Partial Derivatives in Haskell
A while back a friend wanted help with a program that could solve for the roots of functions using Newton's method, and naturally for that I needed some way to calculate the derivative of a function numerically, and this is what I came up with:
deriv f x = (f (x+h) - f x) / h where h = 0.00001
Newton's method was a fairly easy thing to implement, and it works rather well. But now I've started to wonder - Is there some way I could use this function to solve partial derivatives in a numerical manner, or is that something that would require a full-on CAS? I would post my attempts but I have absolutely no clue what to do yet.
Please keep in mind that I am new to Haskell. Thank you!
This is called automatic differentiation and there is a lot of really neat work in this area in Haskell, though I don't know how accessible it is.
From the wiki page:
You can certainly do much the same thing as you already implemented, just with multivariate pertubations. But first (as you should always do with top-level functions) add a type signature:
deriv :: (Double -> Double) -> Double -> Double
(That's not the most general possible signature, but probably sufficiently general for everything you'll need.) I'll call
type ℝ = Double
in the following for brevity, i.e.
deriv :: (ℝ -> ℝ) -> ℝ -> ℝ
Now what you want is, for example in ℝ²
grad :: ((ℝ,ℝ) -> ℝ) -> (ℝ,ℝ) -> (ℝ,ℝ) grad f (x,y) = ((f (x+h,y) - f (x,y)) / h, (f (x,y+h) - f (x,y)) / h) where h = 0.00001
It's awkward having to write out the components individually and making the definition specific to a particular-dimensional vector space. A generic way of doing it:
import Data.VectorSpace import Data.Basis grad :: (HasBasis v, Scalar v ~ ℝ) => (v -> ℝ) -> v -> v grad f x = recompose [ (e, (f (x ^+^ h*^basisValue b) - f x) ^/ h) | (e,_) <- decompose x ] where h = 0.00001
Note that this pre-chosen-step–finite-differentiation is always a tradeoff between inaccuracy from higher-order terms and from floating-point errors, so definitely check out automatic differentiation.