The design of advanced engineering systems generally results in large-scale numerical problems, which require efficient computational electromagnetic (CEM) solutions. Among existing CEM methods, iterative methods have been a popular choice since conventional direct solutions are computationally expensive. The optimal complexity of an iterative solver is *O(NN*_{it}N_{rhs}) with *N* being matrix size, *N*_{it }the number of iterations and *N*_{rhs} the number of right hand sides. How to invert or factorize a dense matrix or a sparse matrix of size *N* in *O(N)* (optimal) complexity with explicitly controlled accuracy has been a challenging research problem. For solving a dense matrix of size *N*, the computational complexity of a conventional direct solution is *O(N*^{3}); for solving a general sparse matrix arising from a 3-D EM analysis, the best computational complexity of a conventional direct solution is *O(N*^{2}). Recently, an *H*^{2}-matrix based mathematical framework has been developed to obtain fast dense matrix algebra. However, existing linear-complexity *H*^{2}-based matrix-matrix multiplication and matrix inversion lack an explicit accuracy control. If the accuracy is to be controlled, the inverse as well as the matrix-matrix multiplication algorithm must be completely changed, as the original formatted framework does not offer a mechanism to control the accuracy without increasing complexity.

In this work, we develop a series of new accuracy controlled fast *H*^{2} arithmetic, including matrix-matrix multiplication (MMP) without formatted multiplications, minimal-rank MMP, new accuracy controlled *H*^{2} factorization and inversion, new accuracy controlled *H*^{2} factorization and inversion with concurrent change of cluster bases, *H*^{2}-based direct sparse solver and new *HSS* recursive inverse with directly controlled accuracy. For constant-rank *H*^{2}-matrices, the proposed accuracy directly controlled *H*^{2} arithmetic has a strict *O(N)* complexity in both time and memory. For rank that linearly grows with the electrical size, the complexity of the proposed *H*^{2} arithmetic is *O(NlogN)* in factorization and inversion time, and *O(N)* in solution time and memory for solving volume IEs. Applications to large-scale interconnect extraction as well as large-scale scattering analysis, and comparisons with state-of-the-art solvers have demonstrated the clear advantages of the proposed new *H*^{2} arithmetic and resulting fast direct solutions with explicitly controlled accuracy. In addition to electromagnetic analysis, the new *H*^{2} arithmetic developed in this work can also be applied to other disciplines, where fast and large-scale numerical solutions are being pursued.