BLAS IV: A BLAS for Rk Matrix Algebra

Basic Linear Algebra Subroutines (BLAS) are well-known low-level workhorse subroutines for linear algebra vector-vector, matrixvector and matrix-matrix operations for full rank matrices. The advent of block low rank (Rk) full wave direct solvers, where most blocks of the system matrix are Rk, an extension to the BLAS III matrix-matrix work horse routine is needed due to the agony of Rk addition. This note outlines the problem of BLAS III for Rk LU and solve operations and then outlines an alternative approach, which we will call BLAS IV. This approach utilizes the thrill of Rk matrix-matrix multiply and uses the Adaptive Cross Approximation (ACA) as a methodology to evaluate sums of Rk terms to circumvent the agony of low rank addition.