The trapezoidal method of integration

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The trapezoidal method of integration

John Kitchin


$$\int_a^b f(x) dx \approx \frac{1}{2}\displaystyle\sum\limits_{k=1}^N(x_{k+1}-x_k)(f(x_{k+1}) + f(x_k))$$

Let's compute the integral of sin(x) from x=0 to $\pi$. To approximate the integral, we need to divide the interval from $a$ to $b$ into $N$ intervals. The analytical answer is 2.0.

We will use this example to illustrate the difference in performance between loops and vectorized operations in Matlab.

a = 0; b = pi;
N = 1000; % this is the number of intervals

h = (b - a)/N; % this is the width of each interval
x = a:h:b; % note there are N+1 elements in this x vector
y = sin(x); % the sin function is already vectorized, so y contains N+1 elements

Trapezoid method using a loop

we use the tic and toc functions to time how long it takes to run.

f = 0;
for k=1:N
    f = f + 0.5*((x(k+1)-x(k))*(y(k+1)+y(k)));
f =


Elapsed time is 0.002520 seconds.

vectorized approach

Xk = x(2:end)-x(1:end-1); % vectorized version of (x(k+1)-x(k))
Yk = y(2:end)+y(1:end-1); % vectorized version of (y(k+1)+y(k))

f = 0.5*sum(Xk.*Yk) % vectorized version of the loop above

% It isn't strictly necessary to calculate Xk this way, since here by
% design every interval is the same width, but this approach would work for
% non-uniform x-values.
f =


Elapsed time is 0.000148 seconds.

total linear algebra way

In the last example, there may be loop buried in the sum command. Lets do one final method, using linear algebra, in a single line. The key to understanding this is to recognize the sum is just the result of a dot product of the x differences and y sums. We have to transpose the y sums to get the vector dimensions to work for the dot product (* is matrix multiplication/dot product in Matlab

f = 0.5*(x(2:end)-x(1:end-1))*(y(2:end)+y(1:end-1))'
f =


Elapsed time is 0.000130 seconds.


The loop method is straightforward to code, and looks alot like the formula that defines the trapezoid method. the vectorized methods are not as easy to read, and take fewer lines of code to write. However, the vectorized methods are much faster than the loop, so the loss of readability could be worth it for very large problems.

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