Sorry, but I don't see it. Using Calculus it says
julia> finite_difference_hessian(sin, 1.0)
ERROR: finite_difference_hessian not defined
and calling hessian alone returns the old, inaccurate result:
julia> hessian(sin, 1.0)
-0.841471649579559
Looking at the definition of hessian, I'd say it calls
"finite_difference_hessian(f, derivative(f), x, :central)"
so basing the calculation on a derivative function and not on a formula
derived from the Taylor series.
What did I misunderstand?
Thanks, Hans Werner
On Tuesday, January 21, 2014 3:52:37 AM UTC+1, Tim Holy wrote:
>
> Most of the second_derivative stuff in derivative.jl could probably be
> ripped
> out, and have it directly call finite_difference_hessian(f, x). Indeed,
> that
> gets you those two extra orders of magnitude in precision, because
> finite_difference_hessian uses exactly this kind of scaling rule.
>
> --Tim
>
> On Monday, January 20, 2014 10:40:28 AM Hans W Borchers wrote:
> > 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!
>
