Impoving NLS Solvers

[avik-pal]

• [x] Add a wrapper over LeastSquaresOptim
• [x] Sparse mul!(A, J', J) seems slower than A .= J' * J (@ChrisRackauckas any idea why?)
• [x] Avoid spurious repeated factorizations
• [x] Add tests

[codecov[bot]]

Codecov Report

Merging #236 (1c19fa7) into master (a6af39c) will increase coverage by 73.03%. The diff coverage is 89.28%.

@@             Coverage Diff             @@
##           master     #236       +/-   ##
===========================================
+ Coverage   20.19%   93.22%   +73.03%     
===========================================
  Files           9       12        +3     
  Lines         832      901       +69     
===========================================
+ Hits          168      840      +672     
+ Misses        664       61      -603     

Files	Coverage Δ
src/gaussnewton.jl	75.00% <100.00%> (+75.00%)	:arrow_up:
src/levenberg.jl	98.42% <100.00%> (+98.42%)	:arrow_up:
ext/NonlinearSolveLeastSquaresOptimExt.jl	95.83% <95.83%> (ø)
src/NonlinearSolve.jl	85.00% <0.00%> (+0.78%)	:arrow_up:
src/utils.jl	77.77% <50.00%> (+19.82%)	:arrow_up:
ext/NonlinearSolveFastLevenbergMarquardtExt.jl	91.66% <91.66%> (ø)
src/algorithms.jl	85.71% <85.71%> (ø)
src/jacobian.jl	86.15% <77.77%> (+5.82%)	:arrow_up:

... and 4 files with indirect coverage changes

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[ChrisRackauckas]

A .= J' * J

What's this actually used for? Generally that's a bad idea

[avik-pal]

For GaussNewton and LM if it is NLLS. For a NLProblem we can get around by dropping the $J^T$ term on both sides.

[avik-pal]

I did that dispatch for the sparse arrays case, because the in place hits the generic matmul code

[avik-pal]

For Matrices it seems to hit the correct BLAS dispatches of syrk, but doesn't do it for Banded Matrices or Sparse Matrices (there seems to be a version in Intel MKL docs but don't know how to access it)

[avik-pal]

Let's hold off a merge on this. We should be specializing the J'J matmul to use symmetric factorizations

[avik-pal]

~Should be good to go!~

While I am at it let me wrap FastLM as well

[avik-pal]

@ChrisRackauckas I will handle the J' * J thing in a later PR. The caching stuff makes the change non-trivial since we have to specialize on which linsolve is being used