Technical note: A successive over-relaxation preconditioner to solve mixed model equations for genetic evaluation

Area of Science:

• Animal Genetics
• Computational Biology
• Numerical Analysis

Background:

• Mixed model equations are crucial for genomic evaluations in animal breeding.
• Iterative methods are often used to solve these large-scale equations.
• Computational efficiency is a key challenge in genomic prediction.

Purpose of the Study:

• To describe a computationally efficient preconditioned conjugate gradient algorithm.
• To evaluate the performance of a symmetric successive over-relaxation (SSOR) preconditioner for solving mixed model equations.
• To assess the computational savings for single-step genomic evaluation in Australian sheep.

Main Methods:

• Development of a preconditioned conjugate gradient algorithm.
• Implementation of a symmetric successive over-relaxation (SSOR) preconditioner.
• Application to single-step genomic evaluation data from Australian sheep.

Main Results:

• The SSOR preconditioner substantially reduced the number of iterations for convergence.
• Significant reductions in overall computing time were observed compared to simpler preconditioners.
• The algorithm demonstrated computational efficiency for large-scale genomic evaluations.

Conclusions:

• The SSOR preconditioned conjugate gradient algorithm offers a computationally efficient solution for mixed model equations.
• This approach provides substantial computational savings for genomic evaluations, particularly in large populations.
• The findings have implications for accelerating genomic selection in livestock breeding programs.