But it is simpler to use the

**differentiation matrix for arithmetic polynomials:**

The 5x5 matrix above is suitable for polynomials up to degree 4. It is possible to create a (n+1)x(n+1) matrix capable of handling polynomials up to degree n.

Proof:

Exercise (hint: use falling powers).

Question: Is there a compact way ( recursive, perhaps ) of describing the matrix capable of handling polynomials up to degree n?

Absolutely. First, notice that the space of polynomials (infinite dimensional, of course), call it V, forms a vector space over your base field, say the field K. Let V(n) be the subspace of V consisting of polynomials of degree <= n. Define differentiation on V(n) in the usual sense. Then differentiation becomes a linear operator on V(n), call it D. Now, V(n) is finite dimensional (of dimension n + 1), so the matrix representation of D is (n + 1)x(n + 1), and will handle any polynomial in V(n).

ReplyDeleteCheers!

William