Asymptotically Optimal Lower Bounds for the Condition Number of a Real Vandermonde Matrix

Lower bounds on the condition number minκp(V ) of a Vandermonde matrix V are established in terms of the dimension n or n and the largest absolute value among all nodes that define the Vandermonde matrix. Many bounds here are asymptotically sharp, and compare favorably to those of the same kind due to Gautschi and Inglese (Numer. Math., 52 (1988), 241–250) who considered either all positive nodes or real symmetric nodes and those due to Tyrtyshnikov (Numer. Math., 67 (1994), 261–269). As consequences our bounds imply that minκp(V ) over all possible real nodes goes to ∞ as fast as (1 + √ 2 )n, and minκp(V ) over all possible nonnegative nodes goes to ∞ as fast as (1 +√2 )2n. Extensions were made to confluent Vandermonde matrices and rectangular Vandermonde matrices, including brief outlines on how to deal with Vandermonde-like matrices and complex Vandermonde matrices. This report is available on the web at http://www.ms.uky.edu/∼math/MAreport/PDF/04-05.pdf. Department of Mathematics, University of Kentucky, Lexington, KY 40506 ([email protected].) This work was supported in part by the National Science Foundation CAREER award under Grant No. CCR9875201. Asymptotically Optimal Lower Bounds For the Condition Number of a Real Vandermonde Matrix Ren-Cang Li ∗ May 2004 Revised July 2004 Abstract Lower bounds on the condition number min κp(V ) of a Vandermonde matrix V are established in terms of the dimension n or n and the largest absolute value among all nodes that define the Vandermonde matrix. Many bounds here are asymptotically sharp, and compare favorably to those of the same kind due to Gautschi and Inglese (Numer. Math., 52 (1988), 241–250) who considered either all positive nodes or real symmetric nodes and those due to Tyrtyshnikov (Numer. Math., 67 (1994), 261–269). As consequences our bounds imply that min κp(V ) over all possible real nodes goes to ∞ as fast as (1 +√2 ), and min κp(V ) over all possible nonnegative nodes goes to ∞ as fast as (1 +√2 ). Extensions were made to confluent Vandermonde matrices and rectangular Vandermonde matrices, including brief outlines on how to deal with Vandermonde-like matrices and complex Vandermonde matrices.Lower bounds on the condition number min κp(V ) of a Vandermonde matrix V are established in terms of the dimension n or n and the largest absolute value among all nodes that define the Vandermonde matrix. Many bounds here are asymptotically sharp, and compare favorably to those of the same kind due to Gautschi and Inglese (Numer. Math., 52 (1988), 241–250) who considered either all positive nodes or real symmetric nodes and those due to Tyrtyshnikov (Numer. Math., 67 (1994), 261–269). As consequences our bounds imply that min κp(V ) over all possible real nodes goes to ∞ as fast as (1 +√2 ), and min κp(V ) over all possible nonnegative nodes goes to ∞ as fast as (1 +√2 ). Extensions were made to confluent Vandermonde matrices and rectangular Vandermonde matrices, including brief outlines on how to deal with Vandermonde-like matrices and complex Vandermonde matrices. ∗Department of Mathematics, University of Kentucky, Lexington, KY 40506 ([email protected].) Supported in part by the National Science Foundation CAREER award under Grant No. CCR-9875201.