Module

for

Padé Approximation

   

Background.

    A Padé rational approximation to f(x) on [a,b] is the quotient of two polynomials [Graphics:Images/PadeApproximationMod_gr_1.gif] and [Graphics:Images/PadeApproximationMod_gr_2.gif] of degrees n and m, respectively. We use the notation [Graphics:Images/PadeApproximationMod_gr_3.gif] to denote this quotient:

        [Graphics:Images/PadeApproximationMod_gr_4.gif].  
        
We attribute much of the founding theory to Henri Eugène Padé (1863-1953).  

 

Theorem (Padé Approximation).  Assume that  [Graphics:Images/PadeApproximationMod_gr_5.gif], and that [Graphics:Images/PadeApproximationMod_gr_6.gif] Maclaurin polynomial expansion of degree at least [Graphics:Images/PadeApproximationMod_gr_7.gif].  Then

    [Graphics:Images/PadeApproximationMod_gr_8.gif],  

where [Graphics:Images/PadeApproximationMod_gr_9.gif] and [Graphics:Images/PadeApproximationMod_gr_10.gif] 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 [Graphics:Images/PadeApproximationMod_gr_11.gif]for [Graphics:Images/PadeApproximationMod_gr_12.gif].  
Solution 1.

 

Example 2.  Find the Padé approximation [Graphics:Images/PadeApproximationMod_gr_75.gif]for [Graphics:Images/PadeApproximationMod_gr_76.gif].  
Solution 2.

 

Various Scenarios and Animations for the Pade Approximation.

Example 3.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_153.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_154.gif], [Graphics:Images/PadeApproximationMod_gr_155.gif], and [Graphics:Images/PadeApproximationMod_gr_156.gif].  
Solution 3 (a).
Solution 3 (b).
Solution 3 (c).

 

Example 4.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_205.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_206.gif].  
Solution 4.

 

Example 5.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_223.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_224.gif], and [Graphics:Images/PadeApproximationMod_gr_225.gif].  
Solution 5 (a).
Solution 5 (b).

 

Example 6.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_258.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_259.gif].  
Solution 6.

 

Example 7.  Find Pade approximations [Graphics:Images/PadeApproximationMod_gr_279.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_280.gif].  
Solution 7.

 

Example 8.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_300.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_301.gif].  
Solution 8.

 

Example 9.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_318.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_319.gif].  
Solution 9.

 

Example 10.  Find Pade approximations [Graphics:Images/PadeApproximationMod_gr_333.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_334.gif].  
Solution 10.

 

Example 11.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_351.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_352.gif].  
Solution 11.

 

Example 12.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_369.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_370.gif].  
Solution 12.

 

Example 13.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_390.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_391.gif].  
Solution 13.

 

Example 14.  Find Pade approximations for [Graphics:Images/PadeApproximationMod_gr_411.gif] expanded about  [Graphics:Images/PadeApproximationMod_gr_412.gif].  
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

 

Return to Numerical Methods - Numerical Analysis

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004