Computing humps of the matrix exponential

This work is devoted to finding maxima of the function Γ(t)=‖exp(tA)‖2 where t≥0 and A is a large sparse matrix whose eigenvalues have negative real parts but whose numerical range includes points with positive real parts. Four methods for computing Γ(t) are considered which all use a special Lanczos method applied to the matrix exp(tA∗)exp(tA) and exploit the sparseness of A through matrix-vector products. In any of these methods the function Γ(t) is computed at points of a given coarse grid to localize its maxima, and then maximized by a standard maximization procedure or via an alternating maximization procedure. Results of such computations with some test matrices are reported and analyzed.