added tufte style margin notes

docs/make.jl

@@ -1,13 +1,17 @@
 push!(LOAD_PATH, "../src")  #make the package available to the script
 using Documenter, ClenshawCurtisQuadrature #load the package and Documenter
 
+doctest(ClenshawCurtisQuadrature, fix=true)
+
 #generate the documentation
 makedocs(
     sitename="ClenshawCurtisQuadrature.jl",
-    doctest = false,    #FIXME: enable once examples are added to the documenation,
-
+    doctest = true,
+    format = Documenter.HTML(assets = ["assets/custom.css"])
 )
 
+
+
 deploydocs(
     repo = "github.com/DavidMSCode/ClenshawCurtisQuadrature.jl.git",
 )

docs/src/assets/custom.css

@@ -0,0 +1,42 @@
+/* Custom CSS for Tufte-style margin note */
+.admonition.is-info {
+    position: relative;
+    float: right;
+    width: var(--margin-width);
+    margin-left: 1em;
+    margin-right: calc(-1*var(--margin-width));
+    font-size: 0.85em;
+    color: #ffffff;
+    background: #ffffff;
+    padding: 0.5em;
+    border-style: none !important;
+    border-width: 0px;
+    border-left: 2px solid #3852C7 !important;
+}
+
+.docstring {
+    clear: right;
+}
+
+.content {
+    margin-right: var(--margin-width);
+    margin-left: 0px;
+}
+
+:root {
+    --margin-width: 300px;  /* Set the desired width of the margin note */
+}
+
+@media screen and (min-width: 1150px) {
+    html.theme--documenter-dark #documenter .docs-main {
+        max-width: 1150px;
+        margin-left: 20rem;
+        padding-right: 1rem;
+    }
+}
+
+@media screen and (max-width: 1150px) {
+    html.theme--documenter-dark #documenter .docs-main {
+        width: 100%;
+    }
+}

docs/src/index.md

@@ -28,17 +28,94 @@ and run
 The purpose of this module is to numerically solve integrals of the form
 
 ```math
-F(\tau) = F(-1) + \int_{-1}^{1}f(\tau)d\tau
+F(\tau) = F(-1) + \int_{-1}^{\tau}f(\xi)d\xi
 ```
-where ``f(\tau)`` is an integrand that is valid on the domain [-1,1] and
-``F(\tau)`` is an antiderivative of the integrand. 
-
 !!! note
     If ``F(\tau)`` is an antiderivative of ``f(\tau)``, then ``F(\tau)+c`` is
     also an antiderivative of ``f(\tau)`` for any constant value ``c``. But, if
     ``F(-1)`` is known, then there is only one antiderivative that satisfies the
     integral.
 
+where ``f(\tau)`` is an integrand that is valid on the domain ``[-1,1]`` and
+``F(\tau)`` is an antiderivative of the integrand. This is performed by
+representing ``F(\tau)`` and ``f(\tau)`` as a summation of Chebyshev
+polynomials on the domain ``[-1,1]``
+
+```math
+F(\tau) \approx \sum_{n=0}^N \alpha_n T_n(\tau)
+```
+
+```math
+f(\tau) \approx \sum_{n=0}^{N-1} a_n T_n(\tau)
+```
+where ``T_n(\tau)`` is the ``n``-th Chebyshev polynomial of the first kind.
+``N`` is the degree of the interpolating polynomial and ``a_n`` and ``\alpha_n``
+are the interpolating coefficients of the integrand and the particular
+antiderivative respectively. Because the integrals of the Chebyshev polynomials
+are related the values of Chebyshev polynomials of higher and lower degrees, we
+can perform linear operations on the integrand's coefficients to calculate
+the corresponding antiderivative's coefficients. The functions in this module
+are designed to generate the matrices required for this quadrature.
+
+## Examples
+If we want to know the integral on the domain ``[-1,1]`` of the function
+``f(\tau)=\tau^2``, we can do so by first sampling the function at set of M+1
+points. Because we know the antiderivative of ``f(\tau)`` is a 3rd order
+polynomial, I'll choose ``N=3`` as our desired interpolating polynomial of the
+solution. We can sample as many points as we want, but because the integrand
+will be represented as a 2nd order polynomial we need at least 3 points to
+calculate the coefficients.
+
+To do so first generate the values of ``\tau`` at cosine spaced nodes.
+!!! note
+    Because we just wanted the value of the integral on the range ``[-1,1]`` 
+    we assume that ``F(-1)=0``. The next examples will demonstrating solving
+    an initial value problem when ``F(-1)\neq 0``.
+
+    The Ta and T1 matrices returned by clenshaw\_curtis\_ivpi() are also
+    interpolating matrices. They specifically contain the values of the
+    chebyshev polynomials needed to interpolate the integrand and
+    antiderivative at each of the ``M+1`` cosine spaced nodes we calculated
+    before. Pre-calculating these matrices can be useful for iterative problems
+    where the integrand is unkown as a function of ``\tau``.
+
+```jldoctest
+using ClenshawCurtisQuadrature
+
+#set the degree variables
+M = 2
+N = 3
+
+#Generate the cosine spaced nodes
+Ms = 0:M
+taus = cos.(pi*Ms/M)
+
+#Sample the integrand
+f = (x) -> x^2
+ys = f.(taus)
+
+#Get the Least Squares Operation and Quadrature matrix
+A,Ta,P1,T1 = clenshaw_curtis_ivpi(N,M)
+
+#Calculate the least squares fit interpolating polynomial coefficients
+a = A*ys
+
+#Apply the quadrature matrix
+alpha = P1*a
+
+#Get the interpolating polynomials at τ=1
+T = interpolate(1,N)
+
+F1 = T*alpha
+
+# output
+
+1-element Vector{Float64}:
+ 0.6666666666666666
+```
+The result is that ``\int_{-1}^1 \tau^2d\tau=\frac{2}{3}`` which is exactly the
+correct answer.
+
 
 ## Methods