Instability and modification of Thiele interpolating continued fractions (Q1110018)

Given distinct points \((x_ i)_{i\in {\mathbb{N}}}\) and function values \((f_ i)_{i\in {\mathbb{N}}}\) one can compute inverse differences \(\phi [x_ 0,...,x_ i]\) and with these construct a Thiele interpolating continued fraction. It is illustrated in this paper how the computation of the inverse differences and the successive convergents of the Thiele continued fraction can be highly unstable. It is therefore important to obtain the value of the Thiele interpolating continued fraction with as few convergents as possible. Several ways to achieve this are given, all based on the use of modifying factors. Since the tail of the Thiele interpolating continued fraction can be written as \((z-x_ n)/\phi [x_ 0,...,x_ n,z],\) modifying factors can be obtained by approximating \(\phi [x_ 0,...,x_ n,z]\). It is shown that, assuming certain conditions, the modified convergents converge more rapidly than the ordinary convergents when the approximation for \(\phi [x_ 0,...,x_ n,z]\) is obtained by applying Aitken's \(\Delta^ 2\)-process or the \(\epsilon\)-algorithm to the sequence \((\phi [x_ 0,...,x_ n,x_{n+1+i}])_{i\in {\mathbb{N}}}\). Several numerical examples are given.