Example 4.  Find the rational approximation for   over the interval [-1,1].  
4 (a).  Use equally spaced interpolation nodes.

Solution 4 (a).

Set up the formula for  .

Calculate the equally spaced values for the   interpolation nodes.   

Form the   ordinates.  

Form the set of   equations to solve and find the solution.
Remark.   Since  Tan[x]  is an odd function we will add the equation   to the list.

Form the rational approximation.

Plot graphs of the function and its rational approximation over the interval  [-1,1].  

Find the error  over the interval  [-1,1].  

Comparison with the Taylor approximation.  

There were 9 coefficients to determine for the rational approximation, and a Maclaurin polynomial of degree 8 requires 9 coefficients.
Compare with the error in a degree Maclaurin polynomial over the interval  .  

We can determine how much smaller the error is for the rational approximation.

Comparison with the Padé approximation.  

We can determine how much smaller the error is for the rational approximation.

(c) John H. Mathews 2004