Final touches.

samuelsonric · Sep 26, 2023 · 11d7c48 · 11d7c48
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diff --git a/docs/literate/pixel.jl b/docs/literate/pixel.jl
@@ -4,9 +4,15 @@ using Catlab
 using UnicodePlots
 
 using Catlab.CategoricalAlgebra.FinRelations: BoolRig
-# Spivak, Dobson, Kumari, and Wu, *Pixel Arrays: A fast and elementary method for solving nonlinear systems.*
+# A pixel array is an array with entries in the boolean semiring. The function defined below
+# constructs a pixel array by discretizing a relation of the form
+# ```math
+# xRy \iff f(x, y) = 0.
+# ```
 #
-# A pixel array is an array with entries in the boolean semiring. The function defined below constructs a pixel array by discretizing a relation of the form ``xRy \iff f(x, y) = 0``.
+# References:
+# - Spivak, David I. et al. "Pixel Arrays: A fast and elementary method for solving
+# nonlinear systems." *arXiv: Numerical Analysis* (2016): n. pag.
 const PixelArray{N} = Array{BoolRig, N}
 
 function PixelArray(f::Function, xdim::NamedTuple, ydim::NamedTuple, tol::Real)
@@ -36,6 +42,15 @@ function Base.isless(x::Int, y::BoolRig)
     y == true ? x < 1 : x < 0
 end;
 # Example 2.4.1 in Spivak et. al.
+# ```math
+# \begin{align*}
+# & \text{Solve relations:}      && \cos(\ln(z^2 + 10^{-3}x)) - x + 10^{-5}z^{-1} = 0 && \text{(Equation 1)} \\
+# &                              && \cosh(w + 10^{-3}y) + y + 10^{-4}w = 2            && \text{(Equation 2)} \\
+# &                              && \tan(x + y)(x - 2)^{-1}(x + 3)^{-1}y^{-2} = 1     && \text{(Equation 3)} \\
+# & \text{Range and resolution:} && w, x, y, z \in [-3, 3)@124                        &&                     \\
+# & \text{Expose variables:}     && (w, z)                                            &&                     \\
+# & \end{align*}
+# ```
 function f₁(x::Real, z::Real)
     0 - (cos(log(z^2 + x / 10^3)) − x + 1 / z / 10^5)
 end
@@ -56,11 +71,11 @@ R₁ = PixelArray(f₁, x, z, tol)
 R₂ = PixelArray(f₂, w, y, tol)
 R₃ = PixelArray(f₃, x, y, tol);
 #
-spy(R₁; title="f₁(x, z) = 0", xlabel="x", ylabel="z")
+spy(R₁; title="Equation 1", xlabel="x", ylabel="z")
 #
-spy(R₂; title="f₂(w, y) = 0", xlabel="w", ylabel="y")
+spy(R₂; title="Equation 2", xlabel="w", ylabel="y")
 #
-spy(R₃; title="f₃(x, y) = 0", xlabel="x", ylabel="y")
+spy(R₃; title="Equation 3", xlabel="x", ylabel="y")
 #
 diagram = @relation (w, z) where (w::n, x::n, y::n, z::n) begin
     R₁(x, z)
@@ -82,5 +97,5 @@ problem = InferenceProblem(diagram, hom_map, ob_map)
 
 R = solve(problem)
 
-spy(R; title="f₁(x, z) = 0 ∧ f₂(w, y) = 0 ∧ f₃(x, y) = 0", xlabel="w", ylabel="z")
+spy(R; title="Solution", xlabel="w", ylabel="z")
 #
diff --git a/docs/src/generated/pixel.md b/docs/src/generated/pixel.md
@@ -12,9 +12,15 @@ using UnicodePlots
 using Catlab.CategoricalAlgebra.FinRelations: BoolRig
 ````
 
-Spivak, Dobson, Kumari, and Wu, *Pixel Arrays: A fast and elementary method for solving nonlinear systems.*
+A pixel array is an array with entries in the boolean semiring. The function defined below
+constructs a pixel array by discretizing a relation of the form
+```math
+xRy \iff f(x, y) = 0.
+```
 
-A pixel array is an array with entries in the boolean semiring. The function defined below constructs a pixel array by discretizing a relation of the form ``xRy \iff f(x, y) = 0``.
+References:
+- Spivak, David I. et al. "Pixel Arrays: A fast and elementary method for solving
+nonlinear systems." *arXiv: Numerical Analysis* (2016): n. pag.
 
 ````@example pixel
 const PixelArray{N} = Array{BoolRig, N}
@@ -54,6 +60,15 @@ nothing #hide
 ````
 
 Example 2.4.1 in Spivak et. al.
+```math
+\begin{align*}
+& \text{Solve relations:}      && \cos(\ln(z^2 + 10^{-3}x)) - x + 10^{-5}z^{-1} = 0 && \text{(Equation 1)} \\
+&                              && \cosh(w + 10^{-3}y) + y + 10^{-4}w = 2            && \text{(Equation 2)} \\
+&                              && \tan(x + y)(x - 2)^{-1}(x + 3)^{-1}y^{-2} = 1     && \text{(Equation 3)} \\
+& \text{Range and resolution:} && w, x, y, z \in [-3, 3)@124                        &&                     \\
+& \text{Expose variables:}     && (w, z)                                            &&                     \\
+& \end{align*}
+```
 
 ````@example pixel
 function f₁(x::Real, z::Real)
@@ -79,15 +94,15 @@ nothing #hide
 ````
 
 ````@example pixel
-spy(R₁; title="f₁(x, z) = 0", xlabel="x", ylabel="z")
+spy(R₁; title="Equation 1", xlabel="x", ylabel="z")
 ````
 
 ````@example pixel
-spy(R₂; title="f₂(w, y) = 0", xlabel="w", ylabel="y")
+spy(R₂; title="Equation 2", xlabel="w", ylabel="y")
 ````
 
 ````@example pixel
-spy(R₃; title="f₃(x, y) = 0", xlabel="x", ylabel="y")
+spy(R₃; title="Equation 3", xlabel="x", ylabel="y")
 ````
 
 ````@example pixel
@@ -113,6 +128,6 @@ problem = InferenceProblem(diagram, hom_map, ob_map)
 
 R = solve(problem)
 
-spy(R; title="f₁(x, z) = 0 ∧ f₂(w, y) = 0 ∧ f₃(x, y) = 0", xlabel="w", ylabel="z")
+spy(R; title="Solution", xlabel="w", ylabel="z")
 ````