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The accumulation of rounding errors in methods used to compute the projection of a point onto a linear manifold is studied. The methods are based on modified Gram-Schmidt, Householder, and Givens orthogonal factorizations. at the least distance from an assigned point p, i.e., the solution to the problem where the n x m real matrix A and the real m-vector b are known, and the n-vector x is unknown. In this paper, the Euclidean norm for the vectors and the corresponding induced norm for thedoi:10.1016/0024-3795(86)90141-2 fatcat:f4d3bue6bbgnhnpwhudh4lse3q