Added lead and update README

Author: scheinerman

README.md

@@ -10,7 +10,8 @@
 
 
 This module defines the `SimplePolynomial` type. These are polynomials
-with exact coefficients (integers, Gaussian integers, or Gaussian rationals).
+with exact coefficients (integers, Gaussian integers, Gaussian rationals,
+or `Mod`s).
 
 
 ## Construction
@@ -33,6 +34,14 @@ julia> SimplePolynomial(3,-2,0,1) == SimplePolynomial(3,-2,0,1,0,0)
 true
 ```
 
+These polynomials support `Mod` coefficients:
+```
+julia> using Mods
+
+julia> p = SimplePolynomial( Mod{7}.(1:5) )
+Mod{7}(1) + Mod{7}(2)*x + Mod{7}(3)*x^2 + Mod{7}(4)*x^3 + Mod{7}(5)*x^4
+```
+
 #### Using `getx`
 
 The function `getx()` is a short cut that returns
@@ -63,6 +72,7 @@ assign to a coefficient; that is, `p[k]=c` does not work.
 * `monic(p)` returns a new `SimplePolynomial` formed by dividing
 all the coefficients by the leading term.
 * `eltype(p)` returns the data type of the coefficients.
+* `lead(p)` returns the coefficient of the highest power of `x` in `p`.
 
 
 ## Arithmetic

src/SimplePolynomials.jl

@@ -4,7 +4,8 @@ using Mods
 import Base: getindex, (==), show, zero, one, eltype, adjoint
 import Polynomials: degree, Polynomial, coeffs, roots, derivative, integrate
 
-export SimplePolynomial, degree, coeffs, getx, Polynomial, roots, derivative, integrate
+export SimplePolynomial, degree, coeffs, getx, Polynomial, roots
+export derivative, integrate, lead
 
 # IntegerX is any sort of real or Gaussian integer
 IntegerX = Union{S,Complex{S},Mod} where S<:Integer
@@ -45,6 +46,13 @@ SimplePolynomial() = SimplePolynomial([0])
 coeffs(p::SimplePolynomial) = copy(p.data)
 eltype(p::SimplePolynomial) = eltype(p.data)
 
+"""
+`lead(p::SimplePolynomial)` returns the coefficient of the highest
+power of `x` in the polynomial `p`. This is nonzero unless `p` is
+the zero polynomial.
+"""
+lead(p::SimplePolynomial) = p.data[end]
+
 
 # conversion to/from Polynomial type
 SimplePolynomial(P::Polynomial) = SimplePolynomial(P.coeffs)

test/runtests.jl

@@ -4,6 +4,7 @@ a = SimplePolynomial(1,1,1)
 @test a(10) == 111
 @test a[0] == 1
 @test a[9] == 0
+@test lead(a) == 1
 
 
 @test a+a == 2a