TY - JOUR

T1 - Afast bjorck-pereyra-type algorithm for solving essenberg-quasiseparable- vandermonde systems

AU - Bella, T.

AU - Eidelman, Y.

AU - Gohberg, I.

AU - Koltracht, I.

AU - Olshevskyl, V.

PY - 2009

Y1 - 2009

N2 - Afast O(n2) algorithm is derived for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix Vr(x) = [rj1(xi)] with polynomials {rk(x)} defined by a Hessenberg matrix with quasiseparable structure. The result generalizes the well-known Bjorck- ereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of Reichel-Opfer for Vr(x) involving Chebyshev polynomials of Higham for Vr(x) involving real orthogonal polynomials, and a recent algorithm of the authors for Vr (x) involving Szegö polynomials. The new algorithm applies to a fairly general new class of (H, k)-quasiseparable polynomials (Hessenberg, order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials, e.g., defined by banded Hessenberg matrices. Numerical experiments are presented that coincide with previous experiences with Björck-Pereyra- type algorithms giving better forward error than Gaussian elimination, and this accuracy is consistent with the so-called Chan-Foulser conditioning of the system.

AB - Afast O(n2) algorithm is derived for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix Vr(x) = [rj1(xi)] with polynomials {rk(x)} defined by a Hessenberg matrix with quasiseparable structure. The result generalizes the well-known Bjorck- ereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of Reichel-Opfer for Vr(x) involving Chebyshev polynomials of Higham for Vr(x) involving real orthogonal polynomials, and a recent algorithm of the authors for Vr (x) involving Szegö polynomials. The new algorithm applies to a fairly general new class of (H, k)-quasiseparable polynomials (Hessenberg, order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials, e.g., defined by banded Hessenberg matrices. Numerical experiments are presented that coincide with previous experiences with Björck-Pereyra- type algorithms giving better forward error than Gaussian elimination, and this accuracy is consistent with the so-called Chan-Foulser conditioning of the system.

KW - Bjorck-pereyra algorithm

KW - Quasiseparable matrices

KW - Vandermonde matrices

UR - http://www.scopus.com/inward/record.url?scp=72449193554&partnerID=8YFLogxK

U2 - 10.1137/060676635

DO - 10.1137/060676635

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AN - SCOPUS:72449193554

SN - 0895-4798

VL - 31

SP - 790

EP - 815

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 2

ER -