e-mail: cuyt@uia.ua.ac.be verdonk@uia.ua.ac.be

**Applications of homogeneous Pad\'e approximants in
Approximation Theory and Linear Algebra**

**Abstract**

In the first part of the talk we show that, when using homogeneous Pad\'e approximants to approximate the simplest hypergeometric function of two variables, namely the Appell function $F_1(a,b,b';c;x,y)$, the homogeneous Pad\'e approximants are normal for $F_1(a,1,b';c;x,y)$ and $F_1(a,b,1;c;x,y)$ with $c>a>0$. This generalizes the results valid for the univariate Pad\'e approximants to the Gauss function.

In the second part of the talk, we consider a matrix interpretation for the denominators of these homogeneous Pad\'e approximants which can be considered as formally orthogonal Hadamard polynomials. In the univariate case, such a matrix interpretation together with the univariate $qd$-algorithm can be put to good use for the computation of the eigenvalues of particular tridiagonal matrices. We investigate the use of the homogeneous $qd$-algorithm for the solution of parameterized eigenvalue problems of some tridiagonal matrices of which the entries are rational functions of one or more parameters. A small-scale numerical example illustrates the new technique.