Some Properties of Chisholm Approximants

The rational fraction approximants to functions of two variablesdefined by Chisholm are shown (a) to factorize into Pad? approximantswhen the given function is a product function and (b) to beunitary when the given function is unitary.     

Practical, Reliable, Rational Interpolation

An algorithm incorporating features essential for practical,reliable, rational interpolation is explained. This algorithmgenerates a Thiele-Werner continued fraction representationof the interpolant. A backward error analysis is presented forthe algorithm, as well as for its special cases of Newton polynomialinterpolation and Thiele rational interpolation. This is madepossible by introducing into the Newton method, Thiele methodand Werner method a strategy for selecting the interpolationpoints in an optimal order.     

Vector-valued Rational Interpolants II

Formulae for rational interpolation of vector data (in a spaceC^[d]) at distinct points are given. Its confluent case of vector-valuedPadé approximation is shown to be equivalent to the Germanpolynomial approximation problem. Formulae are given for thevector of numerator polynomials and for the denominator polynomial.A continued fraction interpolant for vector data is also given.The methods are characterized by their requirement that certaindistinguished directions in the space C^[d] form part of thespecification. The case of matrix Padé approximants forthe partial realization problem is explicitly discussed.     

The Calculation of Some Rational Approximants in Two Variables

Chisholm has shown how rational approximants may be definedfor functions of two variables, which reduce to diagonal Pad?Approximants when one variable is zero. We extend the idea toallow for different maximum powers in numerator and denominator,and state the relevant theorems. We discuss numerical difficultiesand possible singularities occurring in the construction ofthe approximants; approximants to the Beta function are calculatedand discussed in detail.     

Pade Approximants for Linear Integral Equations?

Padé approximation is often used to find a matrix elementof the solution of a Fredholm integral equation of the secondkind. An example is given in which this process is necessarilydivergent, even though the kernel of the integral equation iscompact and the elements lie in a Hilbert space.