Boundary Approximation Method (BAM), or the Collocation Trefftz Method called in the literature, is the most efficient method to solve elliptic boundary value problems with singularities. There are several versions of BAM in practical computation, including the Numerical BAM, Symbolic BAM and their variants. It is known that the Symbolic BAM is much slower than Numerical counterpart. In this thesis, we improve the Symbolic BAM to become the fastest method among all versions of BAM. We prove several important lemmas to reduce the computing time, and a recursive procedure is found to expedite the evaluation of major integrals. Another drawback of the Symbolic BAM is its large condition number. We find a good and easy preconditioner to significantly reduce the condition number. The numerical experiments and comparison are also provided for the Motz problem, a prototype of Laplace boundary value problem with singularity, and the Schiff's Model, a prototype of biharmonic boundary value problem with singularity.