A spray represents a multivariate polynomial so it has some variables. We introduce a class because it will be assigned to the ratios of sprays too.

Whether an object of class HasVariables is constant

Whether an object of class HasVariables is univariate; it is considered that it is univariate if it is constant

Whether an object of class HasVariables is bivariate; it is considered that it is bivariate if it is univariate

Whether an object of class HasVariables is trivariate; it is considered that it is trivariate if it is bivariate

The n-th polynomial variable x_n as a spray; one usually builds a spray by introducing these variables and combining them with the arithmetic operations

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> spray = 2*^x^**^2 ^-^ 3*^y
>>> putStrLn $ prettyNumSpray spray
2*x^2 - 3*y

lone 0 == unitSpray

The n-th polynomial variable for rational sprays; this is just a specialization of lone

The unit spray

spray ^*^ unitSpray == spray

The null spray

spray ^+^ zeroSpray == spray

Constant spray

constantSpray 3 == 3 *^ unitSpray

Scales a spray by a scalar; if you import the Algebra.Module module then it is the same operation as (*>) from this module

Divides a spray by a scalar; you can equivalently use (/>) if the type of the scalar is not ambiguous

Addition of two sprays

Substraction of two sprays

Multiply two sprays

Power of a spray

Pretty form of a spray with monomials displayed in the style of "x.z^2"; you should rather use prettyNumSpray or prettyQSpray if you deal with sprays with numeric coefficients

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettySpray p
(2)*x + (3)*y^2 + (-4)*z^3
>>> putStrLn $ prettySpray (p ^+^ lone 4)
(2)*x1 + (3)*x2^2 + (-4)*x3^3 + x4

prettySpray spray == prettySprayXYZ ["x", "y", "z"] spray

Pretty form of a spray, with monomials shown as "x1.x3^2"; use prettySprayX1X2X3 to change the letter (or prettyNumSprayX1X2X3 or prettyQSprayX1X2X3 if the coefficients are numeric)

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettySpray' p
(2)*x1 + (3)*x2^2 + (-4)*x3^3 

Pretty form of a spray; you will probably prefer prettySpray or prettySpray'

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettySpray'' "x" p
(2)*x^(1) + (3)*x^(0, 2) + (-4)*x^(0, 0, 3)

Pretty form of a spray with monomials displayed in the style of "x.z^2"; you should rather use prettyNumSprayXYZ or prettyQSprayXYZ if your coefficients are numeric

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettySprayXYZ ["X", "Y", "Z"] p
(2)*X + (3)*Y^2 + (-4)*Z^3
>>> putStrLn $ prettySprayXYZ ["X", "Y"] p
(2)*X1 + (3)*X2^2 + (-4)*X3^3

Pretty form of a spray with monomials displayed in the style of "x1.x3^2"; you should rather use prettyNumSprayX1X2X3 or prettyQSprayX1X2X3 if your coefficients are numeric

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> spray = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettySprayX1X2X3 "X" spray
(2)*X1 + (3)*X2^2 + (-4)*X3^3

Prints a spray; this function is exported for possible usage in other packages

Prints a spray, with monomials shown as "x.z^2", and with a user-defined showing function for the coefficients

Prints a spray, with monomials shown as "x.z^2", and with a user-defined showing function for the coefficients; this is the same as the function showSprayXYZ with the pair of braces ("(", ")")

Pretty form of a spray, with monomials shown as "x1.x3^2", and with a user-defined showing function for the coefficients

Pretty form of a spray, with monomials shown as "x1.x3^2", and with a user-defined showing function for the coefficients; this is the same as the function showSprayX1X2X3 with the pair of braces ("(", ")") used to enclose the coefficients

Show a spray with numeric coefficients; this function is exported for possible usage in other packages

Prints a QSpray; for internal usage but exported for usage in other packages

Prints a QSpray'; for internal usage but exported for usage in other packages

Pretty form of a spray with numeric coefficients, printing monomials as "x1.x3^2"

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettyNumSprayX1X2X3 "x" p
2*x1 + 3*x2^2 - 4*x3^3 

Pretty form of a spray with rational coefficients, printing monomials in the style of "x1.x3^2"

>>> x = lone 1 :: QSpray
>>> y = lone 2 :: QSpray
>>> z = lone 3 :: QSpray
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ (4%3)*^z^**^3
>>> putStrLn $ prettyQSprayX1X2X3 "x" p
2*x1 + 3*x2^2 - (4/3)*x3^3 

