Diagonalization of a complex Hermitian matrix with a band structure

Abstract

Presented here is a FORTRAN program called HBAND which reduces to tridiagonal form relatively large complex Hermitian matrices with a band structure. Numerical computations with Hermitian matrices of dimension ∼4000×4000 indicate the execution time is about four times greater than that required to bring an equivalent real symmetric matrix with a band structure to tridiagonal form using the IMSL routine EIGBS. Additional tests performed with much smaller matrices indicate HBAND is about three times faster than the IMSL routine EHOUSH which is used for general complex Hermitian matrices. The algorithm used in HBAND is discussed and an example is given of how a band Hermitian matrix arises by numerically solving, with the finite difference technique, the Schrodinger equation with imaginary terms. A listing of HBAND written in complex double-precision arithmetic is given along with a short program that illustrates how HBAND is used by diagonalizing a simple test matrix.