Module

for

Hermite Polynomial Interpolation

   

Background for the Hermite Polynomial

 

    The cubic Hermite polynomial  p(x)  has the interpolative properties  [Graphics:Images/HermitePolyMod_gr_1.gif]   [Graphics:Images/HermitePolyMod_gr_2.gif]   [Graphics:Images/HermitePolyMod_gr_3.gif]  and  [Graphics:Images/HermitePolyMod_gr_4.gif]  both the function values and their derivatives are known at the endpoints of the interval  [Graphics:Images/HermitePolyMod_gr_5.gif].  Hermite polynomials were studied by the French Mathematician Charles Hermite (1822-1901), and are referred to as a "clamped cubic," where "clamped" refers to the slope at the endpoints being fixed.  This situation is illustrated in the figure below.  

[Graphics:Images/HermitePolyMod_gr_6.gif]

 

Theorem (Cubic Hermite Polynomial).  If  [Graphics:Images/HermitePolyMod_gr_7.gif]  is continuous on the interval  [Graphics:Images/HermitePolyMod_gr_8.gif], there exists a unique cubic polynomial  [Graphics:Images/HermitePolyMod_gr_9.gif]  such that
        [Graphics:Images/HermitePolyMod_gr_10.gif],
        [Graphics:Images/HermitePolyMod_gr_11.gif],
        [Graphics:Images/HermitePolyMod_gr_12.gif],  
        [Graphics:Images/HermitePolyMod_gr_13.gif].

Proof  Hermite Polynomial Interpolation  Hermite Polynomial Interpolation  

 

Remark.  The cubic Hermite polynomial is a generalization of both the Taylor polynomial and Lagrange polynomial, and it is referred to as an "osculating polynomial."  Hermite polynomials can be generalized to higher degrees by requiring that the use of more nodes  [Graphics:Images/HermitePolyMod_gr_14.gif] and the extension to agreement at higher derivatives  [Graphics:Images/HermitePolyMod_gr_15.gif]  for  [Graphics:Images/HermitePolyMod_gr_16.gif]  and    [Graphics:Images/HermitePolyMod_gr_17.gif].  The details are found in advanced texts on numerical analysis    

 

Computer Programs  Hermite Polynomial Interpolation  Hermite Polynomial Interpolation  

 

Example 1.  Find the cubic Hermite polynomial or "clamped cubic" that satisfies  
        [Graphics:Images/HermitePolyMod_gr_18.gif]  

 Solution 1.

 

More Background. The Clamped Cubic Spline

    A clamped cubic spline is obtained by forming a piecewise cubic function which passes through the given set of knots [Graphics:Images/HermitePolyMod_gr_35.gif]  with the condition the function values, their derivatives and second derivatives of adjacent cubics agree at the interior nodes.  The endpoint conditions are [Graphics:Images/HermitePolyMod_gr_36.gif], where [Graphics:Images/HermitePolyMod_gr_37.gif]  are given.

 

Example 2.  Find the "clamped cubic spline" that satisfies  
        [Graphics:Images/HermitePolyMod_gr_38.gif]   
Solution 2.

 

More Background. The Natural Cubic Spline

    A natural cubic spline is obtained by forming a piecewise cubic function which passes through the given set of knots [Graphics:Images/HermitePolyMod_gr_75.gif] with the condition the function values, their derivatives and second derivatives of adjacent cubics agree at the interior nodes.  The endpoint conditions are  [Graphics:Images/HermitePolyMod_gr_76.gif]. The natural cubic spline is said to be "a relaxed curve."

 

Example 3.  Find the "natural cubic spline" that satisfies  
        [Graphics:Images/HermitePolyMod_gr_77.gif]    
Solution 3.

 

Old Lab Project (Hermite polynomial interpolation  Hermite polynomial interpolation).  Internet hyperlinks to an old lab project.  

 

Research Experience for Undergraduates

Hermite Polynomial Interpolation  Hermite Polynomial Interpolation  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Hermite Polynomial Interpolation

 

Return to Numerical Methods - Numerical Analysis

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004