Coded QR Decomposition

QR decomposition of a matrix is one of the essential operations that is used for solving linear equations and finding least-squares solutions. We propose a coded computing strategy for parallel QR decomposition with applications to solving a full-rank square system of linear equations in a high-performance computing system. Our strategy is applied to the parallel Gram-Schmidt algorithm, which is one of the three commonly used algorithms for QR decomposition. Conventional coding strategies cannot preserve the orthogonality of Q. We prove a condition for a checksum-generator matrix to restore the degraded orthogonality of the decoded Q through low-cost post-processing, and construct a checksum-generator matrix for single-node failures. We obtain the minimal number of checksums required for singlenode failures under the "in-node checksum storage setting", where checksums are stored in original nodes, and further adapt the coded QR decomposition to this setting.