The condition of polynomials in power form

Abstract

A study is made of the numerical condition of the coordinate map M n {M_n} which associates to each polynomial of degree ⩽ n − 1 \leqslant n - 1 on the compact interval [a, b] the n-vector of its coefficients with respect to the power basis. It is shown that the condition number ‖ M n ‖ ∞ ‖ M n − 1 ‖ ∞ {\left \| {{M_n}} \right \|_\infty }{\left \| {M_n^{ - 1}} \right \|_\infty } increases at an exponential rate if the interval [a, b] is symmetric or on one side of the origin, the rate of growth being at least equal to 1 + 2 1 + \sqrt 2 . In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.