# poly [![Hackage](https://img.shields.io/hackage/v/poly.svg)](https://hackage.haskell.org/package/poly) [![Stackage LTS](https://www.stackage.org/package/poly/badge/lts)](https://www.stackage.org/lts/package/poly) [![Stackage Nightly](https://www.stackage.org/package/poly/badge/nightly)](https://www.stackage.org/nightly/package/poly) [![Coverage Status](https://coveralls.io/repos/github/Bodigrim/poly/badge.svg)](https://coveralls.io/github/Bodigrim/poly) Haskell library for univariate and multivariate polynomials, backed by `Vector`s. ```haskell > -- Univariate polynomials > (X + 1) + (X - 1) :: VPoly Integer 2 * X > (X + 1) * (X - 1) :: UPoly Int 1 * X^2 + (-1) > -- Multivariate polynomials > (X + Y) * (X - Y) :: VMultiPoly 2 Integer 1 * X^2 + (-1) * Y^2 > (X + Y + Z) ^ 2 :: UMultiPoly 3 Int 1 * X^2 + 2 * X * Y + 2 * X * Z + 1 * Y^2 + 2 * Y * Z + 1 * Z^2 > -- Laurent polynomials > (X^-2 + 1) * (X - X^-1) :: VLaurent Integer 1 * X + (-1) * X^-3 > (X^-1 + Y) * (X + Y^-1) :: UMultiLaurent 2 Int 1 * X * Y + 2 + 1 * X^-1 * Y^-1 ``` ## Vectors `Poly v a` is polymorphic over a container `v`, implementing the `Vector` interface, and coefficients of type `a`. Usually `v` is either a boxed vector from [`Data.Vector`](https://hackage.haskell.org/package/vector/docs/Data-Vector.html) or an unboxed vector from [`Data.Vector.Unboxed`](https://hackage.haskell.org/package/vector/docs/Data-Vector-Unboxed.html). Use unboxed vectors whenever possible, e. g., when the coefficients are `Int`s or `Double`s. There are handy type synonyms: ```haskell type VPoly a = Poly Data.Vector.Vector a type UPoly a = Poly Data.Vector.Unboxed.Vector a ``` ## Construction The simplest way to construct a polynomial is using the pattern `X`: ```haskell > X^2 - 3 * X + 2 :: UPoly Int 1 * X^2 + (-3) * X + 2 ``` (Unfortunately, types are often ambiguous and must be given explicitly.) While being convenient to read and write in REPL, `X` is relatively slow. The fastest approach is to use `toPoly`, providing it with a vector of coefficients (constant term first): ```haskell > toPoly (Data.Vector.Unboxed.fromList [2, -3, 1 :: Int]) 1 * X^2 + (-3) * X + 2 ``` Alternatively one can enable `{-# LANGUAGE OverloadedLists #-}` and simply write ```haskell > [2, -3, 1] :: UPoly Int 1 * X^2 + (-3) * X + 2 ``` There is a shortcut to construct a monomial: ```haskell > monomial 2 3.5 :: UPoly Double 3.5 * X^2 + 0.0 * X + 0.0 ``` ## Operations Most operations are provided by means of instances, like `Eq` and `Num`. For example, ```haskell > (X^2 + 1) * (X^2 - 1) :: UPoly Int 1 * X^4 + 0 * X^3 + 0 * X^2 + 0 * X + (-1) ``` One can also find it convenient to `scale` by a monomial (cf. `monomial` above): ```haskell > scale 2 3.5 (X^2 + 1) :: UPoly Double 3.5 * X^4 + 0.0 * X^3 + 3.5 * X^2 + 0.0 * X + 0.0 ``` While `Poly` cannot be made an instance of `Integral` (because there is no meaningful `toInteger`), it is an instance of `GcdDomain` and `Euclidean` from the [`semirings`](https://hackage.haskell.org/package/semirings) package. These type classes cover the main functionality of `Integral`, providing division with remainder and `gcd` / `lcm`: ```haskell > Data.Euclidean.gcd (X^2 + 7 * X + 6) (X^2 - 5 * X - 6) :: UPoly Int 1 * X + 1 > Data.Euclidean.quotRem (X^3 + 2) (X^2 - 1 :: UPoly Double) (1.0 * X + 0.0,1.0 * X + 2.0) ``` Miscellaneous utilities include `eval` for evaluation at a given point, and `deriv` / `integral` for taking the derivative and an indefinite integral, respectively: ```haskell > eval (X^2 + 1 :: UPoly Int) 3 10 > deriv (X^3 + 3 * X) :: UPoly Double 3.0 * X^2 + 0.0 * X + 3.0 > integral (3 * X^2 + 3) :: UPoly Double 1.0 * X^3 + 0.0 * X^2 + 3.0 * X + 0.0 ``` ## Deconstruction Use `unPoly` to deconstruct a polynomial to a vector of coefficients (constant term first): ```haskell > unPoly (X^2 - 3 * X + 2 :: UPoly Int) [2,-3,1] ``` Further, `leading` is a shortcut to obtain the leading term of a non-zero polynomial, expressed as a power and a coefficient: ```haskell > leading (X^2 - 3 * X + 2 :: UPoly Double) Just (2,1.0) ``` ## Flavours * `Data.Poly` provides dense univariate polynomials with a `Num`-based interface. This is a default choice for most users. * `Data.Poly.Semiring` provides dense univariate polynomials with a `Semiring`-based interface. * `Data.Poly.Laurent` provides dense univariate Laurent polynomials with a `Semiring`-based interface. * `Data.Poly.Sparse` provides sparse univariate polynomials with a `Num`-based interface. Besides that, you may find it easier to use in the REPL because of a more readable `Show` instance, skipping zero coefficients. * `Data.Poly.Sparse.Semiring` provides sparse univariate polynomials with a `Semiring`-based interface. * `Data.Poly.Sparse.Laurent` provides sparse univariate Laurent polynomials with a `Semiring`-based interface. * `Data.Poly.Multi` provides sparse multivariate polynomials with a `Num`-based interface. * `Data.Poly.Multi.Semiring` provides sparse multivariate polynomials with a `Semiring`-based interface. * `Data.Poly.Multi.Laurent` provides sparse multivariate Laurent polynomials with a `Semiring`-based interface. All flavours are available backed by boxed or unboxed vectors. ## Performance As a rough guide, `poly` is at least 20x-40x faster than the [`polynomial`](http://hackage.haskell.org/package/polynomial) library. Multiplication is implemented via the Karatsuba algorithm. Here are a couple of benchmarks for `UPoly Int`: | Benchmark | polynomial, μs | poly, μs | speedup | :---------------------------- | --------------: | -------: | ------: | addition, 100 coeffs. | 45 | 2 | 22x | addition, 1000 coeffs. | 441 | 17 | 25x | addition, 10000 coeffs. | 6545 | 167 | 39x | multiplication, 100 coeffs. | 1733 | 33 | 52x | multiplication, 1000 coeffs. | 442000 | 1456 | 303x Due to being polymorphic by multiple axis, the performance of `poly` crucially depends on specialisation of instances. Clients are strongly recommended to compile with `ghc-options: -fspecialise-aggressively` and suggested to enable `-O2`. ## Additional resources * __Polynomials in Haskell__, MuniHac, 12.09.2020: [slides](https://github.com/Bodigrim/my-talks/raw/master/munihac2020/slides.pdf), [video](https://youtu.be/NAs3ExQZUjA).