Same as prettyQSprayX1X2X3 but for a QSpray' spray

Pretty form of a spray with numeric coefficients, printing monomials as "x.z^2" if possible, i.e. if enough letters are provided, otherwise as "x1.x3^2"

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> w = lone 4 :: Spray Int
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
>>> putStrLn $ prettyNumSprayXYZ ["x","y","z"] p
2*x + 3*y^2 - 4*z^3 
>>> putStrLn $ prettyNumSprayXYZ ["x","y","z"] (p ^+^ w)
2*x1 + 3*x2^2 - 4*x3^3 + x4
>>> putStrLn $ prettyNumSprayXYZ ["a","b","c"] (p ^+^ w)
2*a1 + 3*a2^2 - 4*a3^3 + a4

Pretty form of a spray with rational coefficients, printing monomials in the style of "x.z^2" with the provided letters if possible, i.e. if enough letters are provided, otherwise in the style "x1.x3^2", taking the first provided letter to denote the non-indexed variables

>>> x = lone 1 :: QSpray
>>> y = lone 2 :: QSpray
>>> z = lone 3 :: QSpray
>>> p = 2*^x ^+^ 3*^y^**^2 ^-^ (4%3)*^z^**^3
>>> putStrLn $ prettyQSprayXYZ ["x","y","z"] p
2*x + 3*y^2 - (4/3)*z^3 
>>> putStrLn $ prettyQSprayXYZ ["x","y"] p
2*x1 + 3*x2^2 - (4%3)*x3^3
>>> putStrLn $ prettyQSprayXYZ ["a","b"] p
2*a1 + 3*a2^2 - (4/3)*a3^3

Same as prettyQSprayXYZ but for a QSpray' spray

Pretty printing of a spray with numeric coefficients prop> prettyNumSpray == prettyNumSprayXYZ ["x", "y", "z"]

Pretty printing of a spray with numeric coefficients prop> prettyNumSpray' == prettyNumSprayXYZ [X, Y, Z]

Pretty printing of a spray with rational coefficients prop> prettyQSpray == prettyQSprayXYZ ["x", "y", "z"]

Pretty printing of a spray with rational coefficients prop> prettyQSpray'' == prettyQSprayXYZ [X, Y, Z]

Pretty printing of a spray with rational coefficients prop> prettyQSpray' == prettyQSprayXYZ' ["x", "y", "z"]

Pretty printing of a spray with rational coefficients prop> prettyQSpray''' == prettyQSprayXYZ' [X, Y, Z]

Identify a rational to a A Rational' element

Division of univariate polynomials; this is an application of :% followed by a simplification of the obtained fraction of the two polynomials

Pretty form of a ratio of univariate polynomials

Pretty form of a ratio of univariate polynomials with rational coefficients

Scale a ratio of univariate polynomials by a scalar

Constant univariate polynomial

Univariate polynomial from its coefficients (ordered by increasing degrees)

The variable of a univariate polynomial; it is called "outer" because this is the variable occuring in the polynomial coefficients of a SymbolicSpray

Constant rational univariate polynomial

>>> import Number.Ratio ( (%) )
>>> constQPoly (2 % 3)

constQPoly (2 % 3) == qpolyFromCoeffs [2 % 3]

Rational univariate polynomial from coefficients

>>> import Number.Ratio ( (%) )
>>> qpolyFromCoeffs [2 % 3, 5, 7 % 4]

The variable of a univariate rational polynomial; it is called "outer" because it is the variable occuring in the coefficients of a SymbolicQSpray (but I do not like this name - see README)

outerQVariable == qpolyFromCoeffs [0, 1]

Evaluates a ratio of univariate polynomials

Pretty form of a symbolic spray, using a string (typically a letter) followed by an index to denote the variables

Pretty form of a symbolic spray, using some given strings (typically some letters) to denote the variables if possible, i.e. if enough letters are provided; otherwise this function behaves exactly like prettySymbolicQSprayX1X2X3 a where a is the first provided letter

Pretty form of a symbolic spray; see the definition below and see prettySymbolicSprayXYZ

prettySymbolicSpray a spray == prettySymbolicSprayXYZ a ["x","y","z"] spray

Pretty form of a symbolic spray; see the definition below and see prettySymbolicSprayXYZ

prettySymbolicSpray' a spray == prettySymbolicSprayXYZ a ["X","Y","Z"] spray

Pretty form of a symbolic rational spray, using a string (typically a letter) followed by an index to denote the variables

