We review the theory of condition numbers for the nonsymmetric eigenproblem and give a tabular summary of bounds for eigenvalues, means of clusters of eigenvalues, eigenvectors, invariant subspaces, and related quantities. We describe the design of new algorithms for estimating these condition numbers. Fortran subroutines implementing these algorithms are in the LAPACK library [1].

The goal of the LAPACK project is to design and implement a portable linear algebra library for efficient use on a variety of high-performance computers. The library is based on the widely used LINPACK and EISPACK packages for solving linear equations, eigenvalue problems, and linear least-squares problems, but extends their functionality in a number of ways. The major methodology for making the algorithms run faster is to restructure them to perform block matrix operations (e.g., matrix-matrix multiplication) in their inner loops. These block operations may be optimized to exploit the memory hierarchy of a specific architecture. The LAPACK project is also working on new algorithms that yield higher relative accuracy for a variety of linear algebra problems.>