# What is the standard, extrapolation, and modified version of Richardson iteration method?

I have been studying the iterative methods recently. For classical iterative methods solving $$Ax=b$$, I have seen that the most simplest iteration method is the so-called "Richardson iteration". But I am a little confused that the following names of the "Richardson methods" as follows:

1. Richardson iteration
2. extrapolate Richardson iteration

In addition, I have found there are 3 versions of the Richardson: $$x_{k+1} = x_k+r_k;$$ $$x_{k+1} = x_k +\omega r_k;$$ $$x_{k+1} = x_k+\omega_k r_k;$$ where $$r_k = b-Ax_k$$ is k-th residual vector.

My question is: in different papers, one call different names of the Richarson iteration. For example, someone call the first version of the above 3 the Richardson. Someone call the second the Richardson. And someone call the last version the Richardson iteration. I want to get that which one is the real Richardson iteration, and what is the difference among the 3 iterative versions? In general, how should we call the 3 versions? any suggestions are welcome.

The first version only converges if $$\|I-A\|<1$$.
The second version converges if $$\|I-\omega A\|<1$$, so the parameter $$\omega$$ allows you to use the iteration on a broader class of matrices.
The third version modifies $$\omega$$ at each step to minimize the $$\|r_{k+1}\|_2$$ with respect to $$\omega_k$$. The optimal value is given by $$\omega_k = \frac{r_k^TAr_k}{\|Ar_k\|^2_2}.$$ See here for details.