Computing All or Some Eigenvalues of Symmetric H-Matrices

We use a bisection method, [Par80, p. 51], to compute the eigenvalues of a symmetric Hl-matrix M . The number of negative eigenvalues of M −μI is computed via the LDL T factorisation of M − μI. For dense matrices, the LDL factorisation is too expensive to yield an efficient eigenvalue algorithm in general, but not for Hl-matrices. In the special structure of Hl-matrices there is an LDL T factorisation with linear-polylogarithmic complexity. The bisection method requires only matrix-size independent many iterations to find an eigenvalue up to the desired accuracy, so that an eigenvalue can be found in linear-polylogarithmic time. For all n eigenvalues, O ( n (logn) log (‖M‖2/ǫev) ) flops are needed to compute all eigenvalues with an accuracy ǫev. It is also possible to compute only eigenvalues in a specific interval or the j-th smallest one. Numerical experiments demonstrate the efficiency of the algorithm, in particular for the case where some interior eigenvalues are required.