The author provides a procedure for calculating the initial radius \(r_ 0\) to be used in obtaining the roots of \(p(z)=a_ 0z^ n+...+a_{n- 1}z+a_ n\), \(a_ 0a_ n\neq 0\) by using Durand-Kerner’s method: \(z_ k^{(0)}=r_ 0\exp (2\pi ki/n),\) \(k=0,1,...,n\) \[ z_ k^{(j+1)}=z_ k^{(j)}-p(z_ k^{(j)})/\{a_ 0\prod_{i<k}(z_ k^{(j)}-z_ i^{(j+1)})\prod_{i>k}(z_ k^{(j)}-z_ i^{(j)})\}. \] The procedure is: i) for \(i=1,...,n\) if \(a_ i\neq 0\quad u_ i=2| a_ i/a_ 0|^{1/i}\), for \(j=0,...,n-1\) if \(a_ j\neq 0\quad V_ j=1/(2| a_ j/a_ n|^{1/(n-j)})\); ii) discard \(u_ i\) and \(v_ j\) for which \(u_ i-\bar u\), and \(\bar v-v_ j\) are maximal. Recalculate \(\bar u\) and \(\bar v.\) iii) \(r_ 0=(\bar u+\bar v)/2\). The author gives four examples illustrating that his initial radius \(r_ 0\) gives an efficient procedure in terms of number of evaluations of p(z) and that his strategy is comparable to that of L. W. Ehrlich’s using \(\min \{| z| | p(z)=0\}<r_ 0<\max \{| z| | p(z)=0\}.\)