The condition of polynomials in power form

Abstract:

A study is made of the numerical condition of the coordinate map ${M_n}$ which associates to each polynomial of degree $\leqslant n - 1$ 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 ${\left \| {{M_n}} \right \|_\infty }{\left \| {M_n^{ - 1}} \right \|_\infty }$ 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 $1 + \sqrt 2$. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.