`sequential` examples

This line is intentionally not left blank.

Binomial

In binomial we don't need "sequential", but our change will also make it unnecessary to use a Haskell foldl and the division into individual kernels will not be visible from the Nikola code, but will happen automatically.

finalPut :: Vector G (Exp F) -> Vector G (Exp F) -> Vector D (Exp F)
prevPut :: Vector G (Exp F) -> Vector G (Exp F) -> Exp Int32 -> Vector G (Exp F) -> Exp Int32 -> Vector D (Exp F)

binom = head $ fold (prevPut ...) (finalPut ...) [n..1]
   where
     .....
binomCompiled = compile binom

Sobol

Assume we can generate a sobol sequence given the length and an array of direction vectors:

sobolSequence_ :: Exp Index -> Array D DIM1 (Exp Elem) -> Array D DIM1 (Exp SpecReal)

We now want to generate an n-dimensional sobol sequence / several sobol sequences:

sobolSequences :: Exp Index -> Array D DIM2 (Exp Elem) -> Array D DIM2 (Exp SpecReal)

Currently, we would have to rewrite sobolSequence_ to do that. With the new approach we could do it in two ways:

sobolSequence n dirs = map (sobolSequence_ n) dirs

or

sobolSequence n dirs = map (sequential $ sobolSequence_ n) dirs

Least Squares

We still don't know the best way to parallelize this, but using this approach it would at least be a lot easier to experiment with different methods. For example, when writing polyfit, we would like to write an LU/Cholesky/QR-decomposition algorithm, but only for 3x3 matrices, so that would probably be more efficient to do sequentially.