hspray

Simple multivariate polynomials in Haskell.

---

An object of type Spray a represents a multivariate polynomial whose coefficients are represented by the type a. For example:

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))
putStrLn $ prettyNumSpray poly
-- 8.0*x^4.y^2.z^2 + 4.0*x^3.y.z

This is the easiest way to construct a spray: first introduce the polynomial variables with the lone function, and then use arithmetic operations.

There are numerous functions to print a spray. If you don't like the letters x, y, z in the output of prettyNumSpray, you can use prettyNumSprayXYZ to change them to whatever you want:

putStrLn $ prettyNumSprayXYZ ["A","B","C"] poly
-- 8.0*A^4.B^2.C^2 + 4.0*A^3.B.C

Note that this function does not throw an error if you don't provide enough letters:

putStrLn $ prettyNumSprayXYZ ["A","B"] poly
-- 8.0*A1^4.A2^2.A3^2 + 4.0*A1^3.A2.A3

This is the same output as the one of prettyNumSprayX1X2X3 "A" poly.

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 :: QSpray -- QSpray = Spray Rational
y = lone 2 :: QSpray 
z = lone 3 :: QSpray
poly = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^-^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z))
putStrLn $ prettyQSpray poly
-- (7/6)*x^4.y^2.z^2 - (7/4)*x^3.y.z

Or a = Spray Double:

import Math.Algebra.Hspray
alpha = lone 1 :: Spray Double
x = lone 1 :: Spray (Spray Double)
y = lone 2 :: Spray (Spray Double)
poly = ((alpha *^ x) ^+^ (alpha *^ y))^**^2  
showSprayXYZ' (prettyNumSprayXYZ ["alpha"]) ["x","y"] poly
-- (alpha^2)*x^2 + (2.0*alpha^2)*x.y + (alpha^2)*y^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

Partial evaluation:

import Math.Algebra.Hspray
import Data.Ratio
x1 = lone 1 :: Spray Rational
x2 = lone 2 :: Spray Rational
x3 = lone 3 :: Spray Rational
poly = x1^**^2 ^+^ x2 ^+^ x3 ^-^ unitSpray
putStrLn $ prettyQSprayX1X2X3 "x" poly
-- x1^2 + x2 + x3 - 1
--
-- substitute x1 -> 2 and x3 -> 3
poly' = substituteSpray [Just 2, Nothing, Just 3] poly
putStrLn $ prettyQSprayX1X2X3 "x" poly'
-- x2 + 6

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)
putStrLn $ prettyNumSpray poly
-- 3.0*x^2 + 2.0*x.y.z
--
-- derivative with respect to x
putStrLn $ prettyNumSpray $ derivative 1 poly
-- 6.0*x + 2.0*y.z"

Gröbner bases

As of version 2.0.0, it is possible to compute a Gröbner basis.

import Math.Algebra.Hspray
import Data.Ratio
-- define the elementary monomials
o = lone 0 :: Spray Rational -- same as unitSpray
x = lone 1 :: Spray Rational
y = lone 2 :: Spray Rational
z = lone 3 :: Spray Rational
-- define three polynomials
p1 = x^**^2 ^+^ y ^+^ z ^-^ o -- X² + Y + Z - 1
p2 = x ^+^ y^**^2 ^+^ z ^-^ o -- X + Y² + Z - 1
p3 = x ^+^ y ^+^ z^**^2 ^-^ o -- X + Y + Z² - 1
-- compute the reduced Gröbner basis
gbasis = groebner [p1, p2, p3] True
-- show result
prettyResult = map prettyQSpray gbasis
mapM_ print prettyResult
-- "x + y + z^2 - 1"
-- "y^2 - y - z^2 + z"
-- "y.z^2 + (1/2)*z^4 - (1/2)*z^2"
-- "z^6 - 4*z^4 + 4*z^3 - z^2"

Easier usage

To construct a spray using the ordinary symbols +, -, * and ^, one can hide these operators from Prelude and import them from the numeric-prelude library; constructing a spray in this context is easier:

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 :: QSpray 
y = lone 2 :: QSpray 
z = lone 3 :: QSpray
poly  = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^-^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z))
poly' = ((2%3) *^ (x^3 * y * z) - x^2) * ((7%4) *^ (x * y * z))
poly == poly'
-- True

Note that *> could be used instead of *^ but running lambda *> spray possibly throws an "ambiguous type" error regarding the type of lambda.

