I have $d$ real numbers $x_1,\dotsc,x_d$. I want to efficiently calculate $y_1,\dotsc,y_d$ defined as follows. $$y_e=\sum_{1\le i_1<i_2<\cdots<i_e\le d}x_{i_1}\cdots x_{i_e}.$$ For example, $y_2$ can be calculated efficiently as follows. $$ y_1=x_1+\cdots+x_d,\\ y_2=\big((x_1+\cdots+x_d)^2-(x_1^2+\cdots+x_d^2)\big)/2. $$ Is there any efficient way to calculate $y_3,\dotsc,y_d$?
Asked
Active
Viewed 27 times
1
-
My guess would be no, because (if I am not mistaken), what you are referring to is one of the coefficients in the polynomial $$\prod_{i=1}^d (x - x_i).$$ Assuming that I am right, I have never heard of a shortcut to computing the polynomial by simply multiplying out all of the factors. I am (naturally) outlawing the use of any computer software to aid in the computations. – user2661923 Jan 27 '22 at 20:17