Julia Base Norm Changes

[ChrisRackauckas]

@ranocha JuliaLang/julia#27401 It seems like your default norm matches your recursive norm, so we should change the default on v0.7 to just using norm?

https://github.com/JuliaDiffEq/DiffEqBase.jl/blob/master/src/common_defaults.jl#L2-L7

[ranocha]

In the defaults you cited, there is a division by length(u) which is not included in the new norm. Moreover, there are some additional security means (checks for possible overflows etc.) in norm that make sqrt(sum(abs2, u) / length(u)) faster. Since this part is performance critical and you use @fastmath a lot, sacrificing accuracy for speed, I'm not sure whether the new norm should be used.

ulia> using LinearAlgebra, BenchmarkTools

julia> v = rand(2^10);

julia> @benchmark norm($v)
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     272.331 ns (0.00% GC)
  median time:      272.432 ns (0.00% GC)
  mean time:        272.589 ns (0.00% GC)
  maximum time:     354.695 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     308

julia> @benchmark sqrt(sum(abs2, $v) / length($v))
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     132.341 ns (0.00% GC)
  median time:      133.145 ns (0.00% GC)
  mean time:        133.314 ns (0.00% GC)
  maximum time:     151.387 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     878

julia> v = rand(2^15);

julia> @benchmark norm($v)
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     9.397 μs (0.00% GC)
  median time:      10.060 μs (0.00% GC)
  mean time:        10.076 μs (0.00% GC)
  maximum time:     27.636 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1

julia> @benchmark sqrt(sum(abs2, $v) / length($v))
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     4.811 μs (0.00% GC)
  median time:      4.847 μs (0.00% GC)
  mean time:        4.854 μs (0.00% GC)
  maximum time:     8.283 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     7


julia> v = rand(2^20);

julia> @benchmark norm($v)
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     356.202 μs (0.00% GC)
  median time:      367.955 μs (0.00% GC)
  mean time:        370.762 μs (0.00% GC)
  maximum time:     662.704 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1

julia> @benchmark sqrt(sum(abs2, $v) / length($v))
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     236.158 μs (0.00% GC)
  median time:      239.394 μs (0.00% GC)
  mean time:        240.769 μs (0.00% GC)
  maximum time:     562.734 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1

julia> v = rand(2^30);

julia> @benchmark norm($v)
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     569.887 ms (0.00% GC)
  median time:      571.031 ms (0.00% GC)
  mean time:        570.924 ms (0.00% GC)
  maximum time:     571.777 ms (0.00% GC)
  --------------
  samples:          9
  evals/sample:     1

julia> @benchmark sqrt(sum(abs2, $v) / length($v))
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     465.854 ms (0.00% GC)
  median time:      466.278 ms (0.00% GC)
  mean time:        466.621 ms (0.00% GC)
  maximum time:     468.110 ms (0.00% GC)
  --------------
  samples:          11
  evals/sample:     1

julia> versioninfo()
Julia Version 0.7.0-alpha.27
Commit 58a8931684 (2018-06-06 03:53 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: Intel(R) Core(TM) i7-4770 CPU @ 3.40GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-6.0.0 (ORCJIT, haswell)
Environment:
  JULIA_NUM_THREADS = 4

[ChrisRackauckas]

Yup, the norm is just for a crude error estimate so it definitely wants speed >> accuracy. Looks to be worth keeping. Thanks!