Fast and Flexible Program for Solving Large Mixed Model Equations

A general purpose BLUP-program (PPI) was developed, which allows a wide range of models. The method used to solve the mixed model equations (MME) is preconditioned conjugate gradient (PCG), which was proved to have a higher convergence rate than other commonly used methods, such as Gauss-Seidel method. The preconditioner applied is the block diagonal of coefficient matrix of MME, with all the fixed effects as one block. Evaluated with our testing data sets, PCG with block diagonal preconditioners converged faster than PCG with diagonal preconditioners. Setting all the fixed effects in one block made convergence faster, especially for complicated models, such as multi-trait random regression test-day model. The computational technique of iteration on data (IOD) is employed to solve the storage problem, especially for large data sets. The IOD technique reads in data file at each iteration and can avoid the storage of coefficient matrix. In our program, relationship matrix can be set to additive relationship matrix (A) built using pedigree, genomic relationship matrix (G) built using marker information, and the combined relationship matrix of A and G using both pedigree and marker information. The kinship matrices built using other information are also applicable. Based on the message interface (MPI), the program realized the parallelization of calculation of the preconditioner and PCG. More specifically, we applied shared memory model (SHM), the new approach of MPI-3 standard. SHM takes full advantage of the shared-memory architecture of computers and can accelerate the communication between processes on shared-memory nodes.