Interpolation by polynomial

Question 1

What can you know about the polynomial p_n(x)?

Answer 1

It is of order n. It has n coefficients and the highest power of x is n-1

Question 2

What does m=n+1 mean?

Answer 2

m is the number of data points

n is the order of an aproximation polynomial

n+1 is the number of parameters that need to be found to define the approximation.

Question 3

How do you produce a vandermole matrix to approximate a polynomial?

Answer 3

Question 4

What is the expression for a lagrange interpolent ,p(x), in terms of lagrange polynomials?

Answer 4

Question 5

What is the problem with using Vandermole matrixes to find p(x) with large number of nodes?

Answer 5

The determinant will get closer and closer to 0 and the finite power of computers could mistake it for 0.

Question 6

If you have m data points, what is the order of p(x) for hermite interpolation?

Answer 6

n=2m-1

n=order of p(x)

Question 7

What is Hermite interpolation?

Answer 7

Interpoltion where the interpolent has equal values and gradients at the nodes.

Question 8

If you wish to perform hermite interpolation with 2 nodes, how many coefficients does the aproximating equation need and what is the matrix used?

Answer 8

Question 9

If you produce a gram matrix and the determinant is 0 what does that mean for the solution?

Answer 9

There is no unique solution.

Question 10

If you had three data points and wanted to find an expression for L₁(x) what would it be?

Answer 10