This thesis focuses on some of the numerical aspects of the treatment
of the electronic structure problem, in particular that of determining
the ground state electronic density for the non–equilibrium
Green’s function formulation of two–probe systems and the calculation
of transmission in the Landauer–Büttiker ballistic transport
regime. These calculations concentrate on determining the so–
called Green’s function matrix, or portions thereof, which is the inverse
of a block tridiagonal general complex matrix.
To this end, a sequential algorithm based on Gaussian elimination
named Sweeps is developed and compared to standard Gaussian
elimination, where it is shown to be qualitatively quicker for the
task of determining the block tridiagonal portion of the Green’s
function matrix. The Sweep algorithm is then parallelized via a
straightforward approach in order to enable moderate speedup and
memory distribution.
The well known block cyclic reduction algorithm first developed by
Gene Golub is then presented and analyzed for further expanding
our parallel options, and finally a new hybrid method that combines
block cyclic reduction and a form of Schur complement calculation
is introduced.
The parallel algorithms are then benchmarked and the new hybrid
method is shown to possess promising speedup characteristics for
common cases of problems that need to be modeled.