


Description 
Polynomials and rational functions in a single indeterminate.
Polynomials are represented by a list of coefficients.
All nonzero coefficients are listed, but there may be extra '0's at the end.
Usage:
Say you have the ring of Integer numbers
and you want to add a transcendental element x,
that is an element, which does not allow for simplifications.
More precisely, for all positive integer exponents n
the power x^n cannot be rewritten as a sum of powers with smaller exponents.
The element x must be represented by the polynomial [0,1].
In principle, you can have more than one transcendental element
by using polynomials whose coefficients are polynomials as well.
However, most algorithms on multivariate polynomials
prefer a different (sparse) representation,
where the ordering of elements is not so fixed.
If you want division, you need Number.Ratios
of polynomials with coefficients from a Algebra.Field.
You can also compute with an algebraic element,
that is an element which satisfies an algebraic equation like
x^3x1==0.
Actually, powers of x with exponents above 3 can be simplified,
since it holds x^3==x+1.
You can perform these computations with Number.ResidueClass of polynomials,
where the divisor is the polynomial equation that determines x.
If the polynomial is irreducible
(in our case x^3x1 cannot be written as a nontrivial product)
then the residue classes also allow unrestricted division
(except by zero, of course).
That is, using residue classes of polynomials
you can work with roots of polynomial equations
without representing them by radicals
(powers with fractional exponents).
It is wellknown, that roots of polynomials of degree above 4
may not be representable by radicals.


Synopsis 



Documentation 


Instances  Functor T  C T  C a b => C a (T b)  (C a, C a b) => C a (T b)  (Eq a, C a) => Eq (T a)  (C a, Eq a, Show a, C a) => Fractional (T a)  (C a, Eq a, Show a, C a) => Num (T a)  Show a => Show (T a)  (Arbitrary a, C a) => Arbitrary (T a)  (C a, C a) => C (T a)  C a => C (T a)  C a => C (T a)  C a => C (T a)  C a => C (T a)  (C a, C a) => C (T a)  (C a, C a) => C (T a)  (C a, C a) => C (T a) 













evaluateCoeffVector :: C a v => T v > a > v  Source 

Here the coefficients are vectors,
for example the coefficients are real and the coefficents are real vectors.


evaluateArgVector :: (C a v, C v) => T a > v > v  Source 

Here the argument is a vector,
for example the coefficients are complex numbers or square matrices
and the coefficents are reals.



compose is the functional composition of polynomials.
It fulfills
eval x . eval y == eval (compose x y)




add :: C a => [a] > [a] > [a]  Source 


sub :: C a => [a] > [a] > [a]  Source 




horner :: C a => a > [a] > a  Source 

Horner's scheme for evaluating a polynomial in a ring.


hornerCoeffVector :: C a v => a > [v] > v  Source 

Horner's scheme for evaluating a polynomial in a module.


hornerArgVector :: (C a v, C v) => v > [a] > v  Source 



Multiply by the variable, used internally.




mul :: C a => [a] > [a] > [a]  Source 

mul is fast if the second argument is a short polynomial,
MathObj.PowerSeries.** relies on that fact.


scale :: C a => a > [a] > [a]  Source 


divMod :: (C a, C a) => [a] > [a] > ([a], [a])  Source 


tensorProduct :: C a => [a] > [a] > [[a]]  Source 


tensorProductAlt :: C a => [a] > [a] > [[a]]  Source 


mulShear :: C a => [a] > [a] > [a]  Source 


mulShearTranspose :: C a => [a] > [a] > [a]  Source 




differentiate :: C a => [a] > [a]  Source 


integrate :: C a => a > [a] > [a]  Source 


integrateInt :: (C a, C a) => a > [a] > [a]  Source 

Integrates if it is possible to represent the integrated polynomial
in the given ring.
Otherwise undefined coefficients occur.




alternate :: C a => [a] > [a]  Source 


Produced by Haddock version 2.6.0 