Update 23_test_problems_tests.jl

[ChrisRackauckas]

This update is required due to SciML/DiffEqProblemLibrary.jl#138. Its inherently just a numerical instability. https://github.com/SciML/DiffEqProblemLibrary.jl/pull/138/files#diff-416284a99fdfe358ae2a33c649aa3a715568ff3cf4d0d0cf62e6c7a183aa9fc7R111 this change is effectively random, looking at:

zz = out[1] + 3.0 * x[1] - 2.0 * x[1] * x[2] + 2.0 * x[1]^3
zzz = out[2] + x[2] - x[2]^2 - 1.0

out[1] += x[1] * (3.0 - 2.0 * x[2] + 2.0 * x[1]^2)
out[2] += x[2] * (1.0 - x[2]) - 1.0

if abs(out[1] - zz) > 1e5
    @show out[1] - zz, out[2] - zzz, x[1], x[2]
    @show 2.0 * x[2], 2.0 * x[1]^2
    @show 3.0 * x[1], 2.0 * x[1] * x[2], 2.0 * x[1]^3
    error()
end

I get:

(out[1] - zz, out[2] - zzz, x[1], x[2]) = (-2048.0, 0.0, 1.032129073975807e6, -1.109530258395812e6)
(2.0 * x[2], 2.0 * x[1] ^ 2) = (-2.219060516791624e6, 2.130580850692314e12)
(3.0 * x[1], 2.0 * x[1] * x[2], 2.0 * x[1] ^ 3) = (3.096387221927421e6, -2.2903568762924146e12, 2.1990344404556452e18)

(out[1] - zz, out[2] - zzz, x[1], x[2]) = (-4.503599627370496e15, 0.0, 2.027697193465798e10, -3.714249421601087e10)
(2.0 * x[2], 2.0 * x[1] ^ 2) = (-7.428498843202174e10, 8.223111816778149e20)
(3.0 * x[1], 2.0 * x[1] * x[2], 2.0 * x[1] ^ 3) = (6.0830915803973946e10, -1.5062746256024978e21, 1.6673980752436494e31)

with different thresholds. Basically when x[1] is large the error is floating point noise, and that floating point noise is enough for Broyden to be stable vs unstable.

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.