Mixed-precision block gram schmidt orthogonalization

The mixed-precision Cholesky QR (CholQR) can orthogonalize the columns of a dense matrix with the minimum communication cost. Moreover, its orthogonality error depends only linearly to the condition number of the input matrix. However, when the desired higher-precision is not supported by the hardware, the softwareemulated arithmetics are needed, which could significantly increase its computational cost. When there are a large number of columns to be orthogonalized, this computational overhead can have a dramatic impact on the orthogonalization time, and the mixed-precision CholQR can be much slower than the standard CholQR. In this paper, we examine several block variants of the algorithm, which reduce the computational overhead associated with the softwareemulated arithmetics, while maintaining the same orthogonality error bound as the mixed-precision CholQR. Our numerical and performance results on multicore CPUs with a GPU, as well as a hybrid CPU/GPU cluster, demonstrate that compared to the mixedprecision CholQR, such a block variant can obtain speedups of up to 7:1× while maintaining about the same order of the numerical errors.