In embedded code I want to efficiently compute:
$\sum_{k=1}^{N}y_k \cdot e^{a\cdot k}$,
$\sum_{k=1}^{N}y_k\cdot k \cdot e^{a\cdot k}$ and
$\sum_{k=1}^{N}y_k\cdot k^2 \cdot e^{a\cdot k}$.
Here $a$ is a scalar and $y_k$ is the k-th value in an array with $N$ elements.
These sums needs to be computed for many values of $a$.
Given that the array $y$ remains constant, is there an efficient way to do this without having to recompute the full sum of exponentials for every new value $a$?
One may see the sum $$ \sum_{k=1}^{N}y_k \cdot e^{a\cdot k} $$ as a polynomial function $$ P_N(x)=\sum_{k=1}^{N}y_k \cdot x^{k} $$ valued at $x=e^a$.
If one knows the value of a polynomial at some real number and all its derivatives at the same real number how one may obtain its value at another real number?
One may exploit the Taylor formula $$ P_N(x) = \sum \limits_{k=0}^N \dfrac{P_N^{(k)}(\alpha)(x-\alpha)^k}{k!}, \tag {*} $$ by storing all $P_N^{(k)}(\alpha)$ in a memory only once and use the above identity to get a value of $P_N$ at any $x$.