:10.1016/ · 1.27 Impact Factor
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