We use the angular spectrum representation to compute exactly the Gaussian beam close to the waist (w0) in the case of highly nonparaxial field (w0<λ). The computation is done in the vectorial case for a polarized Gaussian beam. In the area of the waist, the contribution of the propagating and evanescent waves is discussed. Moreover, the Gaussian wave is developed in terms of series, which permits to get analytical expressions for both propagating and evanescent waves when the observation is close to the waist.
The code can be found here plus dependencies. If you use this routine in your research, please reference, as appropriate: P. C. Chaumet, Fully vectorial highly non paraxial beam close to the waist. J. Opt. Soc. Am. A 23, 3197 (2006) and P. C. Chaumet, B. Pouligny, R. Dimova and N. Sojic, Optical tweezers in interaction with an apertureless probe J. Appl. Phys. 102, 024915 (2007).
You will find here several iterative solvers for the linear problem Ax=b. There is no condition on the matrix. Notice that the products Ax is provided by the user. All the codes are written in fortran (double precision). If you use these routines in your research, please reference, as indicated in the routine.