Project Plan

First Milestone

implement required module signature
implement a vanilla implementation
represent a formula using functor
evaluate the formula using ocaml category

Second Milestone

replace additive type with float
- in order to deal with generic zero object issue
- I choose not to have parameterized type and use float only
  - 0.0 is zero object
implement the forward AAD version

Airport Milestone

airport_typeclass: compare haskell type class vs ocaml modular
haskell foundation: type class/type family
basic Haskell type class trick
concat implementation on Additive constraints
hacking solution to introduce additive

X Milestone

represent linear map as generalized matrices
represent linear map as continuation of linear map
RAD implementation

Helper function

linear map: scale (d x r)
scalarD

scalarD f d = D (\x -> let r = f x in (r, scale (d x r)))

scalarR f d = scalarD f (const d)
- const :: a -> b -> a
- d :: (s -> s)
- target :: s -> (s -> s)
- const :: (s -> s) -> s -> (s -> s)
- const d :: s -> (s -> s)
- (const d) x r = d r $$ \frac{d(\frac{1}{x})}{dx} = -(\frac{1}{x})^2 $$
scalarX f d = scalarD f (const . d)
- const :: a -> b -> a
- (.) :: (b -> c) -> (a -> b) -> (a -> c)
- d :: s -> s
- (.) :: (s -> (s -> s)) -> (s -> s) -> (s -> (s -> s))
- const :: s -> (s -> s)
- (const . d) :: s -> (s -> s)
- (const . d) x r = ((const . d) x) r = (const (d x)) r = d x

$$ \frac{d(cos(x))}{x} = - sin(x) $$

$$ \begin{aligned} d(a/b) &= \frac{a'b -ab'}{b^2}\\ D_(x * y) (a, b) (da, db) &= a\times db + b \times da \\ D_(x * \frac{1}{y}) (a, \frac{1}{b}) (da, d(\frac{1}{b})) &= a \times d(\frac{1}{b}) + \frac{1}{b} \times da\\ & = - a \times db \frac{1}{b^2} + \frac{da\times b}{b^2} \\ & = \frac{da\times b - db\times a}{b^2} \end{aligned} $$