CholeskyQR2: A Simple and Communication-Avoiding Algorithm for Computing a Tall-Skinny QR Factorization on a Large-Scale Parallel System

Designing communication-avoiding algorithms is crucial for high performance computing on a large-scale parallel system. The TSQR algorithm is a communication-avoiding algorithm for computing a tall-skinny QR factorization, and TSQR is known to be much faster and as stable as the classical Householder QR algorithm. The Cholesky QR algorithm is another very simple and fast communication-avoiding algorithm, but rarely used in practice because of its numerical instability. Our recent work points out that an algorithm that simply repeats Cholesky QR twice, which we call CholeskyQR2, gives excellent accuracy for a wide range of matrices arising in practice. Although the communication cost of CholeskyQR2 is twice that of TSQR, it has an advantage that its reduction operation is addition whereas that of TSQR is a QR factorization, whose high-performance implementation is more difficult. Thus, CholeskyQR2 can potentially be significantly faster than TSQR. Indeed, in our experiments using 16384 nodes of the K computer, CholeskyQR2 ran about three times faster than TSQR for a 4194304 × 64 matrix.