Before revision: This paper describes and develops a fast and accurate  algorithm that computes the field of values boundary curve for every conceivable complex  square matrix A , i. e., for all matrices, those with or without repeated eigenvalues and with or without Jordan block structures, with ill-conditioned eigenvalues or ill-conditioned ones and so forth.
After revision: This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve for every conceivable complex square matrix A , i. e., for all matrices, those with or without repeated eigenvalues and with or without Jordan block structures, with ill-conditioned eigenvalues or ill-conditioned ones and so forth.