Quadratures with quasi-degree of exactness

Monic orthogonal polynomials satisfy a three-term recurrence relation whose coefficients determine a symmetric tridiagonal Jacobi matrix. To efficiently compute the nodes and weights of Gauss quadrature, the Golub-Welsch algorithm, based on the observations that the nodes are the eigenvalues of the Jacobi matrix and that the weights are proportional to the square of the first components of corresponding normalized eigenvectors, can be applied. The Gauss quadrature with $n$ nodes is exact for all polynomials of degree $\le 2n-1$ and represents a unique optimal interpolatory quadrature rule. After choosing $m$ arbitrary points $x_k$ (at which the integrand is defined), we transform the given integral into a sum of an integral that does not cause a quadrature error and an integral with a property that the points $x_k$ are the zeros of its modified integrand. Then, we approximate the integral of the modified integrand by an $n$-point quadrature $\mathcal G_n$ whose nodes and weights are simply expressed in terms of the nodes and weights of associated $n$-point Gauss quadrature. Formula $\mathcal G_n$ is exact for all polynomials of degree $\le 2n-1+m$ with $m$ fixed zeros $x_k$, but those fixed zeros should not be a disadvantage, since the modified integrand has the same zeros.