The most primitive numerical method of solving the radiative transfer equation forms the basis of an algorithm to generate high quality numerical solutions. We apply convergence acceleration to the discrete ordinates balance equation derived from a diamond difference approximation. Wynn-epsilon and Romberg accelerations then rearrange low order solutions to produce a high order solution. Published semianalytical benchmarks from the C.E. Siewert portfolio establish the credibility of the new converged discrete ordinates method (CDOM).