Matrix methods for quadrature formulas on the unit circle. A survey

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

“…Therefore it is essential to select poles at strategic places much closer to the circle to put more emphasis on regions where the eigenvalues are more important. The effect of the location of the poles can be observed for example in the numerical experiments reported in [6,7,10].…”

“…Indeed, since the odd and even factors C u on and C u en of the unitarily truncated CMV matrix C u n are just block diagonals with all blocks of size at most 2 × 2 with exception of G n+1 which is diagonal. Hence a pencil like (A n C u * en + C u on , C u * en + A * n C u en ) that will allow to compute the quadrature is extremely simple for numerical computations [7,Theorem 7.3].…”

Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circle

“…We may consider methods that are traditionally used to evaluate the nodes ξ n,k and the corresponding weights (see, for example, [1], [2], [5], [6] and [23]). For these methods the starting point are the eigenvalue problems obtained from a well known unitary modifications of the CMV matrices, respectively, of degrees n associated with the Verblunsky coefficients {α n (µ)} n≥0 and of degree n + 1 associated with the Verblunsky coefficients {α n (ν ǫ )} n≥0 .…”

Matrix moment perturbations and the inverse Szegő matrix transformation

“…A nice summary, as well as many references, can be found in [8]. As in the scalar case, matrix orthogonal polynomials have proved to be a useful tool in the analysis of many problems of mathematics such as differential equations [13,28], rational approximation theory [20], spectral theory of Jacobi matrices [1,29], analysis of polynomial sequences satisfying higher order recurrence relations [11,17], quantum random walks [3], and Gaussian quadrature formulas [2,12,16,31], among many others. In this contribution, we are interested in the study of some properties related with a perturbation of a sequence of matrix moments, within the framework of the theory of matrix orthogonal polynomials both on the real line and on the unit circle.…”