Multivariate Frobenius-Padé approximants: properties and algorithms
The definition of the two variable Frobenius-Padé approximant using formal power series expansions in orthogonal polynomials studied in this paper was introduced in [\textit{A. C. Matos} and \textit{J. Van Iseghem}, J. Comput. Appl. Math. 176, No. 2, 231--258 (2005; Zbl 1067.65018)]. Consider the two variable function \[ f(x,y)=\sum_{i=0}^{\infty}\,\sum_{j=0}^{\infty}\,c_{ij}P_i(x)P_j(y), \] with \(\{P_i\}\) a system of orthogonal polynomials on \([a,b]\) with respect to a weight \(w\). Given two non-negative integers \(m,n\in \mathbb N\), find the polynomials \(P(x,y),\,Q(x,y)\) \[ P(x,y)=\sum_{(i,j)\in N}\,a_{ij}P_i(x)P_j(y), \;Q(x,y)=\sum_{(i,j)\in D}\,b_{ij}P_i(x)P_j(y), \] satisfying \[ f(x,y)Q(x,y)-P(x,y)=\sum_{(i,j)\in\mathbb N^2\setminus E}\,d_{ij}P_i(x)P_j(y),\tag \(*\) \] where \(D\subset \mathbb N^2\) has \(m\) elements, \(E\subset \mathbb N^2\) is the set of \(n+m-1\) indices of null terms in \((\ast)\), \(N\subset E\) is a set of \(n\) elements, \(H=E\setminus N\) has \(m-1\) elements. Just as in the case for the one variable Hermite-Padé approximant, the solution is found in two steps -- solve the homogeneous linear system for the \(b_{ij}\), -- calculate with these the numerator coefficients \(a_{ij}\). Then, with the aid of the ordinary three term recurrence relation for the orthogonal polynomials, the author introduces algorithms to calculate the values of a sequence of approximants at a given point. Moreover, for different choices of \(D,\,E\) and \(N\), the author studies -- using the Chebyshev polynomials \(T_n\) -- the problem of solving the homogeneous system for the denominator coefficients using the Hessenberg displacement rank structure of the associated block-matrix. Finally, the so called `mixed' Frobenius-Padé approximant approach is considered. Let \[ f(x,y)=\sum_{j=0}^{\infty}\,\sum_{i=0}^{\infty}\,f_{ij}P_i(y)P_j(x)= \sum_{j=0}^{\infty}\,f_j(y)P_j(x), \] then 1. Consider \[ f_y(x)=\sum_{j=0}^{\infty}\,f_j(y)P_j(x) \] with \(y\) a parameter and \(x\) the variable, and solve \[ Q_y(x)f_y(x)-P_y(x)=O(P_{m+n-1}(x)) \] with \(Q_y(x)\) a polynomial in \(x\) of degree \(m\) with \(Q(0)=1\) and \(P_y(x)\) a polynomial in \(x\) of degree \(n\), both with coefficients depending on \(y\). 2. Then replace the coefficients of \(P_y,\,Q_y\) by polynomials in \(y\) of degree \(N\) resp. \(M\) and solve the coefficient problem in the same manner as for \((\ast)\): first find the solution of a homogeneous system and calculate the other coefficients.
scientific article; zbMATH DE number 5142317