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Applied Linear Algebra - Least Norm Polynomial Interpolation. Attached there is the question - 16.12 and the

question with the solution - 8.7 - that they are referencing to continue...

16.12 Least norm polynomial interpolation. (Continuation of exercise 8.7.) Find the polynomial
of degree 4 that satisfies the interpolation conditions given in exercise 8.7, and minimizes
the sum of the squares of its coefficients. Plot it, to verify that if satisfies the interpolation
conditions.

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* &quot;
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.

As went through the question and followed the solution for 8.7 which says that we will left with two equation given, c_3 +... View the full answer

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