  Numerical methods are developed in the quantum transport calculations for
electron in the waveguides with spin-orbital (Rashba) interaction. The methods
are based on a hybrid mode-matching scheme in which the wavefunctions are
expressed as the superposition of eigenmodes in the lead regions and in the
device region the wavefunction is expressed on the discrete basis. Two versions
are presented for the lead without and with the Rashba interaction. In the
latter case the eigenmodes are obtained from a quadratic eigenproblem
calculation. These methods are suitable for the systems with variable
geometries or arbitrary potential profiles. The computation can be effectively
accelerated by the sparse matrix technique. We also investigate the Fano-Rashba
bound states in the Rashba waveguides by some nonlinear eigenstate calculation.
This calculation is based on a mode-matching method and self-consistent results
are obtained in our calculations.
