adding draft of monte carlo chapter (#115)

* adding draft of monte carlo chapter.

* removing unnecessary tag language tag

* changing images to gif.

* fixing typo

* fixing typo.

* fixing a few mroe smallscale typos

Author: leios

SUMMARY.md

@@ -28,6 +28,7 @@
 * [Tree Traversal](chapters/tree_traversal/tree_traversal.md)
 * [Euclidean Algorithm](chapters/euclidean_algorithm/euclidean.md)
 * [Multiplication](chapters/multiplication/multiplication.md)
+* [Monte Carlo](chapters/monte_carlo/monte_carlo.md)
 * [Matrix Methods](chapters/matrix_methods/matrix_methods.md)
     * [Gaussian Elimination](chapters/matrix_methods/gaussian_elimination/gaussian_elimination.md)
     * [Thomas Algorithm](chapters/matrix_methods/thomas/thomas.md)

chapters/monte_carlo/code/julia/monte_carlo.jl

@@ -0,0 +1,27 @@
+# function to determine whether an x, y point is in the unit circle
+function in_circle(x_pos::Float64, y_pos::Float64, radius::Float64)
+    if (x_pos^2 + y_pos^2 < radius^2)
+        return true
+    else
+        return false
+    end
+end
+
+# function to integrate a unit circle to find pi via monte_carlo
+function monte_carlo(n::Int64, radius::Float64)
+
+    pi_count = 0
+    for i = 1:n
+        point_x = rand()
+        point_y = rand()
+
+        if (in_circle(point_x, point_y, radius))
+            pi_count += 1
+        end
+    end
+
+    pi_estimate = 4*pi_count/(n*radius^2)
+    println("Percent error is: ", signif(100*(pi - pi_estimate)/pi, 3), " %")
+end
+
+monte_carlo(10000000, 0.5)

chapters/monte_carlo/monte_carlo.md

@@ -0,0 +1,100 @@
+# Monte Carlo Integration
+
+Monte Carlo methods were some of the first methods I ever used for research, and when I learned about them, they seemed like some sort of magic.
+Their premise is simple: random numbers can be used to integrate arbitrary shapes embedded into other objects.
+Nowadays, "monte carlo" has become a bit of a catch-all term for methods that use random numbers to produce real results, but it all started as a straightforward method to integrate objects.
+No matter how you slice it, the idea seems a bit crazy at first.
+After all, random numbers are random.
+How could they possibly be used to find non-random values?
+
+Well, imagine you have a square.
+The area of the square is simple, $$\text{Area}_{\text{square}} = \text{length} \times \text{width}$$.
+Since it's a square, the $$\text{length}$$ and $$\text{width}$$ are the same, so the formula is technically just $$\text{Area}_{\text{square}} = \text{length}^2$$.
+If we embed a circle into the square with a radius $$r = \text{length}$$ (shown below), then it's area is $$\text{Area}_{\text{circle}}=\pi r^2$$.
+For simplicity, we can also say that $$\text{Area}_{\text{square}}=4r^2$$.
+
+<p align="center">
+    <img src="res/square_circle.png" width="300"/>
+</p>
+
+Now, let's say we want to find the area of the circle without an equation.
+As we said before, it's embedded in the square, so we should be able to find some ratio of the area of the square to the area of the circle:
+
+$$
+\text{Ratio} = \frac{\text{Area}_{\text{circle}}}{\text{Area}_{\text{square}}}
+$$
+
+This means,
+
+$$
+\text{Area}_{\text{circle}} = \text{Area}_{\text{square}}\times\text{Ratio} = 4r^2 \times \text{ratio}
+$$
+
+So, if we can find the $$\text{Ratio}$$ and we know $$r$$, we should be able to easily find the $$\text{Area}_{\text{circle}}$$.
+The question is, "How do we easily find the $$\text{Ratio}$$?"
+Well, one way is with *random sampling*.
+We basically just pick a bunch of points randomly in the square, and
+each point is tested to see whether it's in the circle or not:
+
+{% method %}
+{% sample lang="jl" %}
+[import:2-8, lang:"julia"](code/julia/monte_carlo.jl)
+{% endmethod %}
+
+If it's in the circle, we increase an internal count by one, and in the end,
+
+$$
+\text{Ratio} = \frac{\text{count in circle}}{\text{total number of points used}}
+$$
+
+If we use a small number of points, this will only give us a rough approximation, but as we start adding more and more points, the approximation becomes much, much better (as shown below)!
+
+<p align="center">
+    <img src="res/monte_carlo.gif" width="400"/>
+</p>
+
+The true power of monte carlo comes from the fact that it can be used to integrate literally any object that can be embedded into the square.
+As long as you can write some function to tell whether the provided point is inside the shape you want (like `in_circle()` in this case), you can use monte carlo integration!
+This is obviously an incredibly powerful tool and has been used time and time again for many different areas of physics and engineering.
+I can gaurantee that we will see similar methods crop up all over the place in the future!
+
+# Example Code
+Monte carlo methods are famous for their simplicity.
+It doesn't take too many lines to get something simple going.
+Here, we are just integrating a circle, like we described above; however, there is a small twist and trick.
+Instead of calculating the area of the circle, we are instead trying to find the value of $$\pi$$, and
+rather than integrating the entire circle, we are only integrating the upper right quadrant of the circle from $$0 < x,y < 1$$.
+This saves a bit of computation time, but also requires us to multiply our output by $$4$$.
+
+That's all there is to it!
+Feel free to submit your version via pull request, and thanks for reading!
+
+{% method %}
+{% sample lang="jl" %}
+### Julia
+[import, lang:"julia"](code/julia/monte_carlo.jl)
+{% endmethod %}
+
+
+<script>
+MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
+</script>
+$$ 
+\newcommand{\d}{\mathrm{d}}
+\newcommand{\bff}{\boldsymbol{f}}
+\newcommand{\bfg}{\boldsymbol{g}}
+\newcommand{\bfp}{\boldsymbol{p}}
+\newcommand{\bfq}{\boldsymbol{q}}
+\newcommand{\bfx}{\boldsymbol{x}}
+\newcommand{\bfu}{\boldsymbol{u}}
+\newcommand{\bfv}{\boldsymbol{v}}
+\newcommand{\bfA}{\boldsymbol{A}}
+\newcommand{\bfB}{\boldsymbol{B}}
+\newcommand{\bfC}{\boldsymbol{C}}
+\newcommand{\bfM}{\boldsymbol{M}}
+\newcommand{\bfJ}{\boldsymbol{J}}
+\newcommand{\bfR}{\boldsymbol{R}}
+\newcommand{\bfT}{\boldsymbol{T}}
+\newcommand{\bfomega}{\boldsymbol{\omega}}
+\newcommand{\bftau}{\boldsymbol{\tau}}
+$$