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.

索博列夫正交多项式是与索博列夫内积正交的多项式，其中出现多项式的导数的内积。它们满足可以用 Hessenberg 矩阵表示的长递归关系。生成 Sobolev 正交多项式的有限序列的问题可以重新表述为矩阵问题，即 Hessenberg 逆特征值问题，其中 Hessenberg 递归矩阵是根据某些已知的光谱信息生成的。通过与 Krylov 子空间的连接，我们表明所需的光谱信息是包含 Hessenberg 矩阵的特征值及其特征向量的归一化第一项的 Jordan 矩阵。使用合适的求积法则，Sobolev 内积被离散化，所得的求积节点形成 Jordan 矩阵，相关的求积权重是特征向量的第一项。我们提出了两种新的数值程序来基于求解等效 Hessenberg 逆特征值问题来计算 Sobolev 正交多项式。