Memoize Coeffs

[oxinabox]

Closes #72.
Now we are faster than we were even on 0.9.2 before I made it slow.
This does part of #61 in particular part of this comment
#61 (comment)

Looks like the cause was #65.
Am going to build a fix shortly

First two timings are as per #72 (comment)

v0.9.5

44.084013 seconds (80.42 M allocations: 4.397 GiB, 3.80% gc time)
29.377420 seconds (28.23 M allocations: 1.830 GiB, 1.86% gc time)
27.364521 seconds (28.23 M allocations: 1.830 GiB, 1.76% gc time)
27.871987 seconds (28.23 M allocations: 1.830 GiB, 1.79% gc time)

v0.9.2

18.984412 seconds (61.60 M allocations: 3.236 GiB, 6.57% gc time)
1.072190 seconds (10.23 M allocations: 737.725 MiB, 10.14% gc time)
1.090849 seconds (10.23 M allocations: 737.725 MiB, 9.55% gc time)
1.072175 seconds (10.23 M allocations: 737.725 MiB, 8.74% gc time)

With this PR

10.143730 seconds (40.32 M allocations: 1.860 GiB, 8.13% gc time)
0.634321 seconds (7.34 M allocations: 243.141 MiB, 7.87% gc time)
0.631055 seconds (7.34 M allocations: 243.141 MiB, 8.24% gc time)
0.661258 seconds (7.34 M allocations: 243.141 MiB, 10.92% gc time)

---

Using timing from #61

using BenchmarkTools
using FiniteDifferences

const _fdm = central_fdm(2,1);
const _fdm5 = central_fdm(5,1);

const xs = collect(1:0.1:200);
f(x) = sum(sin, x);
@btime grad($_fdm, $f, $xs);
@btime grad($_fdm5, $f, $xs);

We get:

v0.9.5

julia> @btime grad($_fdm, $f, $xs);
  226.006 ms (195137 allocations: 9.54 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  744.445 ms (665012 allocations: 23.76 MiB)

v0.9.2

julia> @btime grad($_fdm, $f, $xs);
  221.743 ms (154789 allocations: 4.26 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  440.230 ms (162753 allocations: 5.45 MiB)

With this PR

julia> @btime grad($_fdm, $f, $xs);
  199.677 ms (177218 allocations: 6.90 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  404.316 ms (230974 allocations: 11.58 MiB)

Explain

I think the main reason why this gives much more improvement in the test case from #72 than from #61, is that the #61 benchmark declares the fdm objects outside the the hot-loop.
And that is where _coeffs are calculated.
Except I think they are also calculated when adapt is triggered.
So this helps more there for the _fdm5 case.

[nickrobinson251]

Nice work!

[KristofferC]

Is it worth thinking about thread safety here?

[oxinabox]

Could do, but I don't think it is worth it.
There is much more thread unsafe stuff in this package that this.
(Mostly the history tracking stuff).