Linearly scaling direct method for accurately inverting sparse banded matrices

In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main diagonal, need often to be inverted in order to solve the associated linear system of equations. In this work, we introduce a new O(n) algorithm for solving such a system, being n X n the size of the matrix. We produce the analytical recursive expressions that allow to directly obtain the solution, as well as the pseudocode for its computer implementation. Moreover, we review the different options for possibly parallelizing the method, we describe the extension to deal with matrices that are banded plus a small number of non-zero entries outside the band, and we use the same ideas to produce a method for obtaining the full inverse matrix. Finally, we show that the New Algorithm is competitive, both in accuracy and in numerical efficiency, when compared to a standard method based in Gaussian elimination. We do this using sets of large random banded matrices, as well as the ones that appear when one tries to solve the 1D Poisson equation by finite differences.Comment: 24 pages, 5 figures, submitted to J. Comp. Phy

Common variants on chromosome 2 and risk of primary open-angle glaucoma in the Afro-Caribbean population of Barbados

Primary open-angle glaucoma (POAG) is the second leading cause of blindness worldwide. Although a number of genetic loci have shown association or genetic linkage to monogenic forms of POAG, the identified genes and loci do not appear to have a major role in the common POAG phenotype. We seek to identify genetic loci that appear to be major risk factors for POAG in the Afro-Caribbean population of Barbados, West Indies. We performed linkage analyses in 146 multiplex families ascertained through the Barbados Family Study of Glaucoma (BFSG) and identified a strong linkage signal on chromosome 2p (logarithm of odds score = 6.64 at θ = 0 with marker D2S2156). We subsequently performed case-control analyses using unrelated affected individuals and unaffected controls. A set of SNPs on chromosome 2p was evaluated in two independent groups of BFSG participants, a discovery group (130 POAG cases, 65 controls) and a replication group (122 POAG cases, 65 controls), and a strong association was identified with POAG and rs12994401 in both groups (P < 3.34 E−09 and P < 1.21E−12, respectively). The associated SNPs form a common disease haplotype. In summary, we have identified a locus with a major impact on susceptibility to the common POAG phenotype in an Afro-Caribbean population in Barbados. Our approach illustrates the merit of using an isolated population enriched with common disease variants as an efficient method to identify genetic underpinning of POAG