8.7 Interpolation of polynomial values and derivatives. The 5-vector c represents the coeffi-

cients of a quartic polynomial p(x) = c1 + c2x + c3x2 + c4x3 +c5x4. Express the conditions

P(0) = 0, p'(0) = 0, p(1) =1, p'(1) = 0,

as a set of linear equations of the form Ac = b. Is the system of equations under-

determined, over-determined, or square?

Solution:

Given P (x ) = C, + Cax + (3 x 2+ chx + co* "

If p (0 ) = 0, then @, = 0

P'(0 ) = 0, then ca= 0

P ( 1 ) = 1 , then C, + Cat (3+ Cy + (s= 1

P' ( 1) = 0, then Cat Cat Cut cs = 0

Let A =

O

0

0

O

and C = 1

O

O

O

0

Q

3

4

then Ac= b is

O

-O

1

O

O

O

1

0

3

- co

A is a 4x5 matrix and c is a box1 matrix,

which means b will be a 4x1 matrix. This shows

5 unknowns and 4 equations.

. The system is under determined.