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- W2046806661 abstract "Let f and g be functions defined at the real and distinct nodes $x_k $, and consider the inner product $( f,g ): = sum_{k = 1}^m f ( x_k ) g ( x_k ) w_k^2 $ with positive weights $w_k^2 $. The present paper discusses the computation of orthonormal polynomials $pi _0 ,pi _1 , cdots ,pi _{n - 1} ,nleqq m$, with respect to this inner product, and the use of these polynomials in a fast scheme for computing a QR decomposition of the transpose of Vandermonde-like matrices. Two methods are compared for computing the recurrence coefficients for the polynomials $pi _j $ and their values at the nodes $x_k $: the Stieltjes procedure and a method in which an inverse eigenvalue problem for a tridiagonal symmetric matrix is solved by an algorithm proposed by Rutishauser, Gragg, and Harrod. The latter method is found to generally yield higher accuracy than the Stieltjes procedure if n is close to m, and roughly the same accuracy otherwise. This method for solving an inverse eigenvalue problem is applied in an algorithm for computing a QR decomposition of the transpose of $n times m$ Vandermonde-like matrices. The algorithm so obtained requires only $O ( mn )$ arithmetic operations. This operation count compares favorably with the $O( mn^2 )$ arithmetic operations necessary for the QR decomposition if the structure of Vandermonde-like matrices is ignored." @default.
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- W2046806661 date "1991-07-01" @default.
- W2046806661 modified "2023-09-23" @default.
- W2046806661 title "Fast QR Decomposition of Vandermonde-Like Mmatrices and Polynomial Least Squares Approximation" @default.
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- W2046806661 doi "https://doi.org/10.1137/0612041" @default.
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