Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).


There is a large amount of algorithms for symmetric problems (conjugate gradient) and square non symmetric ones (BICGstab).
But I have difficulties to find a method for both at ones.


I recently saw this artivle <http://dl.acm.org/citation.cfm?id=355989> where the problem is restated as
$$\begin{pmatrix}I & A \\ A^T & -\lambda I \end{pmatrix} \begin{pmatrix}r \\x \end{pmatrix}=\begin{pmatrix}b\\0\end{pmatrix}$$
with a small Tikhonov regularization weight $\lambda$ and $r$ as the residuum.


Does anyone know a better approach for this? I would like to avoid storing the data in $A$ twice.


Thank you very much!



Large scale least squares of non symmetric and non square problems
A widely used iterative method for large scale linear least squares problems that allows for regularization if you want/need it is the LSQR algorithm of Paige and Saunders. See
<http://www.stanford.edu/group/SOL/software/lsqr/>


