Some New Aspects of Rational Interpolation

Abstract

A new algorithm for rational interpolation based on the barycentric formula is developed; the barycentric representation of the rational interpolation function possesses various advantages in comparison with other representations such as continued fractions: it provides, e.g., information concerning the existence and location of poles of the interpolant.1. Introduction.Let r: R - R be the rational function defined by Obviously, lim;_, r(t) =/, if a + 0; thus r is a rational interpolant of the data (/"/,), i = 0(l)n, if a, #0, i = 0(l)n, and t, * t, for /' f j.In order to fix the constants a, i = 0(l)n, one can impose additional conditions on r.If we setthen / = /?/g.Usually, the following assumptions are made:(2) p G nm, q G II, m + k = n, mi k;here II.denotes the space of real polynomials of degree < j. (As is well known, one can treat the case m < k by interpolation of the data (/,, 1//,), i = 0(l)n, such that the degree of the numerator is k and the degree of the corresponding denominator is m; some simple modifications must be made if some of the /,'s vanish.)Note that our point of view differs from the usual one: since our ansatz (1) already ensures the interpolation property (if a =0, i = 0(l)n, at least) we can add any reasonable conditions on p and q, for example (2).The standard approach is to prescribe the form of p and q; the coefficients of p and q are then determined such that p interpolates fq at ?0, tx,..., tn.If q = 1, then (1) represents the barycentric form of the Lagrangian interpolation polynomial; hence the coefficients a,, i = 0(l)n, of (1) can be determined with the same number of operations which is necessary to compute the Newtonian form of the interpolation polynomial via divided differences (cf.[16]).The barycentric representation of rational interpolants has a very remarkable property: Even if the coefficients a, are disturbed, for example, by round off, the exact interpolation property still holds (if the disturbed a,, /' = 0(l)n, do not vanish,