Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part II

In Part I of this report [2], we proposed to build a toolbox for solving the dense nonsymmetric eigenvalue problem on distributed memory parallel computers. Part I described basic building blocks, and their uses in different combinations, for extracting the eigenvalues and/or their corresponding invariant subspaces of a nonsymmetric matrix. However, the numerical accuracy and stability of some of our proposed building blocks, such as the matrix sign function, are poorly understood. Part II of this report focusses on the numerical analysis of these building blocks. We discuss perturbation theory of the matrix sign function, the conditioning of its computation, the numerical stability of the overall algorithm, and iterative refinement schemes. Numerical examples are also presented. An extension of the matrix sign function based algorithm to compute left and right deflating subspaces for the generalized eigenvalue problem of a regular matrix pencil is also described