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swansoft cnc simulator windowsQ:

Efficient decomposition of one in a pair of symmetric matrices

Given $n$ positive real numbers $a_1, \ldots, a_n$, is there a fast way of decomposing any one of them into a sum of two positive reals $c$ and $d$ such that $c^2+d^2 = a^2$? In other words, is there a $O(n \log^2 n)$ algorithm that approximates $a^2$ as a sum of $c$ and $d$ within an additive $O(\log n)$ factor?

This problem arose in a recent paper, and it is mentioned in Section 2.3 as one of five conjectures. Since this problem is pretty basic, I guess there should be an easy solution. Any help would be appreciated.

A:

Define $x_i=a_i^2+a_n-i+1^2$. Then $x_i=c^2+d^2$, and by symmetry we have that

$$a_i=\sqrt{x_

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