The effectiveness of several iterative techniques for solving matrix equations resulting from finite difference approximations to self-adjoint parabolic and elliptic partial differential equations is reviewed. The techniques include Stone's Strongly Implicit Procedure (SIP) and several conjugate gradient algorithms with varying preconditioners. The comparison is made on a vector machine (two-pipe Cyber 205) where vectorization of the code is done primarily by the vector machine compiler available. It is found that of the methods studied, POLCG (Polynominal Preconditioned Conjugate Gradient Method) and MICCG (Modified Incomplete Cholesky Conjugate Gradient Method) appear to require the least amount of central processing time. An advantage of MICCG and POLCG is that it is less sensitive to increasing matrix size. Its disadvantages are that it requires an iteration parameter, has a greater set-up time, and needs more storage than POLCG. (kr)