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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.

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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.

I have heard 2 and 3 both called "modified Richardson iteration" but also just Richardson iteration

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  • $\begingroup$ Thanks for your clear reply, get it. I agree with you that the 2 and 3 are extrapolative version or called modified/damped/ weighted Richardson iteration. The original one must be the first version without parameters. $\endgroup$ – sunshine Dec 5 '19 at 6:09

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