Background for the Hermite Interpolation Polynomial.

Background for the Hermite Interpolation Polynomial. The cubic Hermite polynomial  p(x)  has the interpolative properties          and    both the function values and their derivatives are known at the endpoints of the interval  .  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. 

Example 1.  Find the cubic Hermite polynomial or "clamped cubic" that satisfies 
          

Solution 1.

Enter the formula for a general cubic equation.

Symbolic differentiation (integration too) is permitted with Mathematica.

Set up four equations using the prescribed endpoint conditions. Then find the solution set to this linear system and store it in the variable solset.

Use the solution given above for the coefficients and form the cubic function.  Remember that we must dig out one set of braces using   before we can use the ReplaceAll command.

源自於

http://mathfaculty.fullerton.edu/mathews/n2003/Web/HermitePolyMod/HermitePolyMod.html