Article

A recursive algorithm for computation of the elements of T-1 required for inverting the numerator relationship for genotyped animals

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Abstract

The use of big "omics" data for genomic selection is becoming common place in large scale genetic evaluation systems. The computational requirements of genomic model is prohibitive. More efficient algorithms are required to facilitate integration of "omics" data in genomic prediction models. Reduced linear systems (e.g., Reduced Animal Model - RAM) have proven to be a viable alternative for efficient computation. We propose a recursive algorithm inspired by RAM which provide a computationally efficient strategy for solving local linear systems of the form Zb=y which arise in the inversion of the matrix T used to construct the inverse of the numerator relationship matrix (A_22^(-1)) for genotyped animals. Unlike direct inversion or global matrix factorization (e.g., Cholesky Decomposition) the proposed recursive algorithm decomposes the problem into a sequence of localized absorptions within minimal subsets of related animals (MinSets). Each genotyped animal's path coefficients are obtained by recursively updating conditional relationships, exploiting the inherent sparsity and block structure of the numerator relationship matrix for genotyped animals. This study demonstrates analytically that the proposed recursive algorithm is mathematically equivalent to a block-wise Cholesky factorization of A_22 where each recursive absorption step corresponds to a local elimination in the lower triangular matrix L satisfying A=L^' L. Through symbolic derivations and Schur complement analysis, we show that the recursive update of path coefficients is identical to the forward-backward substitution used in Cholesky-based solvers but implemented dynamically within adaptive sparse blocks. The proposed recursive algorithm is a generalized, block-sparse extension of Cholesky decomposition, optimized for large-scale genomic and pedigree-based computations.

Keywords: 2026

How to Cite:

Maiwashe, A., (2026) “A recursive algorithm for computation of the elements of T-1 required for inverting the numerator relationship for genotyped animals”, World Congress on Genetics Applied to Livestock Production Digital Archive 2026(1): 2286364. doi: https://doi.org/10.31274/wcgalp.24589