Self-scaling fast rotations for stiff and equality-constrained linear least squares problems

We present algorithms which apply self-scaling fast plane rotations to the QR decomposition for stiff least squares problems. We show that both fast and standard Givens rotation-based algorithms produce accurate results, regardless of row sorting and even with extremely large weights, when equality-constrained least squares problems are solved by the weighting method. Numerical test results show that the Householder QR decomposition algorithm is sensitive to row sorting and produces less accurate results when the weights are large, and that the modified Gram-Schmidt algorithm is less sensitive to row sorting. This makes the fast plane rotation a method of choice for the QR decomposition of stiff matrices, since it is also competitive in computational complexity. Based on the above results, an efficient algorithm is also derived for the application where the least squares solutions are required for various constrained matrices for each fixed data matrix.