  Block Floating Point (BFP) arithmetic is currently seeing a resurgence in
interest because it requires less power, less chip area, and is less
complicated to implement in hardware than standard floating point arithmetic.
This paper explores the application of BFP to mixed- and progressive-precision
multigrid methods, enabling the solution of linear elliptic partial
differential equations (PDEs) in energy- and hardware-efficient integer
arithmetic. While most existing applications of BFP arithmetic tend to use
small block sizes, the block size here is chosen to be maximal such that
matrices and vectors share a single exponent for all entries. This is sometimes
also referred to as a scaled fixed-point format. We provide algorithms for
BLAS-like routines for BFP arithmetic that ensure exact vector-vector and
matrix-vector computations up to a specified precision. Using these algorithms,
we study the asymptotic precision requirements to achieve
discretization-error-accuracy. We demonstrate that some computations can be
performed using as little as 4-bit integers, while the number of bits required
to attain a certain target accuracy is similar to that of standard floating
point arithmetic. Finally, we present a heuristic for full multigrid in BFP
arithmetic based on saturation and truncation that still achieves
discretization-error-accuracy without the need for expensive normalization
steps of intermediate results.
