The ThieleInterpolation routine returns the rational function in continued fraction form in variable v that interpolates the points $\{\left(\mathrm{x1},\mathrm{y1}\right),\left(\mathrm{x2},\mathrm{y2}\right),...,\left(\mathrm{xn},\mathrm{yn}\right)\}$.  If v is a numerical value, then the value of the function at this point is returned.  When n is odd, the numerator and denominator polynomials have degree $\frac{n}{2}-\frac{1}{2}$.  When n is even, the numerator has degree $\frac{n}{2}$ and the denominator has degree $\frac{n}{2}-1$.

The ThieleInterpolation routine can be called in two ways.

The first form accepts a list, Array, or Matrix, $\[\[\mathrm{x1},\mathrm{y1}\],\[\mathrm{x2},\mathrm{y2}\],...,\[\mathrm{xn},\mathrm{yn}\]\]$, of data points.

The second form accepts the input data as two lists, two Arrays, or two Vectors. In this form, the first set of data contains the independent values, $\[\mathrm{x1},\mathrm{x2},...,\mathrm{xn}\]$, and the second set of data contains the dependent values, $\[\mathrm{y1},\mathrm{y2},...,\mathrm{yn}\]$.  Each element must be of type algebraic.  All the independent values must be distinct.

In certain situations, the algorithm for computing the Thiele interpolating function produces a denominator of zero.  For example, a division-by-zero error is produced when two successive points have the same dependent value or when three successive points are collinear.  In such cases, perturbing the data points slightly may eliminate the problem.

This function is part of the CurveFitting package, and so it can be used in the form ThieleInterpolation(..) only after executing the command with(CurveFitting).  However, it can always be accessed through the long form of the command by using CurveFitting[ThieleInterpolation](..).

$\mathrm{with}\left(\mathrm{CurveFitting}\right)\:$

$\mathrm{ThieleInterpolation}\left(\left[\left[1\,3\right]\,\left[2\,4\right]\,\left[4\,5\right]\,\left[5\,8\right]\right]\,v\right)$

$3+\frac{v-1}{1+\frac{v-2}{80-19v}}$

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

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