I have created a Gradient Projection Method using Steepest Descent to numerically optimize minimization problems. Demonstrate this Methods effectiveness with solving Constrained Least Squares, a subclass of the minimization of residuals problem.
I also did an investigation into the Damped Spectral Square Root Preconditioning Method.
Please see the associated research paper, Constrained Least Squares Optimizations with Damped Preconditioning Methods for detailed information.
This paper also shows that controlled levels of error added to the Damped Spectral Square Root reduces the convergence rate of the Gradient Projection Method by a statistically significant factor.