  Currently, the dominating constraint in many high performance computing
applications is data capacity and bandwidth, in both inter-node communications
and even more-so in on-node data motion. A new approach to address this
limitation is to make use of data compression in the form of a compressed data
array. Storing data in a compressed data array and converting to standard
IEEE-754 types as needed during a computation can reduce the pressure on
bandwidth and storage. However, repeated conversions (lossy compression and
decompression) introduce additional approximation errors, which need to be
shown to not significantly affect the simulation results. We extend recent work
[J. Diffenderfer, et al., Error Analysis of ZFP Compression for Floating-Point
Data, SIAM Journal on Scientific Computing, 2019] that analyzed the error of a
single use of compression and decompression of the ZFP compressed data array
representation [P. Lindstrom, Fixed-rate compressed floating-point arrays, IEEE
Transactions on Visualization and Computer Graphics, 2014] to the case of
time-stepping and iterative schemes, where an advancement operator is
repeatedly applied in addition to the conversions. We show that the accumulated
error for iterative methods involving fixed-point and time evolving iterations
is bounded under standard constraints. An upper bound is established on the
number of additional iterations required for the convergence of stationary
fixed-point iterations. An additional analysis of traditional forward and
backward error of stationary iterative methods using ZFP compressed arrays is
also presented. The results of several 1D, 2D, and 3D test problems are
provided to demonstrate the correctness of the theoretical bounds.
