Trapezoidal rule to approximate the integral of x^2

I've implemented the trapezoidal rule to compute the integral for a function $x^2$. I would like to see another style of the same code. It seems Matlab hates for a matrix to be expanded without specifying the size.

clear all
clc
n = 5; % number  subdivisions
a = 1.0; % lower limit
b = 2.0; % upper limit
sum = 0.0;
dx = (b-a)/(n-1); % step size

% Generate samples
for i = 1:n
x(i) = a + (i-1)*dx;
end

% Generate function's values
for i = 1:n
y(i) = x(i).^2;
end

% Compute area using Trapz. method
for i = 1:n
if ( i == 1 || i == n) % for the first and last data
sum = sum + y(i)./2;
else
sum = sum + y(i); % for the rest of data
end
end
area = sum * dx;

area

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Your code does not appear to be correct as the integral of $x^2$ from 1 to 2 using 5 areas (4 subdivisions) is supposed to be 2.34 exactly, while your code seems to yield the answer for 4 areas (2.3438) –  mleyfman Aug 21 '14 at 6:07
@mleyfman, don't forget the error. The result 2.3333 not what you said 2.34 is using the analytical form of the function however trapezoidal method is a method to approximate the analytical function. The error here is 0.45 % which means we have some level of uncertainty in the estimated result. –  CroCo Aug 21 '14 at 6:14
I was going by this: nastyaccident.com/calculators/calculus/trapezoidalRule. The integral is indeed 2.333 if done analytically. Trapezoidal rule with n=5 should yield 2.34 exactly, whereas n=4 should yield 2.34375 –  mleyfman Aug 21 '14 at 6:17
@mleyfman, according to the link you gave Answer: 2.34375  which is same of mine. Even though n=5 in my code but the loop start from 0 to 4. Check the implementation of my code. –  CroCo Aug 21 '14 at 6:20
Are you sure you ran it with amount=5? Also see this: wolframalpha.com/input/… –  mleyfman Aug 21 '14 at 6:28

Avoid variable names that collide with built-in functions

sum is a built-in function. Redefining it causes unconventional behavior:

>> sum([1 2 3])

ans =

6

>> sum = 0.0

sum =

0

>> sum([1 2 3])
Index exceeds matrix dimensions.

>> clear all
>> sum([1 2 3])

ans =

6


Idiomatic MATLAB

The whole point of using MATLAB, rather than C, is that the language is designed to work on many values at once. Therefore, you should avoid looping wherever possible, and let MATLAB do its job on vectors.

% Givens
n = 5                                  % number of subdivisions
a = 1.0                                % x coordinate left endpoint
b = 2.0                                % x coordinate right endpoint

x = linspace(a, b, n)                  % x coordinates of samples
y = x .^ 2                             % y coordinates of samples
sum_y = sum(y) - (y(1) + y(end)) ./ 2  % sum y coordinates, taking just half of endpoints
dx = (b - a) / (n - 1)                 % width of each trapezoid
area = sum_y * dx


Reinventing the wheel

For your information, MATLAB has built-in support for numerical integration:

>> integral(@(x) x .^ 2, 1, 2)

ans =

2.3333

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+1 for removing for-loops. –  Yuushi Aug 21 '14 at 6:17
+1 for the point of naming built-in function. This is the approach that I'm looking for. This is absolutely beautiful code. –  CroCo Aug 21 '14 at 6:25

Setting aside that I don't know MATLAB, I would like to mention some things

Naming

If you need a comment to explain a variable then the variable is poorly named.

n = 5; % number subdivided areas
a = 1.0; % lower limit
b = 2.0; % upper limit
dx = (b-a)/(n-1); % step size


should become

subdividedAreaCount = 5;
lowerLimit = 1.0;
upperLimit = 2.0;
stepSize = (upperLimit - lowerLimit) / (subdividedAreaCount -1 );


Refactoring

Now the loops for generating samples and function values would look like this if we also rename x and y to sampleValue and functionValue.

% Generate samples
for i = 1:subdividedAreaCount
sampleValue(i) = lowerLimit + (i-1)*stepSize;
end

% Generate function values
for i = 1:subdividedAreaCount
functionValue(i) = sampleValue(i).^2;
end


The computing of the area would look like

% Compute area using Trapz. method
for i = 1:subdividedAreaCount
if ( i == 1 || i == subdividedAreaCount) % for the first and last data
sum = sum + functionValue(i)./2;
else
sum = sum + functionValue(i); % for the rest of data
end
end


but we can refactor this also to exclude the if condition like

% Compute area using Trapz. method
for i = 2:subdividedAreaCount-1
sum = sum + functionValue(i); % for the rest of data
end
sum = sum + functionValue(2)./2;
sum = sum + functionValue(subdividedAreaCount)./2;

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+1 for naming variables. Is there any benefit of refactoring if statement? I think mine is more readable than the approach you stated. –  CroCo Aug 21 '14 at 6:08
This just depends how big subdividedAreaCount can be. If subdividedAreaCount = 100000 it would be at least faster without using the if . –  Heslacher Aug 21 '14 at 6:18