Merge pull request algorithm-archivists#113 from leios/smallscale_updates

adding smallscale changes to graham implementation, fft, and thomas.

Author: jiegillet

chapters/FFT/code/julia/fft.jl

@@ -5,7 +5,7 @@ function DFT(x)
     # We want two vectors here for real space (n) and frequency space (k)
     n = 0:N-1
     k = n'
-    transform_matrix = exp.(-2im * pi *n *k / N)
+    transform_matrix = exp.(-2im*pi*n*k/N)
     return transform_matrix*x
 
 end
@@ -22,7 +22,6 @@ function cooley_tukey(x)
         x_even = x[2]
     end
     n = 0:N-1
-    #n = n'
     half = div(N,2)
     factor = exp.(-2im*pi*n/N)
     return vcat(x_odd + x_even .* factor[1:half],

chapters/computational_geometry/gift_wrapping/graham_scan/code/julia/graham.jl

@@ -7,26 +7,6 @@ function dist(point1::Point, point2::Point)
     return sqrt((point1.x - point2.x)^2 + (point1.y - point2.y)^2)
 end
 
-function graham_angle(point1::Point, point2::Point, point3::Point)
-    # Find distances between all points
-    a = dist(point3, point2)
-    b = dist(point3, point1)
-    c = dist(point1, point2)
-
-    ret_angle = acos((b*b - a*a - c*c)/(2*a*c))
-
-    if(sign(point1.x - point2.x) != sign(point1.x - point3.x))
-        ret_angle += 0.5*pi
-    end
-
-    if (isnan(ret_angle))
-        exit(1)
-    end
-
-    return ret_angle
-
-end
-
 function ccw(a::Point, b::Point, c::Point)
     return ((b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x))
 end
@@ -38,7 +18,8 @@ function graham_scan(points::Vector{Point})
     sort!(points, by = item -> item.y)
 
     # Sort all other points according to angle with that point
-    other_points = sort(points[2:end], by = item -> graham_angle(Point(points[1].x - 1,points[1].y), points[1], item))
+    other_points = sort(points[2:end], by = item -> atan2(item.y - points[1].y,
+                                                          item.x - points[1].x))
 
     # Place points sorted by angle back into points vector
     for i in 1:length(other_points)
@@ -69,6 +50,7 @@ function graham_scan(points::Vector{Point})
 end
 
 function main()
+    # This hull is just a simple test so we know what the output should be
     points = [Point(2,1.9), Point(1, 1), Point(2, 4), Point(3, 1), Point(2, 0)]
     hull = graham_scan(points)
     println(hull)

chapters/matrix_methods/thomas/thomas.md

@@ -1,9 +1,6 @@
-##### Dependencies
-* [Gaussian Elimination](gaussian_elimination.md)
-
 # Thomas Algorithm
 
-As alluded to in the Gaussian Elimination Chapter, the Thomas Algorithm (or TDMA -- Tri-Diagonal Matrix Algorithm) allows for programmers to **massively** cut the computational cost of their code from $$\sim O(n^3) \rightarrow \sim O(n)$$! This is done by exploiting a particular case of Gaussian Elimination, particularly the case where our matrix looks like:
+As alluded to in the [Gaussian Elimination chapter](../gaussian_elimination/gaussian_elimination.md), the Thomas Algorithm (or TDMA -- Tri-Diagonal Matrix Algorithm) allows for programmers to **massively** cut the computational cost of their code from $$\sim O(n^3) \rightarrow \sim O(n)$$! This is done by exploiting a particular case of Gaussian Elimination, particularly the case where our matrix looks like:
 
 $$
 \left[
@@ -19,7 +16,8 @@ $$
 
 By this, I mean that our matrix is *Tri-Diagonal* (excluding the right-hand side of our system of equations, of course!). Now, at first, it might not be obvious how this helps; however, we may divide this array into separate vectors corresponding to $$a$$, $$b$$, $$c$$, and $$d$$ and then solve for $$x$$ with back-substitution, like before. 
 
-In particular, we need to find an optimal scale factor for each row and use that. What is the scale factor? Well, it is the diagonal $$-$$ the multiplicative sum of the off-diagonal elements. In the end, we will update $$c$$ and $$d$$ to be$$c'$$ and $$d'$$ like so:
+In particular, we need to find an optimal scale factor for each row and use that. What is the scale factor? Well, it is the diagonal $$-$$ the multiplicative sum of the off-diagonal elements.
+In the end, we will update $$c$$ and $$d$$ to be $$c'$$ and $$d'$$ like so:
 
 $$
 \begin{align}