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

Solution 1.

Enter the formula for a general cubic equation.

[Graphics:../Images/HermitePolyMod_gr_19.gif]

[Graphics:../Images/HermitePolyMod_gr_20.gif]

Symbolic differentiation (integration too) is permitted with Mathematica.

[Graphics:../Images/HermitePolyMod_gr_21.gif]

[Graphics:../Images/HermitePolyMod_gr_22.gif]

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.

[Graphics:../Images/HermitePolyMod_gr_23.gif]

[Graphics:../Images/HermitePolyMod_gr_24.gif]

[Graphics:../Images/HermitePolyMod_gr_25.gif]

[Graphics:../Images/HermitePolyMod_gr_26.gif]

[Graphics:../Images/HermitePolyMod_gr_27.gif]

[Graphics:../Images/HermitePolyMod_gr_28.gif]

Use the solution given above for the coefficients and form the cubic function.  Remember that we must dig out one set of braces using [Graphics:../Images/HermitePolyMod_gr_29.gif]  before we can use the ReplaceAll command.

[Graphics:../Images/HermitePolyMod_gr_30.gif]


[Graphics:../Images/HermitePolyMod_gr_31.gif]

[Graphics:../Images/HermitePolyMod_gr_32.gif]
[Graphics:../Images/HermitePolyMod_gr_33.gif]
[Graphics:../Images/HermitePolyMod_gr_34.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004