Allow linsolve to be \

[avik-pal]

For very simple problems with a diagonal Jacobian this is much faster.

Testing on the quadratic problem

julia> @benchmark solve(prob, NLsolveJL())
BenchmarkTools.Trial: 10000 samples with 5 evaluations.
 Range (min … max):  6.047 μs …  2.675 ms  ┊ GC (min … max):  0.00% … 98.90%
 Time  (median):     7.052 μs              ┊ GC (median):     0.00%
 Time  (mean ± σ):   9.251 μs ± 50.774 μs  ┊ GC (mean ± σ):  10.88% ±  1.98%

    ▁█▂                                                       
  ▁▂███▆▃▂▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▂▂▂▃▃▃▄▄▄▃▄▃▃▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁ ▂
  6.05 μs        Histogram: frequency by time        12.6 μs <

 Memory estimate: 8.20 KiB, allocs estimate: 107.

julia> @benchmark solve(prob, NewtonRaphson(; linsolve = \, autodiff = AutoForwardDiff(; chunksize = 2)))
BenchmarkTools.Trial: 10000 samples with 9 evaluations.
 Range (min … max):  2.446 μs …  1.465 ms  ┊ GC (min … max):  0.00% … 98.65%
 Time  (median):     2.976 μs              ┊ GC (median):     0.00%
 Time  (mean ± σ):   4.062 μs ± 27.179 μs  ┊ GC (mean ± σ):  13.24% ±  1.98%

     ▄█▇▅▁                                                    
  ▁▂▅██████▆▃▃▃▂▂▂▂▃▂▃▃▃▃▂▂▂▂▂▁▂▂▂▂▃▃▃▃▃▃▃▄▄▃▃▂▂▂▂▂▂▂▂▂▁▁▁▁▁ ▂
  2.45 μs        Histogram: frequency by time        5.89 μs <

 Memory estimate: 5.09 KiB, allocs estimate: 54.

julia> @benchmark solve(prob, NewtonRaphson(; linsolve = LUFactorization(), autodiff = AutoForwardDiff(; chunksize = 2)))
BenchmarkTools.Trial: 10000 samples with 8 evaluations.
 Range (min … max):  3.409 μs …  1.503 ms  ┊ GC (min … max): 0.00% … 98.89%
 Time  (median):     3.759 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   4.631 μs ± 25.575 μs  ┊ GC (mean ± σ):  9.47% ±  1.71%

   ▄▆██▇▇▇▆▅▄▃▃▂▁ ▁             ▁▁▁▁▂▁▂▂▂▂▃▂▃▃▂▃▂▁▂▁▁▁       ▂
  █████████████████▇▅▆▆▆▅▆▆▇▇▇███████████████████████████▇▇▇ █
  3.41 μs      Histogram: log(frequency) by time     6.77 μs <

 Memory estimate: 5.22 KiB, allocs estimate: 55.

julia> @benchmark solve(prob, NewtonRaphson(; autodiff = AutoForwardDiff(; chunksize = 2)))
BenchmarkTools.Trial: 10000 samples with 7 evaluations.
 Range (min … max):  4.846 μs …  1.846 ms  ┊ GC (min … max):  0.00% … 98.97%
 Time  (median):     5.304 μs              ┊ GC (median):     0.00%
 Time  (mean ± σ):   6.414 μs ± 40.806 μs  ┊ GC (mean ± σ):  14.11% ±  2.21%

   ▃▅▇███▇▆▅▄▃▂▂▂                                            ▂
  █████████████████▇▆▇▇▆▆▆▇▆▇▆▅▆▄▅▆▅▄▄▅▅▄▆▅▇▆▆▇▇▇▆███▇█▇▇▇▇▇ █
  4.85 μs      Histogram: log(frequency) by time     9.25 μs <

 Memory estimate: 8.25 KiB, allocs estimate: 100.

[codecov]

Codecov Report

Attention: 1 lines in your changes are missing coverage. Please review.

Comparison is base (e989b72) 86.23% compared to head (d840c17) 86.19%.

Files	Patch %	Lines
src/internal/linear_solve.jl	75.00%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #358      +/-   ##
==========================================
- Coverage   86.23%   86.19%   -0.05%     
==========================================
  Files          44       44              
  Lines        2609     2608       -1     
==========================================
- Hits         2250     2248       -2     
- Misses        359      360       +1     

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[avik-pal]

Failure is due to SciML/SciMLBase.jl#605.