My german numerical-calculus-book gives an example of integration and derivation in polynomial-spaces. But I do not understand the approach.
Question 14.2
We have a polynomial with degree n and monomial basis p(x) = $\sum_{i=0}^n c_i x^i = (1,x,x^2,\ldots)\,\mathbf{c}$, $\mathbf{c} \in \mathbb{R}^{n+1}$ as well as a derivation p'(x) and an antiderivative P(x) where the constant of integration equals 0, represented by a coefficient vector for the monomial basis $\mathbf{c}, \mathbf{c'}$ and $ \mathbf{C}.$
I need to show that the coefficient vector $ \mathbf{c}'$ and $\mathbf{C}$ for $p'(x)$ and P(x) can be calculated with the coefficient vector $\mathbf{c}$ by matrix multiplication with $A_n$
and respectively $T_n$ . We get $p'(x)=(1,x,x^2,\ldots)\,\mathbf{c'} $
with $\mathbf{c'}= \mathbf{A}_n \mathbf{c}$ and respectively $P(x)=(1,x,x^2,\ldots)\,\mathbf{C}$ with $\mathbf{C}= \mathbf{T}_n \mathbf{c}$
More concrete I need help with the following steps:
- Find matrix $A_3 \in R^{3x4}$ for derivation and $T_2 \in R^{4x3}$ for integration.
- Are $A_n$ and $T_{n-1}$ in some kind related?
- Find $∫^1_0 p(x)$ as a linear vector of c. Find moreover the vector, which dot-product with c equals the integral.
How can I find the vector d so that $\mathbf{c}^\mathsf{T}\mathbf{d} = \int_0^1 p(x)$?
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Please note that my level of mathematics is still "in development": I am learning without a teacher.