Fast Plane Rotations with Dynamic Scaling

This paper presents fast plane rotations for orthogonal similarity and orthogonal one-sided transformations. Fast rotations have the advantage that they reduce the number of square roots and multiplications. The authors’ new rotations have further advantages over the existing fast rotations: they obviate the rescaling that has been necessary to guard against underflow or overflow and they give higher efficiency, especially on vector processors. An error analysis, in the case of the $QR$ decomposition, and computational results that illustrate the effects of the dynamic scaling are presented.