  Sobolev orthogonal polynomials are polynomials orthogonal with respect to a
Sobolev inner product, an inner product in which derivatives of the polynomials
appear. They satisfy a long recurrence relation that can be represented by a
Hessenberg matrix. The problem of generating a finite sequence of Sobolev
orthogonal polynomials can be reformulated as a matrix problem. That is, a
Hessenberg inverse eigenvalue problem, where the Hessenberg matrix of
recurrences is generated from certain known spectral information. Via the
connection to Krylov subspaces we show that the required spectral information
is the Jordan matrix containing the eigenvalues of the Hessenberg matrix and
the normalized first entries of its eigenvectors. Using a suitable quadrature
rule the Sobolev inner product is discretized and the resulting quadrature
nodes form the Jordan matrix and associated quadrature weights are the first
entries of the eigenvectors. We propose two new numerical procedures to compute
Sobolev orthonormal polynomials based on solving the equivalent Hessenberg
inverse eigenvalue problem.
