https://github.com/ocramz/sd
Symbolic multivariate differentiation in Haskell
https://github.com/ocramz/sd
Last synced: 23 days ago
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Symbolic multivariate differentiation in Haskell
- Host: GitHub
- URL: https://github.com/ocramz/sd
- Owner: ocramz
- License: bsd-3-clause
- Created: 2017-01-23T15:13:28.000Z (over 8 years ago)
- Default Branch: master
- Last Pushed: 2017-01-26T15:15:46.000Z (over 8 years ago)
- Last Synced: 2025-03-27T06:24:32.689Z (6 months ago)
- Language: Haskell
- Homepage:
- Size: 43.9 KB
- Stars: 2
- Watchers: 3
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# sd
[](https://travis-ci.org/ocramz/sd)
Symbolic multivariate differentiation in Haskell
## Example
In `sd`, free variables are indexed by integer numbers:
> ix = 0
> iy = 1
> x = Var ix
> y = Var iy
Now we can define a function of `x` and `y` such as `3x^2+y^3` :
> f = 3 * x**2 + y**3
> f
((3.0 * x^2.0) + y^3.0)
and a suitable binding environment, relating the free variables to their value:
> env0 = fromListEnv [(ix, 5.0), (iy, 2.0)]
We can then require the _gradient_ of `f` as follows:
> g = grad env0 f
> g
fromList [(0,(6.0 * x)),(1,(3.0 * y^2.0))]
Behind the scenes, `grad` differentiates the expression with respect to each of the variables in the binding, which in turn means applying mechanically a number of rewriting and simplification passes. The result is a new IntMap which contains the _symbolic_ expressions for the partial derivatives.
Finally, the _numerical value_ of the partial derivatives is obtained by substitution of the bindings:
> eval env0 <$> g
fromList [(0,30.0),(1,12.0)]
## Credits and inspiration
I started `sd` after reading this blog by Ben Kovach (from which I also shamelessly ripped the AST and the simplification code for the `Numeric.SD` module):
* https://5outh.blogspot.se/2013/05/symbolic-calculus-in-haskell.html
This can be enhanced in multiple ways, for example by factoring out the recursion from the gradient and evaluation algorithm, using the Fix recursive type and _recursion schemes_ :
* https://jtobin.io/ad-via-recursion-schemes
and perhaps by augmenting the abstract syntax with `Let` constructors to achieve _sharing of common sub-expressions_ [1, 2]:
* https://jtobin.io/sharing-in-haskell-edsls
Alternatively, the free monad route lets us use `do` notation as well:
* http://dlaing.org/cofun/posts/free_and_cofree.html
## References
[1] Ankner, J. and Svenningsson, J., An EDSL Approach to High Performance Haskell Programming
[2] Oliveira, B. C. d. S. and Löh, A., Abstract Syntax Graphs for Domain Specific Languages