Calculate nodes and weights at GaussLegendre construction time

[mikeingold]

Changes

• Adds nodes and weights as stored fields to GaussLegendre <: IntegrationRule. These are generated at instantiation-time instead of at integral-time so that the results can be re-used.
  • Closes Possible to cache Gauss-Legendre nodes and weights #148

Discussion

The Gauss-Legendre nodes and weights are purely a function of order n, so I think it makes sense to just calculate them when constructing the rule itself. This will lead to some moderate performance benefits for rules used more than once, or where the integration occurs in some performance-sensitive user code block.

Currently, FastGaussQuadrature.jl doesn't allow us to request nodes and weights in a particular eltype, and instead always returns Vector{Float64}s. If that support ever materializes then I think it could make sense to add a parameter GaussLegendre{T} to handle this, but for now I don't think it's necessary.

[github-actions]

Benchmark Results

	main	faaecf3...	main/faaecf35e3ef9a...
Differentials/Differential	0.24 ± 0.001 μs	0.24 ± 0.001 μs	1
Differentials/Jacobian	0.207 ± 0.0001 μs	0.205 ± 0.0001 μs	1.01
Integrals/Segment/Scalar GaussKronrod	1.19 ± 0.004 μs	1.19 ± 0.006 μs	1
Integrals/Segment/Scalar GaussLegendre	6.97 ± 0.13 μs	4.16 ± 0.01 μs	1.67
Integrals/Segment/Scalar HAdaptiveCubature	1.43 ± 0.15 μs	1.46 ± 0.15 μs	0.981
Integrals/Segment/Vector GaussKronrod	3.46 ± 0.056 μs	3.42 ± 0.068 μs	1.01
Integrals/Segment/Vector GaussLegendre	22.8 ± 0.52 μs	20.3 ± 0.42 μs	1.12
Integrals/Segment/Vector HAdaptiveCubature	4.32 ± 0.079 μs	4.39 ± 0.092 μs	0.984
Integrals/Sphere/Scalar GaussKronrod	0.0823 ± 0.002 ms	0.0825 ± 0.00094 ms	0.997
Integrals/Sphere/Scalar GaussLegendre	2.26 ± 0.0089 ms	2.25 ± 0.0098 ms	1.01
Integrals/Sphere/Scalar HAdaptiveCubature	0.0584 ± 0.00016 ms	0.0577 ± 0.00011 ms	1.01
Integrals/Sphere/Vector GaussKronrod	0.115 ± 0.00088 ms	0.117 ± 0.0011 ms	0.991
Integrals/Sphere/Vector GaussLegendre	3.61 ± 0.079 ms	3.68 ± 0.075 ms	0.981
Integrals/Sphere/Vector HAdaptiveCubature	0.109 ± 0.00092 ms	0.108 ± 0.0011 ms	1
Rules/GaussLegendre	2.79 ± 0.009 ns	21.5 ± 0.51 μs	0.00013
Specializations/Scalar GaussLegendre/BezierCurve	0.251 ± 0.003 ms	0.246 ± 0.00034 ms	1.02
Specializations/Scalar GaussLegendre/Line	11.1 ± 0.41 μs	8.15 ± 0.08 μs	1.36
Specializations/Scalar GaussLegendre/Plane	0.905 ± 0.0089 ms	0.9 ± 0.0071 ms	1.01
Specializations/Scalar GaussLegendre/Ray	10.4 ± 0.45 μs	7.45 ± 0.071 μs	1.39
Specializations/Scalar GaussLegendre/Rope	0.211 ± 0.0015 ms	0.13 ± 0.00012 ms	1.63
Specializations/Scalar GaussLegendre/Tetrahedron	0.288 ± 0.0022 s	0.315 ± 0.0067 s	0.915
Specializations/Scalar GaussLegendre/Triangle	0.729 ± 0.0083 ms	0.727 ± 0.0057 ms	1
time_to_load	1.54 ± 0.0059 s	1.57 ± 0.034 s	0.979

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR.
Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 100.00%. Comparing base (3c06d76) to head (faaecf3).
Report is 1 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff            @@
##              main      #149   +/-   ##
=========================================
  Coverage   100.00%   100.00%           
=========================================
  Files           18        18           
  Lines          166       165    -1     
=========================================
- Hits           166       165    -1     

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

[JoshuaLampert]

Nice! This can also be helpful if you want to integrate multiple functions even in potentially different geometries because you can reuse the rule object, i.e. also in this case you only need to compute the nodes and weight once.