Rational interpolation ($q(x_i)/r(x_i)=f_i$)

Rational interpolation seeks a rational function $q/r$ with prescribed bounds on the polynomial degrees of its numerator and denominator and prescribed values at distinct interpolation nodes. A rational interpolant must have a well-defined value at each interpolation node. With a specified numerator-denominator representation, this requires $r(x_i)\ne 0$ for every $i$. A normalization such as a monic polynomial denominator removes the common scaling freedom but does not guarantee existence or uniqueness. Multiplying by a zero denominator can create spurious solutions to the rational interpolation linearization.