Merge pull request #768 from hly1204/master

Oleksandr Kulkov · web-flow · commit d617dd81f9f9 · 2022-01-09T13:13:58.000+01:00
add another method of czt
diff --git a/src/algebra/polynomial.md b/src/algebra/polynomial.md
@@ -201,6 +201,11 @@ Now if you need to evaluate a polynomial in the points $x_r = z^{2r+1}$ you can
 
 It gives us an $O(n \log n)$ algorithm when you need to compute values in powers of $z$, thus you may compute the DFT for non-powers of two.
 
+Another observation is that $kr = \binom{k+r}{2} - \binom{k}{2} - \binom{r}{2}$. Then we have
+
+$$\boxed{A(z^r) = z^{-\binom{r}{2}}\sum\limits_{k=0}^n \left(a_k z^{-\binom{k}{2}}\right)z^\binom{k+r}{2}}$$
+
+The coefficient of $x^{n+r}$ of the product of the polynomials $A_0(x) = \sum\limits_{k=0}^n a_{n-k}z^{-\binom{n-k}{2}}x^k$ and $A_1(x) = \sum\limits_{k\geq 0}z^{\binom{k}{2}}x^k$ equals $z^{\binom{r}{2}}A(z^r)$. You can use the formula $z^\binom{k+1}{2}=z^{\binom{k}{2}+k}$ to calculate the coefficients of $A_0(x)$ and $A_1(x)$.
 
 ### Multi-point Evaluation
 Assume you need to calculate $A(x_1), \dots, A(x_n)$. As mentioned earlier, $A(x) \equiv A(x_i) \pmod{x-x_i}$. Thus you may do the following:
