poly
Haskell library for univariate and multivariate polynomials, backed by Vector
s.
>  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
or an unboxed vector from Data.Vector.Unboxed
. Use unboxed vectors whenever possible, e. g., when the coefficients are Int
s or Double
s.
There are handy type synonyms:
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
:
> 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):
> toPoly (Data.Vector.Unboxed.fromList [2, 3, 1 :: Int])
1 * X^2 + (3) * X + 2
Alternatively one can enable {# LANGUAGE OverloadedLists #}
and simply write
> [2, 3, 1] :: UPoly Int
1 * X^2 + (3) * X + 2
There is a shortcut to construct a monomial:
> 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,
> (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):
> 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
package. These type classes
cover the main functionality of Integral
, providing division with remainder and gcd
/ lcm
:
> 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:
> 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):
> unPoly (X^2  3 * X + 2 :: UPoly Int)
[2,3,1]
Further, leading
is a shortcut to obtain the leading term of a nonzero polynomial,
expressed as a power and a coefficient:
> 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.
As a rough guide, poly
is at least 20x40x faster than the 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 ghcoptions: fspecialiseaggressively
and suggested to enable O2
.
Additional resources
 Polynomials in Haskell, MuniHac, 12.09.2020:
slides,
video.