The Laguerre-and-sums-of-powers algorithm for the efficient and reliable approximation of all polynomial roots

We prove the first sufficient convergence criterion for Laguerre's root-finding algorithm, which by empirical evidence is highly efficient. The criterion is applicable to simple roots of polynomials with degree greater than 3. The “Sums of Powers Algorithm” (SPA), which is a reliable iterative root-finding method, can be used to fulfill the condition for each root. Therefore, Laguerre's method together with the SPA is now an efficient and reliable algorithm (LaSPA). In computational mathematics these results solve a central task which was first attacked by L. Euler 266 years ago.