We present a novel approach to the calculation of the Coulomb and exchange contributions to the total electronic energy in self consistent field and density functional theory. The numerical procedure is based on the Cholesky decomposition and involves decomposition of specific Hadamard product matrices that enter the energy expression. In this way, we determine an auxiliary basis and obtain a dramatic reduction in size as compared to the resolution of identity (RI) method. Although the auxiliary basis is determined from the energy expression, we have complete control of the errors in the gradient or Fock matrix. Another important advantage of this method specific Cholesky decomposition is that the exchange energy and Fock matrix can be evaluated with a linear scaling effort contrary to the RI method or standard Cholesky decomposition of the two-electron integral matrix. The methods presented show the same scaling properties as the so-called local density fitting methods, but with full error control.