Fast non-allocating calculations of gradients, Jacobians, and Hessians with sparsity support
This package is for calculating derivatives, gradients, Jacobians, Hessians, etc. numerically. This library is for maximizing speed while giving a usable interface to end users in a way that specializes on array types and sparsity. Included is:
If you want the fastest versions, create a cache and repeatedly call the
differencing functions at different x
values (or with different f
functions),
while if you want a quick and dirty numerical answer, directly call a differencing
function.
For analogous sparse differentiation with automatic differentiation, see SparseDiffTools.jl.
FiniteDiff.jl and FiniteDifferences.jl are similar libraries: both calculate approximate derivatives numerically. You should definitely use one or the other, rather than the legacy Calculus.jl finite differencing, or reimplementing it yourself. At some point in the future they might merge, or one might depend on the other. Right now here are the differences:
The general structure of the library is as follows. You can call the differencing functions directly and this will allocate a temporary cache to solve the problem with. To make this non-allocating for repeat calls, you can call the cache construction functions. Each cache construction function has two possibilities: one version where you give it prototype arrays and it generates the cache variables, and one fully non-allocating version where you give it the cache variables. This is summarized as:
See the documentation for details on the API.
In all functions, the inplace form is f!(dx,x)
while the out of place form is dx = f(x)
.
Coloring vectors are allowed to be supplied to the Jacobian routines, and these are
the directional derivatives for constructing the Jacobian. For example, an accurate
NxN tridiagonal Jacobian can be computed in just 4 f
calls by using
colorvec=repeat(1:3,N÷3)
. For information on automatically generating coloring
vectors of sparse matrices, see SparseDiffTools.jl.
Hessian coloring support is coming soon!