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 rational numbers. You can represent this polynomial by a Spray (Spray Rational) spray as follows:

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

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^2 + y^2) + ((2 *^ b) /^ 3) *^ z 
putStrLn $ 
  showSprayXYZ' (prettyQSprayXYZ ["a","b"]) ["X","Y","Z"] poly
-- (a)*X^2 + (a)*Y^2 + ((2/3)*b)*Z

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)]]

These Spray (Spray a) sprays can be very useful. They represent polynomials whose coefficients depend on some parameters, with a polynomial dependence. For example, the coefficients of the Gegenbauer polynomials are polynomials in their parameter \(\alpha\) (this is clear from the recurrence relation). Here is their implementation in hspray:

gegenbauerPolynomial :: Int -> Spray (Spray Rational) 
gegenbauerPolynomial n 
  | n == 0 = unitSpray
  | n == 1 = (2.^a) *^ x
  | otherwise = 
    (2.^(n'' ^+^ a) /^ n') *^ (x ^*^ gegenbauerPolynomial (n - 1))
    ^-^ ((n'' ^+^ 2.^a ^-^ unitSpray) /^ n') *^ gegenbauerPolynomial (n - 2)
  where 
    x = lone 1 :: Spray (Spray Rational)
    a = lone 1 :: Spray Rational
    n'  = toRational n
    n'' = constantSpray (n' - 1)

Let's try it:

n = 3
g = gegenbauerPolynomial n
putStrLn $ 
  showSprayXYZ' (prettyQSprayXYZ ["alpha"]) ["X"] g
-- ((4/3)*alpha^3 + 4*alpha^2 + (8/3)*alpha)*X^3 + (-2*alpha^2 - 2*alpha)*X

Let's check the differential equation:

g'  = derivative 1 g
g'' = derivative 1 g'
alpha = lone 1 :: Spray Rational
x     = lone 1 :: Spray (Spray Rational)
nAsSpray = constantSpray (toRational n)
shouldBeZero = 
  (unitSpray ^-^ x^**^2) ^*^ g''
  ^-^ (2.^alpha ^+^ unitSpray) *^ (x ^*^ g')
  ^+^ n.^(nAsSpray ^+^ 2.^alpha) *^ g
putStrLn $ prettySpray shouldBeZero
-- 0

Now, how to substitute a value to the parameter \(\alpha\)? The package provides the function evalSpraySpray to perform this task:

putStrLn $ 
  prettyQSpray'' $ evalSpraySpray g [1]
-- 8*X^3 - 4*X

This is a Spray Rational spray.

The SymbolicSpray type

If you have only one symbolic coefficient, you can deal with the sprays of type SymbolicSpray a. These are sprays whose coefficients are ratios of univariate polynomials, so this allows more possibilities than a Spray (Spray a). Since the variable of these univariate polynomials occurs in the coefficients of such a spray, I call it the outer variable sometimes, although I do not very like this name (see below). And I say that the variables of the symbolic spray are the inner variables or the main variables, though I would prefer to simply call them the variables. Assume you want to deal with the polynomial 4/5 * a/(a² + 1) * (x² + y²) + 2a/3 * yz. Then you define it as follows:

import           Prelude hiding ((*), (+), (-), (/), (^), (*>))
import qualified Prelude as P
import           Algebra.Additive              
import           Algebra.Module            
import           Algebra.Ring
import           Algebra.Field
import           Math.Algebra.Hspray
import           Number.Ratio       ( (%), T ( (:%) ) )
x = lone 1 :: SymbolicQSpray 
y = lone 2 :: SymbolicQSpray 
z = lone 3 :: SymbolicQSpray 
a = outerQVariable  
sSpray 
  = ((4%5) *. (a :% (a^2 + one))) *> (x^2 + y^2)  +  (constQPoly (2%3) * a) *> (y * z)
putStrLn $ prettySymbolicQSpray' "a" sSpray
-- { [ (4/5)*a ] %//% [ a^2 + 1 ] }*X^2 + { [ (4/5)*a ] %//% [ a^2 + 1 ] }*Y^2 + { (2/3)*a }*Y.Z

There are three possible evaluations of a symbolic spray:

-- substitute a value for 'a':
putStrLn $ 
  prettyQSpray''' $ evalSymbolicSpray sSpray (6%5)
-- (24/61)*X^2 + (24/61)*Y^2 + (4/5)*Y.Z

-- substitute a value for 'a' and some values for 'X', 'Y', 'Z':
evalSymbolicSpray' sSpray (6%5) [2, 3, 4%7]
-- 13848 % 2135

-- substitute some values for 'X', 'Y', 'Z':
putStrLn $ 
  prettyRatioOfQPolynomials "a" $ evalSymbolicSpray'' sSpray [2, 3, 4%7]
-- [ (8/7)*a^3 + (404/35)*a ] %//% [ a^2 + 1 ]

Although it does not make sense to replace the main variables (X, Y, Z) of a symbolic spray with some fractions of univariate polynomials, this feature is not provided. We rather consider that a SymbolicSpray K spray defines a multivariate polynomial on the field K whose coefficients lie in K but depend on a parameter, the so-called outer variable ("a"). By the way I am not a fan of this name, and maybe the parameter would be a better name? And then parametric spray would be a better name than symbolic spray? Do not hesitate to open a Github issue to leave some comments if you want!

