Background

    A rational approximation to f(x) on the interval [a,b] is obtained by forming the quotient of two polynomials and of degrees n and m, respectively.  We use the notation to denote this quotient:

        .  

The polynomials used in the construction are:

        
    and
        .

Proof  Rational Approximation  Rational Approximation  

    The Pade approximation is a form of rational approximation which is analogous to the Taylor approximation because it is based on the derivatives of  f(x) at x=0.  The Padé approximation is very accurate near the center of expansion.  However, the error increases as one moves away from the center of expansion.  More accurate rational approximations are obtained if "interpolation nodes" are used, and we permit the error to be spread out more evenly over the entire interval.  The process is similar to Lagrange and Chebyshev interpolation.

Computer Programs  Rational Approximation  Rational Approximation  

Example 1.  Find the rational approximation for over the interval [-1,1].
1 (a).  Use equally spaced interpolation nodes.
1 (b).  Use Chebyshev interpolation nodes.
1 (c).  Use Mathematica's built in  MiniMaxApproximation  procedure.
Solution 1 (a).
Solution 1 (b).
Solution 1 (c).

Example 2.  Find the rational approximation for over the interval [-1,1].
2 (a).  Use equally spaced interpolation nodes.
2 (b).  Use Chebyshev interpolation nodes.
2 (c).  Use Mathematica's built in  MiniMaxApproximation  procedure.
Solution 2 (a).
Solution 2 (b).
Solution 2 (c).

Exercise 3.  Find the rational approximation for   over the interval [-1,1].  
3 (a).  Use equally spaced interpolation nodes.
3 (b).  Use Chebyshev interpolation nodes.
Solution 3 (a).
Solution 3 (b).

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

Example 5.  Find the rational approximation for   over the interval [-1,1].  
5 (a).  Use equally spaced interpolation nodes.
5 (b).  Use Chebyshev interpolation nodes.
Solution 5 (a).
Solution 5 (b).

Research Experience for Undergraduates

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(c) John H. Mathews 2004