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 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 row of monomials, the monomial support of the numerator
  • denBasis, a matrix, a row of monomials, the monomial support of the numerator
  • numDenBasis, a matrix, a row of monomials, the monomial support 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

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  + y )

o6 : Sequence

Caveat

See also

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