EFFICIENT REDUCTION OF BANDED HERMITIAN POSITIVE DEFINITE GENERALIZED EIGENVALUE PROBLEMS TO BANDED STANDARD EIGENVALUE PROBLEMS

We present a method for reducing the generalized eigenvalue problem Ax = Bx lambda with banded hermitian matrices A, B, and B positive definite to an equivalent standard eigenvalue problem Cy = y lambda, such that C again is banded. Our method combines ideas of an algorithm proposed by Crawford in 1973 and LAPACK's reduction routines _{SY,HE}GST and retains their respective advantages, namely, being able to rely on matrix-matrix operations (Crawford) and to handle split factorizations and different bandwidths b(A) and b(B) (LAPACK). In addition, it includes two algorithmic parameters (block size, n(b), and width of blocked orthogonal transformations, w) that can be adjusted to optimize performance. We also present a heuristic for automatically determining suitable values for these parameters. Numerical experiments confirm the efficiency of our new method.