Hello, I have just been given a question that I don't understand and after looking over my course notes I can't find anything resembling it. So could anyone could show me how to solve it, explain what exactly this question is about, or point me in the direction of a site that explains it clearly without too much reliance on other mathematics topics? I mainly just want to know the technique so I can do the this thing.

**Any help whatsoever** would be

__greatly__ appreciated!

Let

Let

,

be two distinct points at which the function

and its first derivative

are defined and assume that the second derivative

is also defined at the point

. A Hermite interpolating quartic polynomial of the form

can be constructed for the function

from the given five data values by determining the unknown coefficients

using the conditions:

and

Construct a Hermite quartic polynomial that interpolates the function

at the eigenvalues of

by solving an appropriately defined linear system whose solution provides the coefficients

,

. Use this polynomial to construct a matrix function approximation for

.