Rational Function Interpolation

 

There may be times when the use of polynomials may not be desirable for the approximation of smooth curves, because of rapidly changing slopes around their many extrema. For these cases, the use of interpolating rational functions may be a better choice. For this case, we pose the problem of finding an interpolating rational function, , in the form:

 

,        (1)

 

for a set of given interpolation points , , with and for , where and are polynomials of degree and , respectively, and for all . Where must meet the interpolation conditions:

 

       (2)

 

for , and where and are independent on coefficients. But by normalizing one of or to have a leading coefficient set to  1, then is set to depend on only coefficients. Furthermore, if  interpolates at the points , then

 

for  .        (3)

 

Conversely, if all the equations in (3) are satisfied, then Eq. (2) holds provided .  If  , then Eq. (3) implies that .

 

 

Werner & Schaback (1979)

 

Approach this problem by defining inverse difference quotients for the points ,  , as:

 

,        (4)

 

wherefrom we obtain,

 

       (5)

 

for  .

 

As appreciated, Eq. (4) implies that

 

       (6)

 

By replacing the inverse difference quotients of Eq. (6) in Eq. (5), takes the form of a continued fraction:

 

       (7)

 

By stopping short of the last fractional term of Eq. (7), marked therein with an asterisk, *, we obtain to the rational function, , for which    is satisfied for the interpolation points:

 

 

 

Stoer & Bulirsch (2002)

 

Have a slightly different approach. Here they allow  and solve Eq. (3) as an homogeneous system of linear equations.