In research of the RepRap Mendel, I found the resolution of this printer to be 100 microns. How many cycles of the stepper motor are required to reach this resolution, and can this be reduced through the software to improve the resolution of the device?

Replacing the steppers with 0.9° versions would double up the resolution.
Using smaller GT2-pulleys where possible would help, too ( 16 tooth vs 20 tooth )
Or leadscrews with finer pitch.
Then there are stepper drivers with upto 128 microstepping vs 16 microsteps, but that's no reliable improvement.
You have to make sure, your mechanics run without play, backlash or binding to help steppers to really reach ( and hold ) these fine microsteps.

Edited 1 time(s). Last edit at 11/25/2018 03:17AM by o_lampe.

@LVerh

X or Y steps/mm = {motor steps per rev * driver microstep} / {belt pitch * pulley number of teeth)

Most reprap steppers are 200 steps/revolution (1.8 degree)
Most use 16 microstep
belt pitch is 2mm
pully with 20 teeth

(200 * 16) / (2 * 20) = 80.0 steps/mm

so theoretical resolution of 1/80mm ie 0.0125 per step. But you can't get accurate positioning with microstepping. Use a max of 1/4 micro stepping for accurate positioning.

no this cannot be improved with software, you need to change the hardware

use 0.9 degree steppers, or geared steppers
use pulleys with less teeth eg 16