During these weeks I worked on improving CGS, which I rewrite from scratch. Then I implemented TFQMR, which is not available in Octave. Finally I made a revision on the codes that I improved the past weeks.

All the file revisioned (gmres, pcg, bicg, bicgstab) and the new ones (cgs and tfqmr) are available here:

https://CrisDorigo@bitbucket.org/CrisDorigo/socis16-octave-iter_meth

in the folder definitive_codes

**CGS**

As written in my second post, the actual Octave's CGS implementation is not very clear. Indeed I didn't understand

**most of the code (I was not able to recognize how the is was adapted). Moreover I tested it and the results are not correct (in my second post there is an example of this).**

Then I decided to rewrite if from scratch following closely the algorithm given by Saad in the reference book suggested.

I wrote it following the pattern that I used in the other codes (i.e. to wrote three subscripts, each for the different cases: all matrices, all functions handle or mixed). Iit has also the same characteristics of the other methods improved, for example: the approximation returned is the one with the minimum residual, there are five types of flag (0 convergence, 1 maximum number of iterations reached, 2 preconditioner singular, 3 stagnation, 4 division by zero during the process)... I don't go in deep about this details to make less boring this post, since they are already explained in my previous posts.

**TFQMR**

In my last post I wrote that the mentors told me to implement a new algorithm not available in Octave. Since I had no suggestion about which method write, and the community had no particular needs, I chose the transpose-free qmr algorithm because it is the only methods treated in the Saad book.

Also with this algorithm, I use the same pattern of the others (i.e. the three subscripts) and synchronized in such a way it has the same type of input and output as the others.

One mention to the number of iterations effectively performed. In the Saad book, the method has different behaviours if the iteration number is even or if it is odd, but at every iteration the approximation is computed. Making some test in Matlab with its tfmqr, I noticed that if the method converges, the vector of the residuals in long two times the number of iteration that Matlab tells that are necessary to convergence. Then I think that Matlab counts as one iteration the block (odd iteration + even iteration), i.e. I think that Matlab_it = odd_Saad_it + even_Saad_it. In this way the iterations performed by Matlab is two times the effective iterations performed. I decided to count the iterations in this way to be compatible.

**LAST REVISION**

Finally I checked (one more time) the codes that I improved in these weeks (i.e. pcg, gmres, bicg, bicgstab).

Indeed I updated the documentation (for example: making all similar, giving more details about the preconditioning, adding the reference book,...).

Then I checked the codes and I tried to make it more readable and to adapt it at the Octave conventions.

Then I added some tests in all the codes, mostly to check if all the subscripts works (i.e. checked every combination of A matrix/function, M1 matrix/function, M2 matrix/function), but also to check that the flags works well (I checked only the flag 0, 1 and 2, since the others are difficult to reproduce) and also to check that these methods solves also complex linear systems.

The mentors suggest me to make a patch with the methods improved (pcg, gmres, bicg, bicgstab and cgs), in such a way it can be easily the review of them and if it is in time (maybe) they can be included in Octave 4.2.

This patch is available here

https://CrisDorigo@bitbucket.org/CrisDorigo/socis16-octave-iter_meth

and it is the file SoCiS16_Improve_iterative_methods.diff

I made this patch from the Octave changeset 22406:dc4e2203cd15 of the Octave 4.1.0+ version.

**THANKS TO OCTAVE**

Since the official deadline for the SoCiS is tomorrow (the 31-th of August), this is my last post in this blog.

To make a small summary, during this summer of codes I make an improving on

most of the iterative methods to solve linear systems. In particular I improved pcg, gmres, bicg, bicgstab and cgs.

I wrote for all of these methods three subscripts to make them more efficient, dividing the particular cases "matrix", "function handle" and "mixed", in such a way that, according to the input data, the method uses the appropriate subscripts to make computations.

I tried to make these methods more compatible with Matlab (for example noticing that Matlab gives as output the approximation with the minimum residual and not the last performed, as used in Octave, or adding the possibility of "flag =2", i.e. singular preconditioner).

I tried to fix some "non-clear" situation that I noticed, for example pcg does not noticed if the matrix was not Hermitian positive definite if it was a complex matrix, or to face the problem of a division by zero in bicg, bicgstab and cgs.

Then I implemented tfqmr which is available in Matlab, but not in Octave.

Unfortunately, there are other two methods that needs an improve (pcr and qmr), but due to unexpected problems with the above methods I had not enough time to improve also these. When my scholastic commitments are not so big, I want to try to improve also these two methods.

I want to thank all the Octave community that gives me this fantastic opportunity to face a big project as this and then to learn much theoretically and practical things about these iterative solvers for linear systems.

I want to thank specially the mentors, who were always available to give suggestions, advices and ideas of how to proceed in the work, mostly when came some unexpected problems.

I hope that this improving will be useful to Octave, and to contact me if there is something not clear or wrong.

I also hope to continue to help the Octave community in the following times.

One more thanks to all of you.