mhodlr: Matrix computations in HODLR representation

Differential equations often result in rank-structured matrices associated with low-rank off-diagonal blocks. These matrices are often represented in a hierarchical format, and their operation often results in fast arithmetic, e.g., matrix-vector product. The hierarchical matrix [2] is a class of dense rank-structured matrices with a hierarchical low-rank off diagonal block structure, which frequently arises from finite element discretization of an elliptic PDE, radial basis function interpolation, and boundary integral equations. Hierarchical Off-Diagonal Low-Rank (HODLR) matrix, as a typical hierarchical matrix, is formulated by hierarchically partitioning the matrix in terms of a binary cluster tree and all off-diagonal blocks of each level of the tree are represented as low-rank matrices.

It is widely known that low precision can reduce data communication and be more energy- and storage-efficient. Regarding the IEEE standard for floating point, single precision arithmetic can be twice as fast as double precision on specific hardware and the half-precision arithmetic achieves 4 times speedup over double precision. Using the software mhodlr, one can know what precisions are required for the HODLR matrix construction by evaluating their reconstruction error and computations error by simulating various precisions. This repository is concerned with HODLR matrix construction as well as basic matrix computations with HOLDR matrices, which aims to provide a convenient API for HODLR operations and efficient simulations for mixed-precision and adaptive precision HODLR matrix computing [1]. Our low precision arithmetic is simulated in terms of [4].