rationalInterpolation -- numerically interpolate rational functions

Synopsis

Usage:

(n,d) = rationalInterpolation(pts, vals, numBasis, denBasis)

(n,d) = rationalInterpolation(pts, vals, numDenBasis)
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
- numBasis, a matrix, a one-row matrix of monomials, the monomial support of the numerator
- denBasis, a matrix, a one-row matrix of monomials, the monomial support of the numerator
- numDenBasis, a matrix, a one-row matrix of monomials, the monomial supports of both the numerator and denominator
Optional inputs:
- Tolerance => a real number, default value .000001, default value 1e-6
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 polynomials $g$ and $h$ have monomial support numBasis and denBasis respectively.

i1 : R = CC[x,y]

o1 = R

o1 : PolynomialRing

i2 : pts = {point{{1,0}}, point{{0,1}}, point{{1,1}}, point{{-1,-1}}, point{{-1,0}}}

o2 = {{1, 0}, {0, 1}, {1, 1}, {-1, -1}, {-1, 0}}

o2 : List

i3 : vals = {1, 0, 1/2, -1/2, -1}

            1    1
o3 = {1, 0, -, - -, -1}
            2    2

o3 : List

i4 : numBasis = matrix{{x,y}}

o4 = | x y |

             1      2
o4 : Matrix R  <-- R

i5 : denBasis = matrix{{x^2,y^2}}

o5 = | x2 y2 |

             1      2
o5 : Matrix R  <-- R

i6 : rationalInterpolation(pts, vals, numBasis, denBasis)

            2     2
o6 = (1x, 1x  + 1y )

o6 : Sequence

The output corresponds to the function $x / (x^2 + y^2)$. If no fitting rational function is found, the method returns an error.

The method rationalInterpolation(List,List,Ring) can be used to choose monomial supports automatically.

Caveat

The method uses the first point to remove the rational functions whose numerator and denominator would both evaluate to 0 on the point. Because of this, the first entry of val should be non-zero.

Ways to use rationalInterpolation :

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

For the programmer