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

Solution 4 (b).

Set up the formula for  .

Calculate the values for the   Chebyshev 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 Chebyshev rational approximation.

Plot graphs of the function and its Chebyshev rational approximation over the interval  [-1,1].  But we will draw the graphs over [-2,2].

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 Chebyshev rational approximation.

Comparison with the Padé approximation.  

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

(c) John H. Mathews 2004