Rational function

Rational function expansion

A rational function is defined as the ratio of two polynomials

$R\left( x \right) = \frac{P\left( x \right)}{Q\left( x \right)}$

The expansion of the rational function is computed recursively. More explicitly, if

$P\left( x \right) = p_0+p_1 x +\cdots + p_m x^m$

and

$Q\left( x \right) = q_0+q_1 x +\cdots + q_n x^n \qquad , \: q_0 \neq 0$

then

$R\left( x \right) = r_0+r_1 x + r_2 x^2 +\cdots$

is computed by equating the coefficients of

$R\left( x \right) Q\left( x \right) \quad and \quad P\left( x \right)$

we get

$r_0 = \frac{p_0} {q_0}$
$r_1 = \frac{ p_1 - r_0 q_1} { q_0 }$
$\cdots$
$r_k=\frac{p_k-\sum_{i=1}^{k} {r_{k-i} q_i}}{q_0} \quad,\:k\le n$

$r_k=\frac{p_k-\sum_{i=1}^{n} {r_{k-i} q_i}}{q_0} \quad,\:k\gt n$

Rational functions of polynomials with real coefficients are implemented in the class demetra.maths.polynomials.RationalFunction