rationalInterpolation(List,List,Ring) -- numerically interpolate rational functions

Synopsis

Function: rationalInterpolation
Usage:

(n,d) = rationalInterpolation(pts, vals, R)
Inputs:
- pts, a list, of one-row matrices corresponding to points at which the rational function was evaluated
- vals, a list, of numbers corresponding to evaluations of the rational function
- R, a ring, the polynomial ring in which the numerator and denominator are sought
Optional inputs:
- Tolerance => ..., default value .000001, optional argument for numerical tolerance
Outputs:
- n, a ring element, the numerator of the rational function
- d, a ring element, the denominator of the rational function

Description

Given a list of points $pts = \{p_1,\dots,p_k\}$ and values $vals = \{v_1,\dots,v_k\}$, attempts to find a rational function $f = g/h$, such that $f(p_i) = v_i$. The method first tries to find polynomials $g,h$ of degree 0; if this fails, it tries to find $g,h$ of degree 1 and so on. This procedure stops when there are not enough points to compute the next degree, in which case an error will be thrown.

i1 : R = CC[x]

o1 = R

o1 : PolynomialRing

i2 : pts = {point{{0}},point{{1}},point{{2}}, point{{3}}, point{{4}}}

o2 = {{0}, {1}, {2}, {3}, {4}}

o2 : List

i3 : vals = {-1, 1/2, 1, 5/4, 7/5}

          1     5  7
o3 = {-1, -, 1, -, -}
          2     4  5

o3 : List

i4 : rationalInterpolation(pts, vals, R)
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)

o4 = (2x - 1, 1x + 1)

o4 : Sequence

Ways to use this method: