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README
poly
Haskell library for univariate and multivariate polynomials, backed by Vector
.
>  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 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 coefficients are Int
or Double
.
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 convenient to scale
by 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 semirings
package. These type classes
cover 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 value of indeterminate,
and reciprocals deriv
/ integral
:
> 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 withNum
based interface. This is a default choice for most users.Data.Poly.Semiring
provides dense univariate polynomials withSemiring
based interface.Data.Poly.Laurent
provides dense univariate Laurent polynomials withSemiring
based interface.Data.Poly.Sparse
provides sparse univariate polynomials withNum
based interface. Besides that, you may find it easier to use in REPL because of a more readableShow
instance, skipping zero coefficients.Data.Poly.Sparse.Semiring
provides sparse univariate polynomials withSemiring
based interface.Data.Poly.Sparse.Laurent
provides sparse univariate Laurent polynomials withSemiring
based interface.Data.Poly.Multi
provides sparse multivariate polynomials withNum
based interface.Data.Poly.Multi.Semiring
provides sparse multivariate polynomials withSemiring
based interface.Data.Poly.Multi.Laurent
provides sparse multivariate Laurent polynomials withSemiring
based interface.
All flavours are available backed by boxed or unboxed vectors.
Performance
As a rough guide, poly
is at least 20x40x faster than polynomial
library.
Multiplication is implemented via Karatsuba algorithm.
Here is 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 