Update README.md

Author: dlfivefifty

README.md

@@ -1,2 +1,38 @@
 # PiecewiseOrthogonalPolynomials.jl
 A Julia package for piecewise orthogonal polynomials which can be used in p-FEM
+
+
+We can make piecewise integrated-Legendre bases using the quasi-matrix `ContinuousPolynomial{1}`:
+```julia
+using PiecewiseOrthogonalPolynomials, Plots
+
+𝐗 = range(-1,1; length=4)
+C = ContinuousPolynomial{1}(𝐗)
+plot(C[:,Block(1)])
+```
+```julia
+plot(C[:,Block.(2:3)])
+```
+
+The mass matrix can be constructed via:
+```julia
+M = C'C
+```
+We can also construct the stiffness matrix:
+```julia
+Δ = weaklaplacian(C)
+```
+
+We can truncate as follows:
+```julia
+N = 10
+KR = Block.(Base.OneTo(N))
+Mₙ = M[KR,KR]
+Δₙ = Δ[KR,KR]
+```
+
+We can compute the reverse Cholesky in optimal complexity:
+```julia
+using MatrixFactorizations
+L = reversecholesky(Symmetric(-Δₙ + Mₙ)).L
+```