Pretty form of a symbolic rational spray, using some given strings (typically some letters) to denote the variables if possible, i.e. if enough letters are provided; otherwise this function behaves exactly like prettySymbolicQSprayX1X2X3 a where a is the first provided letter

Pretty form of a symbolic rational spray, using "x", "y" and "z" for the variables if possible; i.e. if the spray does not have more than three variables, otherwise "x1", "x2", ... are used to denote the variables

prettySymbolicQSpray a == prettySymbolicQSprayXYZ a ["x","y","z"]

Pretty form of a symbolic rational spray, using X, Y and Z for the variables if possible; i.e. if the spray does not have more than three variables, otherwise X1, X2, ... are used

prettySymbolicQSpray' a = prettySymbolicQSprayXYZ a ["X","Y","Z"]

Simplifies the coefficients (the fractions of univariate polynomials) of a symbolic spray

Substitutes a value to the outer variable of a symbolic spray (the variable occuring in the coefficients)

Substitutes a value to the outer variable of a symbolic spray as well as some values to the main variables of this spray

Substitutes some values to the main variables of a symbolic spray

A RatioOfSprays a object represents a fraction of two multivariate polynomials whose coefficients are of type a which represents a field. These two polynomials are represented by two Spray a objects. Generally we do not use this constructor to build a ratio of sprays: we use the %//% operator instead, because it always returns an irreducible ratio of sprays, meaning that its corresponding fraction of polynomials is irreducible, i.e. its numerator and its denominator are coprime. You can use this constructor if you are sure that the numerator and the denominator are coprime. This can save some computation time, but unfortunate consequences can occur if the numerator and the denominator are not coprime. An arithmetic operation on ratios of sprays always returns an irreducible ratio of sprays under the condition that the ratios of sprays it involves are irreducible. Moreover, it never returns a ratio of sprays with a constant denominator other than the unit spray. If you use this constructor with a constant denominator, always set this denominator to the unit spray (by dividing the numerator by the constant value of the denominator).

Irreducible ratio of sprays from numerator and denominator

Division of a ratio of sprays by a spray

Whether a ratio of sprays is constant; this is an alias of isConstant

Wheter a ratio of sprays actually is polynomial, that is, whether its denominator is a constant spray (and then it should be the unit spray)

The null ratio of sprays

The null ratio of sprays

The unit ratio of sprays

The unit ratio of sprays

Constant ratio of sprays

Coerces a spray to a ratio of sprays

Evaluates a ratio of sprays; this is an alias of evaluate

Substitutes some variables in a ratio of sprays; this is an alias of substitute

Jacobi polynomial

Converts a ratio of polynomials to a ratio of sprays

Converts a ratio of rational polynomials to a ratio of rational sprays; this is not a specialization of fromRatioOfPolynomials because RatioOfQPolynomials is RatioOfPolynomials a with a = Rational', not with a = Rational

General function to print a RatioOfSprays object

Prints a ratio of sprays with numeric coefficients

Prints a ratio of sprays with rational coefficients

Prints a ratio of sprays

Prints a ratio of sprays

Prints a ratio of sprays

Prints a ratio of sprays

Prints a ratio of sprays with rational coefficients

Prints a ratio of sprays with rational coefficients, printing the monomials in the style of "x1^2.x2.x3^3"

Prints a ratio of sprays with rational coefficients

prettyRatioOfQSprays rOS == prettyRatioOfQSpraysXYZ ["x","y","z"] rOS

Prints a ratio of sprays with rational coefficients

prettyRatioOfQSprays' rOS == prettyRatioOfQSpraysXYZ ["X","Y","Z"] rOS

Prints a ratio of sprays with numeric coefficients

Prints a ratio of sprays with numeric coefficients, printing the monomials in the style of "x1^2.x2.x3^3"

Prints a ratio of sprays with numeric coefficients

prettyRatioOfNumSprays rOS == prettyRatioOfNumSpraysXYZ ["x","y","z"] rOS

Prints a ratio of sprays with numeric coefficients

prettyRatioOfNumSprays' rOS == prettyRatioOfNumSpraysXYZ ["X","Y","Z"] rOS

Get coefficient of a term of a spray

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = 2 *^ (2 *^ (x^**^3 ^*^ y^**^2)) ^+^ 4*^z ^+^ 5*^unitSpray
>>> getCoefficient [3, 2, 0] p
4
>>> getCoefficient [0, 4] p
0

