Numerically solving polynomial systems in the past meant computing approximations to all isolated solutions. Based on the blackbox solver of PHCpack, new tools were developed to compute a numerical irreducible decomposition. These toolboxes implement the algorithms which define the field of numerical algebraic geometry. Nowadays, solving a polynomial system numerically includes giving accurate and reliable descriptions of positive dimensional solution sets and their multiplicities.
Interactions between Classical and Numerical Algebraic Geometry. A conference in honor of Andrew J. Sommese. University of Notre Dame, 22-24 May 2008.