Important: The thiele function has been deprecated. Use the superseding function CurveFitting[ThieleInterpolation] instead.  A call to thiele automatically generates a call to CurveFitting[ThieleInterpolation].

The thiele function computes the rational function of variable v (or evaluated at numerical value v) in continued fraction form which interpolates the points {(x[1], y[1]), (x[2], y[2]), ..., (x[n], y[n])}.  If n is odd then the numerator and denominator polynomials will have degree $\frac{n}{2}-\frac{1}{2}$. Otherwise, n is even and the degree of the numerator is $\frac{n}{2}$ and the degree of the denominator is $\frac{n}{2}-1$.

If the same x-value is entered twice, it is an error, whether the same y-value is entered.  All independent values must be distinct.

$\mathrm{thiele}\left(\left[1\,2\,a\right]\,\left[3\,4\,5\right]\,z\right)$

$3+\frac{z-1}{1+\frac{z-2}{\frac{1-a}{3-a}+1}}$

The function Thiele uses Thiele's interpolation formula involving reciprocal differences.  For more information, refer to:

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1965. Chap. 25 p. 881, Formula 25.2.50.