I looked into the *Calculus* package and its derivative functions. First, I
got errors when running examples from the README file:
julia> second_derivative(x -> sin(x), pi)
ERROR: no method eps(DataType,)
in finite_difference at
/Users/HwB/.julia/Calculus/src/finite_difference.jl:27
in second_derivative at /Users/HwB/.julia/Calculus/src/derivative.jl:67
Then I was a bit astonished to see not too accurate results such as
julia> abs(second_derivative(sin, 1.0) + sin(1.0))
6.647716624952338e-7
while, when applying the standard central formula for second derivatives,
(f(x+h) - 2*f(x) + f(x-h)) / h^2 with the (by theory) suggested step length
eps^0.25 (for second derivatives) will result in a much better value:
julia> h = eps()^0.25;
julia> f = sin; x = 1.0;
julia> df = (sin(x+h) - 2*sin(x) + sin(x-h)) / h^2
-0.8414709866046906
julia> abs(df + sin(1.0))
1.7967940468821553e-9
The functions for numerical differentiation in *Calculus* look quite
involved, maybe it would be preferable to apply known approaches derived
from Taylor series. Even the fourth order derivative will in this case lead
to an absolute error below 1e-05!
