avoid integrand evaluations at endpoints

[lxvm]

This pr concerns issues #86, #82, and #39, and addresses errors arising from evaluation of the integrand at the endpoints of the interval. This pr handles the endpoints by:

• throwing a DomainError if the roundoff happens on an initial segment
• silently returning the best integral and error estimate if the roundoff happens during refinement

I think it would be better to warn the user of the following situations in which quadgk silently returns a possibly unconverged result

• endpoint roundoff error
• maxevals reached

Should we do this and add return codes?

[codecov]

Codecov Report

Patch coverage: 100.00% and project coverage change: +0.04% 🎉

Comparison is base (10bc1a1) 98.16% compared to head (173eff3) 98.20%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #91      +/-   ##
==========================================
+ Coverage   98.16%   98.20%   +0.04%     
==========================================
  Files           6        6              
  Lines         599      614      +15     
==========================================
+ Hits          588      603      +15     
  Misses         11       11              

Files Changed	Coverage Δ
src/adapt.jl	96.99% <100.00%> (+0.32%)	⬆️
src/batch.jl	99.17% <100.00%> (+0.01%)	⬆️

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[stevengj]

Should we do this and add return codes?

The problem is that adding return codes to quadgk is breaking.

Of course, you can always tell from the error estimate — if the final error estimate is greater than the requested tolerance, you know that it must have returned early.

[stevengj]

This introduces a type-instability into evalrule. Does that affect performance?

[lxvm]

Of course, you can always tell from the error estimate — if the final error estimate is greater than the requested tolerance, you know that it must have returned early.

Right, that's a good point

This introduces a type-instability into evalrule. Does that affect performance?

Thanks for pointing this out, so I refactored a bit and put the type instability into the refine routine.

As for performance, the endpoint check has an overhead of about 3 ns, which should be negligible. Here is a benchmark:

master

julia> using BenchmarkTools

julia> using QuadGK

julia> @benchmark quadgk(x -> x, 0, 1) # zero-cost integrand, no subdivisions
BenchmarkTools.Trial: 10000 samples with 993 evaluations.
 Range (min … max):  34.597 ns … 133.580 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     35.161 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   36.762 ns ±   6.744 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

  █▅▆                                                          ▁
  ███▆▅▅▃▅▅▆▇▇█▆▆▆▄▆▆▆▆▆▆▆▅▆█▇▅▄▇█▅▅▄▆▅▅▄▆▅▄▄▄▅▅▄▁▅▅▅▅▆▆▅▅█▅▄▃ █
  34.6 ns       Histogram: log(frequency) by time      72.5 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark quadgk(x -> sin(100x), 0, 1) # low-cost integrand with subdivisions
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  11.684 μs … 63.263 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     11.858 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   12.611 μs ±  1.834 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

  ▆█▅▂             ▃▆▆▄▃▁             ▁▂▁                     ▂
  █████▆▅▄▁▁▁▁▁▁▄▁▁███████▇▆▆▇▇▇▇▅▆▆▄▅███▇▇▆▄▁▁▃▁▃▃▄▅▄▁▃▅▁▄▄▅ █
  11.7 μs      Histogram: log(frequency) by time      16.9 μs <

 Memory estimate: 1.69 KiB, allocs estimate: 3.

this pr

julia> using BenchmarkTools

julia> using QuadGK

julia> @benchmark quadgk(x -> x, 0, 1) # zero-cost integrand, no subdivisions
BenchmarkTools.Trial: 10000 samples with 992 evaluations.
 Range (min … max):  37.661 ns … 132.844 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     37.885 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   39.574 ns ±   7.290 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

  █  ▁                                                         ▁
  █▇▃█▃▃▃▄▅▄▄▅▅▅▄▂▃▄▂▂▄▆▆▅▅▄▅▅▅▄▅▄▅▅▄▄▄▅▅▅▅▄▄▃▅▅▅▆▆▇▆▆▆▅▅▄▄▄▄▄ █
  37.7 ns       Histogram: log(frequency) by time      75.2 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark quadgk(x -> sin(100x), 0, 1) # low-cost integrand with subdivisions
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  11.616 μs … 79.306 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     11.986 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   12.831 μs ±  2.304 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

  ▄██▄▃     ▂▆▇▄▃▁      ▂▂                                    ▂
  █████▇▁▃▃▄██████▇███▅▇███▇▆▆▅▆▆▄▄▅▄▃▄▅▆▇▇▅▆▃▄▄▄▁▄▁▄▄▄▃▁▄▃▁▄ █
  11.6 μs      Histogram: log(frequency) by time      20.5 μs <

 Memory estimate: 1.69 KiB, allocs estimate: 3.