Get the constant term of a spray

getConstantTerm p == getCoefficient [] p

Whether a spray is constant; this is an alias of isConstant

Terms of a spray

Evaluates a spray; this is an alias of evaluate

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> spray = 2*^x^**^2 ^-^ 3*^y
>>> evalSpray spray [2, 1]
5

Substitutes some variables in a spray by some values; this is an alias of substitute

>>> x1 = lone 1 :: Spray Int
>>> x2 = lone 2 :: Spray Int
>>> x3 = lone 3 :: Spray Int
>>> p = x1^**^2 ^-^ x2 ^+^ x3 ^-^ unitSpray
>>> p' = substituteSpray [Just 2, Nothing, Just 3] p
>>> putStrLn $ prettyNumSprayX1X2X3 "x" p'
-x2 + 6 

Sustitutes the variables of a spray with some sprays (e.g. change of variables)

>>> x = lone 1 :: Spray Int
>>> y = lone 2 :: Spray Int
>>> z = lone 3 :: Spray Int
>>> p = x ^+^ y
>>> q = composeSpray p [z, x ^+^ y ^+^ z]
>>> putStrLn $ prettyNumSpray' q
X + Y + 2*Z

Evaluates the coefficients of a spray with spray coefficients; see README for an example

Division of a spray by a spray

Remainder of the division of a spray by a list of divisors, using the lexicographic ordering of the monomials

Gröbner basis, always minimal and possibly reduced

groebner sprays True == reduceGroebnerBasis (groebner sprays False)

Reduces a Groebner basis

Elementary symmetric polynomial

>>> putStrLn $ prettySpray' (esPolynomial 3 2)
(1)*x1x2 + (1)*x1x3 + (1)*x2x3

Power sum polynomial

Whether a spray is a symmetric polynomial, an inefficient algorithm (use the function with the same name in the jackpolynomials package if you need efficiency)

Resultant of two sprays

Resultant of two sprays with coefficients in a field; this function is more efficient than the function resultant

Resultant of two univariate sprays

Subresultants of two sprays

Subresultants of two univariate sprays

Greatest common divisor of two sprays with coefficients in a field

Determinant of a matrix with entries in a ring by using Laplace expansion (this is slow); the numeric-prelude package provides some stuff to deal with matrices over a ring but it does not provide the determinant

Determinant of a matrix over a ring by using Laplace expansion; this is the same as detLaplace but for a matrix from the numeric-prelude package

Characteristic polynomial of a square matrix

>>> import Data.Matrix (Matrix, fromLists)
>>> m = fromLists [ [12, 16, 4]
>>> , [16, 2, 8]
>>> , [8, 18, 10] ] :: Matrix Int
>>> spray = characteristicPolynomial m
>>> putStrLn $ prettyNumSpray spray
-x^3 + 24*x^2 + 268*x - 1936

Scale by an integer (I do not find this operation in numeric-prelude)

3 .^ x == x Algebra.Additive.+ x Algebra.Additive.+ x

Divides by a scalar in a module over a field

Creates a spray from a list of terms

Spray as a list

Converts a spray with rational coefficients to a spray with double coefficients (useful for evaluation)

Leading term of a spray

Whether a spray can be written as a polynomial of a given list of sprays (the sprays in the list must belong to the same polynomial ring as the spray); this polynomial is returned if this is true

>>> x = lone 1 :: Spray Rational
>>> y = lone 2 :: Spray Rational
>>> p1 = x ^+^ y
>>> p2 = x ^-^ y
>>> p = p1 ^*^ p2

isPolynomialOf p [p1, p2] == (True, Just $ x ^*^ y)

Bombieri spray (for internal usage in the 'scubature' library)

Whether two sprays are equal up to a scalar factor

Gegenbauer polynomials; we mainly provide them to give an example of the Spray (Spray a) type

>>> gp = gegenbauerPolynomial 3
>>> putStrLn $ showSprayXYZ' (prettyQSprayXYZ ["alpha"]) ["X"] gp
((4/3)*alpha^3 + 4*alpha^2 + (8/3)*alpha)*X^3 + (-2*alpha^2 - 2*alpha)*X
>>> putStrLn $ prettyQSpray'' $ evalSpraySpray gp [1]
8*X^3 - 4*X