About iterative refinement to the solution of the linear equations I want to know what is iterative refinement for improving the solution to the linear equations? How they improve solutions and what are the various techniques for the iterative refinements? 
Any references and basic ideas will be helpful for me to understand this concept.
Thank you very much for your time and suggestions.
 A: It seems intuitive that if $\tilde x$ is an approximation to the solution $x$ of $Ax = b$ and the residual vector $r = b -A \tilde x$ has the property that $||r||$ is small, then $||x-\tilde x||$ would be small too. This is often true, but for certain systems that occur frequently in practice, fail to have this property.
If we are solving the system:
$$Ay = r$$
The solution to this system can be readily approximated, since the multipliers for the Gaussian have already been calculated and presumably retained. In fact $\tilde y$, the approximate solution of $Ay = r$, satisfies:
$$\displaystyle \tilde y \approx A^{-1}r = A^{-1}(b - A \tilde x) = A^{-1}b-A^{-1}A \tilde x = x - \tilde x$$
So, $\tilde y$ is an estimate of the error in approximating the solution to the original system. This provides an approximate number for the condition number, given by:
$$K(A) \approx \frac{||\tilde y||}{||\tilde x||} 10^{t}$$
Note, the proof for the $10$ term can be found in Forsythe and Moler.
Iterative Refinement is used with several different Gaussian elimination methods that can include:


*

*Iterative Refinement Algorithm 

*Crout and Dolittle Factorization

*Dolittle Factorization

*Factorization With Pivoting

*Crout Factorization

*Solutions to systems using Triangularization


You should be able to find more details in any book on Numerical Analysis, like "Analysis of Numerical Methods" by Isaacson and Keller. You can also look into the references in the Wiki Iterative Refinement page.
