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
- By Mkwadee at English Wikipedia, CC BY-SA 3.0, Link
The formula for this method is the following:
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!
References
Understanding the trapezoidal rule
Trapezoidal Rule
How to approximate area with the trapezoid rule
Top comments (4)
Great job but I'm not sure why using functional-first language to use it OO-style. To me using mutable in F# is a flag that something is done in a non-functional fashion.
How about considering the code below?
The key tricks here are
Array.mapiwhich allows us to work with tuple (element,index) instead of just an element; and substitution of iterative increment of xi by h to multiplicationindex*h+aSince this was my first time with a functional programming language I think I hadn't fully grasped the concepts. I ended up using recursion in order to remove the mutable variables; which I updated in my post. Thank you very much for your feedback!
This is so true. The more trapezoids you use the more exact the integral tends to be. Also, it is very simple to use compared to other methods. Great article!
Really cool approach