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We present parallel and sequential dense QR factorization algorithms that areboth optimal (up to polylogarithmic factors) in the amount of communicationthey perform, and just as stable as Householder QR. Our first algorithm, TallSkinny QR (TSQR), factors m-by-n matrices in a one-dimensional (1-D) blockcyclic row layout, and is optimized for m >> n. Our second algorithm, CAQR(Communication-Avoiding QR), factors general rectangular matrices distributedin a two-dimensional block cyclic layout. It invokes TSQR for each block columnfactorization.

Communication-optimal parallel and sequential QR and LU factorizations: theory and practice