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