The success and importance of dense linear algebra libraries such as BLAS and LAPACK in high performance computing can be seen from Jack Dongarra’s receipt of the ACM Turing award. However, in the modern era of low/mixed-precision computing, it does not make sense to compute such dense linear algebra operations with exact algorithms that require O(N³) compute and O(N²) memory. The purpose of this research project is to replace BLAS and LAPACK with approximate dense linear algebra methods using H-matrices, which can reduce the compute and memory complexity to O(N).