Professor Higham refers to continuous least squares approximation by polynomials, which is not the same thing as polynomial least squares fitting. The entries of the matrix are (definite) integrals of the monomials, when the integration interval is (0,1).

The paper (from 1894) by David Hilbert

“Ein Beitrag zur Theorie des Legendre’schen Polynoms”, Acta Mathematica, 18: 155–159

https://link.springer.com/article/10.1007/BF02418278

seems to be the origin of the name “Hilbert matrix”.

]]>Further details can be found in many numerical analysis textbooks. Try Googling “hilbert matrix normal equations”.

]]>https://nhigham.com/2020/06/02/what-is-bfloat16-arithmetic/ and

https://nhigham.com/2020/05/07/what-is-ieee-standard-arithmetic/ ]]>

Thanks – corrected.

]]>Good point, Cleve. I don’t think this is an issue as regards very large growth – see the note at https://nickhigham.files.wordpress.com/2020/06/note_rhonest.pdf

]]>