The nice point regarding these ratios of univariate polynomials is that they are automatically "simplified" (i.e. written as irreducible fractions). For example:

polyFrac = (a^8 - one) ^/^ (a - one)
putStrLn $ prettyRatioOfQPolynomials "a" polyFrac
-- a^7 + a^6 + a^5 + a^4 + a^3 + a^2 + a + 1

Maybe you prefer the fractional form, but it is nice to see that this ratio of polynomials actually is a polynomial. Note that I used ^/^ here and not :%. That's because :% does not simplify the fraction, it just constructs a fraction with the given numerator and denominator. Whenever an arithmetic operation is performed on a fraction, the result is always simplified. So the ^/^ operator simply constructs a fraction with :% and then it multiplies it by one to get the simplification.

As of version 0.2.7: RatioOfSprays

So far we have good stuff to deal with symbolic coefficients: the Spray (Spray a) sprays and the SymbolicSpray a sprays. The SymbolicSpray a sprays are successfully used in the jackpolynomials package. However this is not enough. For example we cannot implement the Jacobi polynomials with symbolic parameters, because they have two parameters and their recurrence relation involves some divisions of their coefficients.

We need a new type, similar to SymbolicSpray a but allowing multivariate fractions of polynomials for the coefficients.

A first step in this direction has been achieved in version 0.2.7: the type RatioOfSprays a, whose objects represent ratios of sprays, has been introduced. Thus it suffices to introduce the type Spray (RatioOfSprays a) now.

Thus the Spray (RatioOfSprays a) sprays are more general than the SymbolicSpray a sprays, which are restricted to univariate fractions of polynomials. But it is possible that the Spray (RatioOfSprays a) sprays will be less efficient than the SymbolicSpray a sprays in the univariate case. I will have to benchmark in order to get an answer to this question.

To construct a ratio of sprays, apply %//% between its numerator and its denominator:

import Math.Algebra.Hspray
x = qlone 1 -- shortcut for  lone 1 :: Spray Rational
y = qlone 2 
rOS = (x ^-^ y) %//% (x^**^2 ^-^ y^**^2)
putStrLn $ prettyRatioOfQSprays rOS
-- [ 1 ] %//% [ x + y ]

The %//% operator always returns an irreducible fraction.

The RatioOfSprays a type makes sense when a has a field instance, and then it has a field instance too. To use the field operations, import the necessary modules from numeric-prelude, and hide these operations from the Prelude module (then you can also use the numeric-prelude operations for sprays, instead of using ^+^, ^-^, ^*^, ^**^):

import Prelude hiding ((+), (-), (*), (/), (^), (*>), (<*))
import qualified Prelude as P
import Algebra.Additive              
import Algebra.Module
import Algebra.RightModule
import Algebra.Ring
import Algebra.Field              
import Math.Algebra.Hspray
x = qlone 1  
y = qlone 2 
p = x^2 - 3*^(x * y) + y^3 
q = x - y
rOS1 = p^2 %//% q
rOS2 = rOS1 + unitRatioOfSprays
rOS = rOS1^2 + rOS1*rOS2 - rOS1/rOS2 + rOS2 -- slow!
(rOS1 + rOS2) * (rOS1 - rOS2) == rOS1^2 - rOS2^2
-- True

The RatioOfSprays a type also has left and right module instances over a and over Spray a as well. That means you can multiply a ratio of sprays by a scalar and by a spray, by using, depending on the side, either *> or <*:

import Data.Ratio ( (%) )
rOS' = (3%4::Rational) *> rOS^2  +  p *> rOS
rOS' / rOS' == unitRatioOfSprays
-- True

You can also divide a ratio of sprays by a spray with %/%:

p *> (rOS' %/% p) == rOS'
-- True
rOS1 %/% p == p %//% q
-- True

When a has a field instance, both a Spray a spray and a RatioOfSprays a ratio of sprays can be divided by a scalar with the /> operator:

k = 3 :: Rational
(p /> k) *> rOS == p *> (rOS /> k)
-- True

Use evalRatioOfSprays to evaluate a ratio of sprays:

import Data.Ratio ( (%) )
f :: Algebra.Field.C a => a -> a -> a
f u v = u^2 + u*v - u/v + v
rOS == f rOS1 rOS2
-- True
values = [2%3, 7%4]
r1 = evalRatioOfSprays rOS1 values
r2 = evalRatioOfSprays rOS2 values
evalRatioOfSprays rOS values == f r1 r2
-- True

Other features

Resultant and subresultants of two polynomials, and greatest common divisor of two polynomials with coefficients in a field.