hspray: Multivariate polynomials.

[ algebra, gpl, library, math ] [ Propose Tags ]

Manipulation of multivariate polynomials on a ring.


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Versions [RSS] 0.1.0.0, 0.1.1.0, 0.1.2.0, 0.1.3.0
Change log CHANGELOG.md
Dependencies base (>=4.7 && <5), containers (>=0.6.4.1), hashable (>=1.3.4.0), numeric-prelude (>=0.4.4), text (>=1.2.5.0), unordered-containers (>=0.2.17.0) [details]
License GPL-3.0-only
Copyright 2022 Stéphane Laurent
Author Stéphane Laurent
Maintainer laurent_step@outlook.fr
Category Math, Algebra
Home page https://github.com/stla/hspray#readme
Source repo head: git clone https://github.com/stla/hspray
Uploaded by stla at 2023-08-30T13:21:55Z
Distributions NixOS:0.1.3.0
Reverse Dependencies 3 direct, 0 indirect [details]
Downloads 136 total (5 in the last 30 days)
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Status Docs available [build log]
Last success reported on 2023-08-30 [all 1 reports]

Readme for hspray-0.1.3.0

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hspray

Stack-lts Stack-nightly

Simple multivariate polynomials in Haskell.


import Math.Algebra.Hspray
x = lone 1 :: Spray Double
y = lone 2 :: Spray Double
z = lone 3 :: Spray Double
poly = (2 *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ (4 *^ (x ^*^ y ^*^ z))
prettySpray show "X" poly
-- "(4.0) * X^(3, 1, 1) + (8.0) * X^(4, 2, 2)"

More generally, one can use the type Spray a as long as the type a has the instances Eq and Algebra.Ring (defined in the numeric-prelude library). For example a = Rational:

import Math.Algebra.Hspray
import Data.Ratio
x = lone 1 :: Spray Rational
y = lone 2 :: Spray Rational
z = lone 3 :: Spray Rational
poly = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z))
prettySpray show "X" poly
-- "(7 % 4) * X^(3, 1, 1) + (7 % 6) * X^(4, 2, 2)"

Or a = Spray Double:

import Math.Algebra.Hspray
p = lone 1 :: Spray Double
x = lone 1 :: Spray (Spray Double)
y = lone 2 :: Spray (Spray Double)
poly = ((p *^ x) ^+^ (p *^ y))^**^2  
prettySpray (prettySpray show "a") "X" poly
-- "((1.0) * a^(2)) * X^(0, 2) + ((2.0) * a^(2)) * X^(1, 1) + ((1.0) * a^(2)) * X^(2)"

Evaluation:

import Math.Algebra.Hspray
x = lone 1 :: Spray Double
y = lone 2 :: Spray Double
z = lone 3 :: Spray Double
poly = 2 *^ (x ^*^ y ^*^ z) 
-- evaluate poly at x=2, y=1, z=2
evalSpray poly [2, 1, 2]
-- 8.0

Differentiation:

import Math.Algebra.Hspray
x = lone 1 :: Spray Double
y = lone 2 :: Spray Double
z = lone 3 :: Spray Double
poly = 2 *^ (x ^*^ y ^*^ z) ^+^ (3 *^ x^**^2)
-- derivate with respect to x
prettySpray show "X" $ derivSpray 1 poly
-- "(2.0) * X^(0, 1, 1) + (6.0) * X^(1)"

Easier usage

To construct a polynomial using the ordinary symbols +, * and -, one can hide these operators from Prelude and import them from the numeric-prelude library:

import Prelude hiding ((*), (+), (-))
import qualified Prelude as P
import Algebra.Additive              
import Algebra.Module                
import Algebra.Ring                  
import Math.Algebra.Hspray

Or, maybe better (I didn't try yet), follow the "Usage" section on the Hackage page of numeric-prelude.

Symbolic coefficients

Assume you have the polynomial a * (x² + y²) + 2b/3 * z, where a and b are symbolic coefficients. You can define this polynomial as a Spray as follows:

import Prelude hiding ((*), (+), (-))
import qualified Prelude as P
import Algebra.Additive              
import Algebra.Module                
import Algebra.Ring                  
import Math.Algebra.Hspray
import Data.Ratio

x = lone 1 :: Spray (Spray Rational)
y = lone 2 :: Spray (Spray Rational)
z = lone 3 :: Spray (Spray Rational)
a = lone 1 :: Spray Rational
b = lone 2 :: Spray Rational

poly = a *^ (x*x + y*y) + ((2%3) *^ b) *^ z 
prettySpray (prettySpray show "a") "X" poly
-- "((2 % 3) * a^(0, 1)) * X^(0, 0, 1) + ((1 % 1) * a^(1)) * X^(0, 2) + ((1 % 1) * a^(1)) * X^(2)"

The prettySpray function shows the expansion of the polynomial. You can extract the powers and the coefficients as follows:

l = toList poly
map fst l
-- [[0,0,1],[2],[0,2]]
map toList $ map snd l
-- [[([0,1],2 % 3)],[([1],1 % 1)],[([1],1 % 1)]]