On direct elimination methods for solving the equality constrained least squares problem

Abstract

Two closely related methods for solving the least squares problem with equality constraints (LSE) are considered. The first is the direct elimination (DE) method that is implemented using Modified Gram-Schmidt (MGS) for minimizing ‖Ax − c‖2 subject to the constraints ‖Bx − d‖2 = min. The second is the MGS for solving the following weighted least squares problem where we take the limit τ → ∞ analytically. The two methods are proved to be equivalent. We provide the backward and forward round-off error analysis for the DE method via MGS, and illustrate our results with numerical experiments.