Background.

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

        .  
        
We attribute much of the founding theory to Henri Eugène Padé (1863-1953).  

Theorem (Padé Approximation).  Assume that  , and that Maclaurin polynomial expansion of degree at least .  Then

    ,  

where and are polynomials of degree n and m, respectively.  

Proof  Padé Approximation  Padé Approximation  

Animations (Padé Approximation  Padé Approximation).  Internet hyperlink to animations.

Computer Programs  Padé Approximation  Padé Approximation  

Example 1.  Find the Padé approximation for .  
Solution 1.

Example 2.  Find the Padé approximation for .  
Solution 2.

Various Scenarios and Animations for the Pade Approximation.

Example 3.  Find Pade approximations for expanded about  , , and .  
Solution 3 (a).
Solution 3 (b).
Solution 3 (c).

Example 4.  Find Pade approximations for expanded about  .  
Solution 4.

Example 5.  Find Pade approximations for expanded about  , and .  
Solution 5 (a).
Solution 5 (b).

Example 6.  Find Pade approximations for expanded about  .  
Solution 6.

Example 7.  Find Pade approximations expanded about  .  
Solution 7.

Example 8.  Find Pade approximations for expanded about  .  
Solution 8.

Example 9.  Find Pade approximations for expanded about  .  
Solution 9.

Example 10.  Find Pade approximations expanded about  .  
Solution 10.

Example 11.  Find Pade approximations for expanded about  .  
Solution 11.

Example 12.  Find Pade approximations for expanded about  .  
Solution 12.

Example 13.  Find Pade approximations for expanded about  .  
Solution 13.

Example 14.  Find Pade approximations for expanded about  .  
Solution 14.

Animations (Pade Approximation  Pade Approximation).  Internet hyperlink to animations.

Research Experience for Undergraduates

Padé Approximation  Padé Approximation  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Pade Approximation

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(c) John H. Mathews 2004