# monomial basis conditioning

In this talk delivered by professor N. Trefethen it is stated that the condition number of a Vandermonde matrix of degree n verifies: $$\kappa\sim(1+\sqrt{2})^{n}$$ It is based on this paper by Gautschi. Unfortunately, I do not have access to that paper. How can I justify this approximation ?

• That does not seem correct as you stated it; $\kappa$ should depend on the choice of nodes. Jan 5, 2021 at 14:04

Gautschi's paper can be downloaded from here. You may find An elementary proof of the exponential conditioning of real Vandermonde matrices more easily understandable. For real distinct nodes the condition number has the lower bound $$\kappa\geq \sqrt{\frac{2}{n+1}}\bigl(1+\sqrt 2\bigr)^{n-1}.$$