A Recursive Method for Solving Haplotype Frequencies in Multiple Loci Linkage Analysis

Multiple loci analysis has become popular with the advanced development in biological experiments. A lot of studies have been focused on the biological and the statistical properties of such multiple loci analysis. In this paper, we study one of the important computational problems: solving the probabilities of haplotype classes from a large linear system Ax = b derived from the recombination events in multiple loci analysis. Since the size of the recombination matrix A increases exponentially with respect to the number of loci, fast solvers are required to deal with a large number of loci in the analysis. By exploiting the nice structure of the matrix A, we develop an efficient recursive algorithm for solving such structured linear systems. In particular, the complexity of the proposed algorithm is of O(mlogm) operations and the memory requirement is of O(m) locations where m is the size of the matrix A. Numerical examples are given to demonstrate the effectiveness of our efficient solver.