The ability of the transfer-matrix methodology to efficiently simulate quantum-mechanical field emission from nanotips is demonstrated. This general methodology, well suited to the peculiar geometry of field emission systems where electrons tunnel from an emitter to a collector, is limited in use when numerical instabilities are not controlled. In this article, the method is extended to deal with representations implying rectangular transfer matrices and to provide results with an explicit accuracy evaluation. The usefulness of the layer addition procedure for improving the accuracy and the range of validity of the method is then clearly established. This accuracy evaluation extension enables computations with optimal accuracy. For the specific field emission situation, this methodology is applied with an adequate wave-function representation that takes advantage of a possible n-fold symmetry axis. Field emission computations are presented. The electronic source is represented by a pyramidal protrusion that consists of four atomic layers. A circular aperture is placed on the collector, facing the tip. The modifications in the resulting diffraction figures, induced by successive removal of the constitutive layers of the tip, reveal the necessity of a monoatomic termination for applications in the Fresnel projection microscope.