Roundoff error analysis of the Cholesky QR2 algorithm

We consider the QR decomposition of an m × n matrix X with full column rank, where m × n. Among the many algorithms available, the Cholesky QR algorithm is ideal from the viewpoint of high performance computing since it consists entirely of standard level 3 BLAS operations with large matrix sizes, and requires only one reduce and broadcast in parallel environments. Unfortunately, it is well-known that the algorithm is not numerically stable and the deviation from orthogonality of the computed Q factor is of order O((κ₂(X))²u), where κ₂(X) is the 2-norm condition number of X and u is the unit roundoff. In this paper, we show that if the condition number of X is not too large, we can greatly improve the stability by iterating the Cholesky QR algorithm twice. More specifically, if κ₂(X) is at most O(u^-1/2 ), both the residual and deviation from orthogonality are shown to be of order O(u). Numerical results support our theoretical analysis.