Desired features

[pnkraemer]

All of the following should be possible with pnfindiff at some point of the future, ideally, documented in example notebooks.

Basic scalar derivatives:

Evaluate the numerical derivative f'(x_0) of a function f: R -> R at point x_0

• Central, forward, backward schemes
• Variable derivative order
• Variable method order
• Numerical noise
• Custom grid
• Batching over x_0
• Batching over x_0 with a non-uniform grid
• Batching over f
• Polynomial vs exponentiated quadratic kernel

Basic multivariate derivatives

Evaluate the numerical derivative (D f) (x_0) of a function f: R^n -> R^m at point x_0

• Partial derivatives via schemes
• Gradients via a tensor-product approach and using all of the above
• Jacobians and Hessians via a tensor-product approach and using all of the above
• Gradients, via an n-dimensional prior
• (Mixed) partial derivatives for n-dimensional priors
• Jacobians (forward and reverse mode?), Hessians, (JVPs?) via a n-dimensional prior

Advanced features

• Uncertainty calibration
• Control over stencils -> which point is on boundary, etc. (https://github.com/maroba/findiff)
• Computation of (L f)(Xn) with FD. This is not the same as batched evaluation, because for instance, boundary nodes may use forward schemes, and central nodes use central schemes. (https://numpy.org/doc/stable/reference/generated/numpy.gradient.html)
• Support for symbolic computation
• Support for sparse matrices
• A bunch of kernels: matern, (inverse) multiquadrics, thin-plate splines
• Boundary conditions?
• Advanced differential operators: Divergence, Laplacian, curl
• Polar coordinates
• complex differentials? integer-valued derivatives? (e.g. grad(f, allow_int=True, holomorphic=True))
• Unsymmetric and symmetric collocation (with examples and docs)