In this paper a novel preconditioning strategy is presented that is designed to improve the convergence rates of the Generalized Minimal Residual (GMRES) method when applied to dense linear systems of boundary element equations of the form Hz ----c. The GMRES method is applied to the preconditioned system (D + L)-1 12Hx = (D + L)-112c, where D --diag(H), L is the strictly lower triangular part of 12H and 12 is a sparsely populated upper triangular matrix. The coefficients in 12 are determined via the minimization of the square of the Frobenius norm II u + D-D II F, where U is the strictly upper triangular part of 12H and D --diag(12H). Several proofs are given to demonstrate that minimizing II u + D-D II2 provides for improved conditioning and consequently faster convergence rates. Numerical experiments are performed on systems of boundary element equations generated by three-dimensional potential and elastostatic problems. Computation times are determined and compared against those for Jacobi preconditioned GMRES, preconditioned Gauss-Seidel and Gaussian elimination. Moreover, condition numbers are noted and up to 100-fold reductions are observed for the systems tested. ยข 1997 Elsevier Science B.V.