A generalized-Laguerre-Hermite pseudospectral method is proposed for
computing symmetric and central vortex states in Bose-Einstein
condensates (BECs) in three dimensions with cylindrical symmetry.  The
new method is based on the properly scaled generalized-Laguerre \&
Hermite functions and a normalized gradient flow.  It enjoys three
important advantages: (i) it reduces a three dimensional (3D) problem
with cylindrical symmetry into an effective two-dimensional (2D)
problem; (ii) it solves the problem in the whole space instead of in a
truncated artificial computational domain; and (iii) it is spectrally
accurate. Extensive numerical results for computing symmetric and
central vortex states in BECs are presented for one-dimensional (1D)
BEC, 2D BEC with radial symmetry and 3D BEC with cylindrical symmetry.