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

The ratinterp function computes a rational function u/v that satisfies

with u and v being polynomials in z having degrees at most m and n. In most (but not all) cases, this is equivalent to finding a rational function in z having certain degree constraints that interpolate the points (x0,y0), (x1,y1), ..., (xk,yk). Here $\mathrm{mn}=\left[m\,n\right]$. At least m+n+1 points are required, with all the independent points distinct.

Sometimes the linearized equation gives u and v satisfying the linearized equation, but with at least one of the xi a common zero. In this case, the resulting rational function does not interpolate the point xi (called an ``unattainable'' point), since common factors are removed from the rational function. In such cases ratinterp finds the rational function that interpolates a maximum number of points with the given degree constraints. The case where there are unattainable points is recognizable by the fact that the degrees of both the numerator and the denominator are less than the degree constraints.

Two procedures are available in the case of exact input: one that avoids singularities in the corresponding table of interpolants by a ``look-around'' procedure, and another that jumps singularities by a ``look-ahead'' procedure. Both algorithms are fraction-free. The look-ahead procedure uses a recusion similar to that used for subresultant computation. The second procedure can be specified by using a fifth argument, subresultant. The procedure produces an error message in the case of inexact data.

$\mathrm{xpoints}≔\left[0\,1\,2\,3\right]\:$$\mathrm{ypoints}≔\left[0\,3\,1\,3\right]\:$

$\mathrm{f1}≔\mathrm{ratinterp}\left(\mathrm{xpoints}\,\mathrm{ypoints}\,x\,\left[1\,2\right]\right)$

$\mathrm{f1}≔-\frac{3x}{4x^2-17x+12}$

$$\begin{gathered} \mathbf{for}i\mathbf{to}4\mathbf{do} \\ \mathrm{normal}\left(\mathrm{eval}\left(\mathrm{f1}\,x=\mathrm{xpoints}\left[i\right]\right)-\mathrm{ypoints}\left[i\right]\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$0$

$\mathrm{xpoints}≔\left[0\,1\,2\,3\,4\,-1\right]\:$$\mathrm{ypoints}≔\left[0\,3\,1\,3\,a\,\frac{1}{11}\right]\:$

$\mathrm{f2}≔\mathrm{ratinterp}\left(\mathrm{xpoints}\,\mathrm{ypoints}\,x\,\left[2\,3\right]\right)$

$\mathrm{f2}≔-\frac{3x}{4x^2-17x+12}$

$$\begin{gathered} \mathbf{for}i\mathbf{to}6\mathbf{do} \\ \mathrm{normal}\left(\mathrm{eval}\left(\mathrm{f1}\,x=\mathrm{xpoints}\left[i\right]\right)-\mathrm{ypoints}\left[i\right]\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$0$

Beckermann, B., and Labahn, G. Fraction-free Computation of Matrix GCD's and Rational Interpolants. CS Tech Report, University of Waterloo, 1997.

See also: SIAM Journal on Matrix Analysis and Applications, 2000.

Beckermann, B., and Labahn, G. Numeric and Symbolic Computation of Problems Defined by Structured Linear Systems. CS Tech Report, University of Waterloo, 1999.

See also: Reliable Computing, 2000.