The condition number of a root $r$ of a polynomial $p$ is $$ \kappa := \frac{\left\| p \right\|}{|rp'(r)|} $$ There is some arbitrariness in the choice of norm. This suggests taking the largest coefficient of the polynomial. This is straightforward to compute, in Python or any language for that matter. A more sophisticated norm considers perturbations of the ...


Quadrature points $x_1,x_2$ are also unknown. You need two more equations corresponding to $x^2$ and $x^3$. Then you solve for $a_1, a_2, x_1, x_2$.


The highest degree of polynomial is represented in sequence : (n^-3* n^n, n^-2n^n, n^-1n^n, n^0 n^n, n^1n^n, n^2 n^n, n^5n^n, n^7 *n^n,... etc), where k is the exponent. K is variable odd increasing function depends on the value of n which is K = 1,3,5,7,9.. etc for (n=4,5,6,7,.. etc) respectively.


As for how to create a linear index for the polynomial terms, let's consider an arrangement of terms that works nice for deduction. The terms for each dimension are enumerated as $a,b,c,\dots$. A one dimensional polynomial is straightforward $$ \begin{matrix} 0 & 1 & 2 & 3 & 4 & \dots \\ \hline 1 & a & a^2 & a^3 & a^4 &...

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