The performance of conjugate gradient (CG) algorithms for the solution of the system of linear equations that results from the finite-differencing of the neutron diffusion equation was analyzed on SIMD, MIMD, and mixed-mode parallel machines. A block preconditioner based on the incomplete Cholesky factorization was used to accelerate the conjugate gradient search. The issues involved in mapping both the unpreconditioned and preconditioned conjugate gradient algorithms onto the mixed-mode PASM prototype, the SIMD MasPar MP-1, and the MIMD Intel Paragon XP/S are discussed. On PASM, the mixed-mode implementation outperformed either SIMD or MIMD alone. Theoretical performance predictions were analyzed and compared with the experimental results on the MasPar MP-1 and the Paragon XP/S. Other issues addressed include the impact on execution time of the number of processors used, the effect of the interprocessor communication network on performance, and the relationship of the number of processors to the quality of the preconditioning. Applications studies such as this are necessary in the development of software tools for mapping algorithms onto either a single parallel machine or a heterogeneous suite of parallel machines.