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 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, 3rd 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.