QMR

[platawiec]

I've finished a bare-bones QMR implementation. No pre-conditioning or look-ahead Lanczos (yet). Please have a look and let me know what you think.

I've mostly followed the style for MINRES. I split the Lanczos bi-orthogonalization process from the QMR iteration, in keeping with how the algorithms are abstracted in the original papers.

This algorithm should work with GPUs though it remains untested.

[codecov-io]

Codecov Report

Merging #255 into master will increase coverage by <.01%. The diff coverage is 90.56%.

@@            Coverage Diff             @@
##           master     #255      +/-   ##
==========================================
+ Coverage   90.52%   90.53%   +<.01%     
==========================================
  Files          17       18       +1     
  Lines        1077     1130      +53     
==========================================
+ Hits          975     1023      +48     
- Misses        102      107       +5

Impacted Files	Coverage Δ
src/qmr.jl	90.56% <90.56%> (ø)

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[platawiec]

Ref Issue #1

Added a bit of documentation and renamed the iterable construction to keep in line with the syntax for other methods.

[mschauer]

Thank you. How would you assess this code yourself? Do you think it is good to go once checked?

[platawiec]

Yes I think so. I haven't built the documentation so I don't know if there will be any issues there, but the core code works with minimal allocations. I have used it successfully where idrs was failing.

As I said, it is missing preconditioners, but so are other methods.

There could be an argument made for eliminating the (very small) vectors storing the relevant Hessenberg and right-hand-side values (H and g in my implementation) and just storing them directly in QMRIterable, but I chose to follow the pattern from MINRES for consistency (fields H and rhs in MINRESIterable). Overhead for any of these smaller design decisions is trivial compared to the matrix-vector product anyways.

One potentially dicey issue is the convergence for low-precision dense matrices. In the tests for dense matrices, I relaxed the tolerance by a factor of 10. Even though the solve would show up as converged, the residual test when calculated directly for Float32 and Complex{Float32} would sometimes fail. I chalk it up to numerical issues and the fact that we are not explicitly tracking the residual. If this is a bigger issue than I think it is we can discuss a strategy to mitigate.

[mschauer]

Thank you, this is really helpful

[platawiec]

Can this be merged? Are there any open items left to address?