Lagrange Polynomials

Lagrange Polynomials

Background.

    We have seen how to expand a function   in a Maclaurin polynomial about involving the powers and a Taylor polynomial about involving the powers .  The Lagrange polynomial of degree passes through the points    for    and were investigated by the mathematician Joseph-Louis Lagrange (1736-1813).   

Theorem ( Lagrange Polynomial ).  Assume that   and   for    are distinct  values.  Then

    ,
    
where is a polynomial that can be used to approximate  ,

      

and we write  

    .

The Lagrange polynomial goes through the points  ,  i.e.

        for   .  

The remainder term   has the form

    ,

for some value that lies in the interval .  

    The cubic curve in the figure below illustrates a Lagrange polynomial of degree n = 3, which passes through the four points for  .  
    

Theorem.  (Error Bounds for Lagrange Interpolation, Equally Spaced Nodes)  Assume that    defined on ,  which contains the equally spaced nodes  .  Additionally, assume that      and the derivatives of    up to the order    are continuous and bounded on the special subintervals  , , , , and , respectively;  that is,

    ,  

for  .  The error terms corresponding to these three cases have the following useful bounds on their magnitude  

(i).       is valid for  ,  

(ii).       is valid for  ,  

(iii).       is valid for  ,  

(iv).       is valid for  ,  

(v).       is valid for  .  

Algorithm ( Lagrange Polynomial ).  To construct the Lagrange polynomial  

      
    
of degree ,  based on the points for  .  The Lagrange coefficient polynomials    for degree are:  

      for  .

 

 

You can use the first Mathematica subroutine that does things in the "traditional way" or you are welcome to use the second subroutine that illustrates  "Object Oriented Programming."  

Mathematica Subroutine (Lagrange Polynomial). Traditional programming.

The above algorithm is sufficient for understanding and/or constructing the Lagrange polynomial.  

Object Oriented Programming.  Welcome to the brave new world of "Object Oriented Programming."  Use the following Mathematica subroutine which is "programmed" using the "mathematical objects"  .  Templates for the objects are located by going to "File" then select "Palettes", then select "BasicInput."  

Mathematica Subroutine (Lagrange Polynomial). Object oriented programming.

Mathematica Subroutine (Lagrange Polynomial). Compact object oriented programming.