  Laguerre's rootfinding algorithm is highly recommended although most of its
properties are known only by empirical evidence. In view of this, we prove the
first sufficient convergence criterion. It is applicable to simple roots of
polynomials with degree greater than 3. The "Sums of Powers Algorithm" (SPA),
which is a reliable iterative rootfinding method, can be used to fulfill the
condition for each root. Therefore, Laguerre's method together with the SPA is
now a reliable algorithm (LaSPA). In computational mathematics these results
solve a central task which was first attacked by L. Euler 266 years ago. In
order to study convergence properties, we eliminate the polynomial and its
derivatives in the definition of the Laguerre iteration, replacing them by
sums, only depending on the roots and the iterated values. For this iteration
with roots, the above criterion of convergence represents an efficient stopping
condition. In this way, we visualize convergence properties by coloring small
neighbourhoods of each starting point in squares.
