  Since numbers in the computer are represented with a fixed number of bits,
loss of accuracy during calculation is unavoidable. At high precision where
more bits (e.g. 64) are allocated to each number, round-off errors are
typically small. On the other hand, calculating at lower precision, such as
half (16 bits), has the advantage of being much faster. This research focuses
on experimenting with arithmetic at different precision levels for large-scale
inverse problems, which are represented by linear systems with ill-conditioned
matrices. We modified the Conjugate Gradient Method for Least Squares (CGLS)
and the Chebyshev Semi-Iterative Method (CS) with Tikhonov regularization to do
arithmetic at lower precision using the MATLAB chop function, and we ran
experiments on applications from image processing and compared their
performance at different precision levels. We concluded that CGLS is a more
stable algorithm, but overflows easily due to the computation of inner
products, while CS is less likely to overflow but it has more erratic
convergence behavior. When the noise level is high, CS outperforms CGLS by
being able to run more iterations before overflow occurs; when the noise level
is close to zero, CS appears to be more susceptible to accumulation of
round-off errors.
