Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A₁ is presented. From the eigenpairs of some real matrix A₀, the eigenpairs of A(t) ≡ (1-t) A₀ + t A₁ are followed at successive "times" from $t=0$ to $t=1$ using continuation. At $t=1$, the eigenpairs of the desired matrix A₁ are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: bifurcation, ill conditioning due to nonorthogonal eigenvectors, jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stabilityof a solution of a PDE. Some numerical results will be presented.