Module for the Pade Approximation

Check out the new Numerical Analysis Projects page.

Background. A Pade 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:

        .  

    The method is attributed to the French mathematician Henri Eugène Padé (1863-1953), and requires that f(x) and its derivatives be continuous at x = 0. There are two reasons for the arbitrary choice of x = 0. First, it makes the manipulations simpler. Second, a change of variable can be used to shift the calculations over to the interval that contains zero. The polynomials used in the construction are:

        
    and
        .

    The polynomials are constructed so that f(x) and agree at x = 0 and their derivatives up to n + m agree at x = 0.  In the case , the approximation is just the Maclaurin expansion for  f(x). For a fixed value of n + m the error is usually smallest when and have the same degree or when had degree one higher than .

    Notice that the constant coefficient of is . This is permissible, because it cannot be 0 and is not changed when both and are divided by the same constant. Hence the rational function has n + m + 1 unknown coefficients. Sometimes a specific choice for n and m will yield an inconsistent set of equations to solve, so one of the values n or m must be changed up or down.

    Assume that f(x) is analytic and has the Maclaurin series expansion:

          

We proceed to construct the rational function .  Observe that we want , so that .
To start, we form the difference and call it Z(x):

        .

We assume that Z(x) has a Maclaurin expansion too:

        .  

Now expand the former equation in its series form  

        .  

    If Z(x) were identically zero, then f(x) would be identically equal to the rational function . Since this does not happen in general, the best that we can hope for is that a significant number of the leading terms in the Maclaurin expansion for Z(x) are zero. Recall that the rational function has n + m + 1 unknown constants. We attempt to construct and under the additional assumption that ,...,. The equation that we must solve is

        .  

The lower index i = n + m + 1 in the summation on the right side of the equation is chosen because the first n + m derivatives of f(x) and are to agree at x = 0.

    When the left side of the above equation is expanded and the coefficients of the powers are set equal to zero for i=0,1,...,m+n, the result is a system of n+m+1 linear equations:

            ,  
            ,  
            ,  
            ,  
            ...  
            ,  
            ...
            ,  
            ,  
            ...
            ,  
            .  

    Notice that in each equation the sum of the subscripts on the factors of each product is the same, and this sum increases consecutively from 0 to n+m. The last m equations involve only the unknowns and must be solved first. Then the known values can be substituted in the first n+1 equations.  Then the first n+1 equations are used successively to find  .  

Animations (Pade Approximation).  Internet hyperlink to animations.

Example 1.  Find the Pade approximation for .  

Solution 1.

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

Research Experience for Undergraduates

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

Downloads (Pade Approximation Pade Approximation).  
Download this Mathematica notebook.  

(c) John H. Mathews 2003