Numerical is approximate; analytical is exact.
A numerical method is a tool used to find a numerical solution to a problem which can be difficult to solve analytically.
Mostly done by computers, this approach is a complete set of instructions that are executed in order to find a solution. With this being said,
the Trapezoidal Rule is a numerical method used to find definite integrals. The most common application for this method is to find the
area under a function in a given set of intervals. Finding this area is typically done using Riemann sums, but this method proves to be more exact because it uses trapezoids instead of rectangles.
- Reimann sum
- Trapezoidal Rule
In this case, I implemented this numerical method using F# which is a functional programming language. First, I chose the function to integrate; in this case sin(x):
let f x = System.Math.Sin(x)
The algorithm has to receive the intervals (a,b) through which it will find the area and a number (N) of trapezoids to form.
With these parameters, it can now calculate the length of each trapezoid:
let h = (b - a)/N
I created two functions in order to calculate the formula above:
let rec sum_recursive (xi:float) (h:float) f (N:float) (iter:float) = if iter = N then xi else let xi_1 = xi + h*f(h*iter) sum_recursive xi_1 h f N (iter+1.0) let trapezoidal (a:float) (b:float) f (N:float) = let h = (b - a)/N let left_side = h/2.0*(f(a)+f(b)) left_side + sum_recursive 0.0 h f N 0.0
This method can be used for things like: finding how much fluid is flowing through a specific area, calculating the work needed to move an object at a certain speed, or even finding the center of mass of an object.
Overall, learning to program a numerical method has enabled me to relate mathematical concepts, that are normally just seen in class; with programming, which is something I actually use in my day to day life. Furthermore, this was also my first time working with a functional programming language such as F#; which I ended up enjoying.
Here's my complete code
Hope you find it useful!