Add Monte Carlo integration and update README

This commit adds Monte Carlo integration via `sxm_monte_carlo`.

This commit also updates the README.

Author: jmcph4

README.md

@@ -1,14 +1,31 @@
 # Sane Extensions to MATLAB #
 ---
 
-**S**ane **E**xtensions to **M**ATLAB (sxm) is a library of MATLAB functions and classes which perform some useful tasks in an arguably saner way.
+**S**ane **E**xtensions to **M**ATLAB (sxm) is a library of MATLAB functions and classes which perform some useful tasks in a much saner way than MATLAB does natively.
 
 A brief overview of features:
 
  - ADTs
 	 - List
 	 - Set
- - Wrappers around common MATLAB operations (e.g. plotting, solving linear systems, etc.)
+ - Wrappers around common MATLAB operations
+	 - Plotting
+	 - Solving BVPs
+ - Numerical methods
+	 - Numerical differentiation
+		 - Forward difference
+		 - Backward difference
+		 - Central difference
+ 	 - Linear systems of equations
+	 	 - Naive Gaussian elimination
+	 - IVP solving
+		 - Euler's method
+	 - Monte Carlo methods
  - Performance testing extensions
- - Various mathematical utilities of historical interest (e.g. [Egyptian multiplication](https://en.wikipedia.org/wiki/Ancient_Egyptian_multiplication))
+	 - Raising square matrices to arbitrary powers
+	 - Summing arbitrarily large vectors
+	 - Naive Gaussian solution of arbitrarily large Hilbert matrices
+	 - Euler's method for arbitrary number of steps
+ - Various mathematical utilities of historical interest
+	 - [Egyptian multiplication](https://en.wikipedia.org/wiki/Ancient_Egyptian_multiplication)
 

sxm_monte_carlo.m

@@ -0,0 +1,53 @@
+% SXM_MONTE_CARLO  Integrate arbitrary shapes using Monte Carlo method.
+%   area = sxm_monte_carlo(domain, shapefunc, n) finds the area or volume
+%   (or other, higher-dimensional equivalent) of the shape defined by
+%   shapefunc. Note that domain must *totally* enclose the shape defined by
+%   shapefunc. Also note that accuracy is proportional to the number of
+%   points, n.
+%
+%   Example
+%       area = sxm_monte_carlo([-5 5; -5 5], @(vec)(vec(1)^2+vec(2)^2<=1),
+%           10000);
+%       finds the area of the unit circle centred at the origin by using
+%       10000 points in the square region bounded by x=-5, x=5, y=-5, y=5.
+function area = sxm_monte_carlo(dom, mem, n)
+    if n == 0 % bounds check the number of points
+        error('nonpositive number of points')
+    end
+    
+    sz = size(dom);
+    
+    dim = sz(1);
+    vec = zeros(dim);
+    hits = 0;
+    points = zeros([n 2]);
+    
+    for i = 1:n
+        % construct vector elementwise
+        for j = 1:dim
+            a = dom(j,1);
+            b = dom(j,2);
+            vec(j) = (b-a).*rand + a;
+        end
+        
+        % check for membership
+        if mem(vec)
+            hits = hits + 1; % increment hit counter
+        end
+        
+        points(i,1) = vec(1);
+        points(i,2) = vec(2);
+    end
+    
+    % calculate area of n-box
+    total_area = abs(dom(1,1) - dom(1,2));
+    
+    for i = 2:dim
+        a = dom(i,1);
+        b = dom(i,2);
+        total_area = total_area * abs(b - a);
+    end
+    
+    p = hits / n;
+    area = p * total_area;
+end