## rejtrix^{}

## Technical References

## By Rondall E. Jones, Ph.D.

1. The
automatic regularization methods in use in the package are based on determining
when the Picard condition is violated, and estimating the error in the right
hand side from the rejected equations of the SVD version of the system. The
details are in this paper from IPES 2006.

2. See
this abstract and PowerPoint presentation for a presentation on the whole
solution process for least squares, equality constraints, and inequality
constraints.

3. This
whole package is based on the excellent matrix decompositions provided by the MathWorks and NIST (see here), but modified to invert
the object-oriented design to a more typical matrix-centric form, with the
decompositions packaged as operations on a matrix, not as free standing
classes, where feasible. Also, the Matrix class is elevated to maximum
functionality rather than to a low level utility.

4. P.
C Hansens' regtools package
for MATLAB is an excellent playground for developing and tuning regularization
algorithms. See http://www.matlab.com/matlabcentral
and then search for regtools. The main automatic
regularization algorithm in rejtrix.h is available in
MATLAB form from the author of rejtrix.h. The default
auto-regularization parameters in rejtrix.h were
largely chosen by parametric comparisons with other regularization methods
(especially L-curve based methods) using regtools.

5. *Matrix Computations*, 3^{rd}
Edition, by Golub and Van Loan, is an excellent resource, and was much relied
upon in this development.

6. The
reference for the GMRES algorithm as coded herein, was *GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymetric
Linear Systems, *by Youcef Saad amd Martin Schultz, Siam J. Sci. Stat. Comput., July 1986.