Finding the Smallest Eigenvalue by Properties of Semidefinite Matrices

We consider the smallest eigenvalue problem for symmetric or Hermitian matrices by properties of semidefinite matrices. The work is based on a floating-point Cholesky decomposition and takes into account all possible computational and rounding errors. A computational test is given to verify that a given symmetric or Hermitian matrix is not positive semidefinite, so it has at least one negative eigenvalue. This criterion helps us to find the smallest eigenvalue and singular value. Computational examples show that these results can be quite accurate.