IR fix and example

[fritzgoebel]

This PR introduces the following fix to IR:
When using the advanced apply for the inner solver, the initial guess will be x. This way, the initial residual of the inner solver is r - Ax which will generally not be small but close to Ax as we try to get r to be zero. This causes very poor to no convergence when e.g. using CG with tolerance 1e-2 as inner solver. By kind of manually doing the advanced apply, we can use 'r' as initial guess, which gives much better results.

This PR also introduces a small example on iterative refinement.

[pratikvn]

Maybe we can move this from all the specific examples directories to a common directory. Now we just duplicate this matrix everywhere.

[pratikvn]

Some minor stuff needed for complex support.

[pratikvn]

LGTM!

[codecov]

Codecov Report

Merging #522 into develop will increase coverage by 0.03%.
The diff coverage is 100.00%.

@@             Coverage Diff             @@
##           develop     #522      +/-   ##
===========================================
+ Coverage    88.40%   88.44%   +0.03%     
===========================================
  Files          271      271              
  Lines        17153    17132      -21     
===========================================
- Hits         15164    15152      -12     
+ Misses        1989     1980       -9     

Impacted Files	Coverage Δ
core/solver/ir.cpp	95.12% <100.00%> (+0.38%)	⬆️
include/ginkgo/core/matrix/ell.hpp	68.75% <0.00%> (-31.25%)	⬇️
include/ginkgo/core/base/composition.hpp	89.28% <0.00%> (-3.31%)	⬇️
reference/components/format_conversion.hpp	100.00% <0.00%> (ø)
include/ginkgo/core/base/polymorphic_object.hpp	98.43% <0.00%> (+3.19%)	⬆️
include/ginkgo/core/base/lin_op.hpp	78.57% <0.00%> (+20.23%)	⬆️

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[yhmtsai]

LGTM. Could you add some comments or description like the inner solver will use residual as initial guess if needed or something else?

[hartwiganzt]

@fritzgoebel I think this is ready to merge...

[upsj]

@fritzgoebel Maybe before merging, can you do an interactive rebase onto develop and clean up the history a bit? Especially the "Update examples/*" files. In case you haven't done an interactive rebase before:

git fetch   # copy changes from remote
git rebase -i origin/develop

git then opens an editor and shows from top to bottom all changes that you did. change every one of the Update ... commit lines except for the first from pick to fixup or f, and they will be merged into a single commit. Save the file and close. While the rebase is running, you will probably be shown another editor with the first commit message of the commits you are merging, you can just close that like when committing directly.

The rebase command rewrites the git history so all changes are applied after the changes on develop. This gives us a nice clean linear history instead of many merges.

Don't worry about breaking anything, you can always reset your local branch to the remote branch that is stored on github ;)

[sonarcloud]

SonarCloud Quality Gate failed.

0 Bugs
0 Vulnerabilities (and 0 Security Hotspots to review)
10 Code Smells

6.5% Coverage
32.3% Duplication