Vectorization

Overview

Teaching: 15 min
Exercises: 15 min

Questions

• What is vectorization in MATLAB?

• What are the advantages and disadvantages of vectorization?

Objectives

• To learn to vectorize your code in MATLAB.

Vectorization

Vectorization in computer programming is the act of transforming loop-based scalar operations into operations over vectors (or arrays). This should not be confused with automatic vectorization.

It is debatable whether vectorization should be part of a course on High-Performance Computing in MATLAB. However, if your objective is to improve the performance of your MATLAB code, vectorization is the place to start.

What is Vectorization in MATLAB?

To understand the concept of vectorization in MATLAB, let us take the example of adding two square matrices of rank n. In scalar languages such as C, this can be done using a for-loop:

// scalar operators of array variables using for-loop
//
for(i=0; i<n; i++)
{
  for(j=0; j<n; j++)
    C[i][j] = A[i][j] + B[i][j];
}

The syntax for the same operation in MATLAB is:

% scalar operators of array variables using for-loop

for i:n
    for j:n
        C(i,j) = A(i,j) + B(i,j);
    end
end

Unlike C, MATLAB and Fortran are array languages; they support some basic operations such as addition, subtraction, multiplication and division by a scalar to variables of type scalars and arrays. In short, vectorization generalises some operations to scalars and arrays.

The vectorized code for the addition of two variables in MATLAB is:

C = A + B;

The above code is valid irrespective of whether the variables A, B and C are scalars or arrays. The only requirement for array type variables is that all the arrays are of the same size and shape, and contain compatible data types.

Reasons for using Vectorization

• Vectorized code is often faster than the code with loops. MATLAB optimises the vectorized code using optimised libraries and multi-threading.
• Vectorized code is concise. Therefore, it is easier to understand and debug.

An exercise on vectorization

The following MATLAB code adds two square matrices of size n:

n = 100;
A = rand(n,n);
B = rand(n,n);
tic
for i=1:n
    for j=1:n
        C(i,j) = A(i,j) + B(i,j);
    end
end
toc

Append the code with a vectorized version of the for-loop. Measure the time to check if you gain any computational benefits. Repeat this by increasing the value of n to 1000 and 10000.

Solution

% vectorized version
tic
C = A + B;
toc

Elapsed times vary from computer to computer. However, you should see significant improvements between the for-loop code and vectorized code.

Period operator

The period operator (.) transforms the scalar operators multiplication (*), division (/) and power (^) into array operators.

Suppose, we want to calcualte the square of each element in an array. Using the scalar operations and for-loop, we write:

n = 100;
x = 1:n;
y = x;
for i=1:n
    y(i) = x(i)*x(i)
end

The above code can be vectorized as:

n = 100;
x = 1:n;
y = x.*x;

An exercise on array operators

The following code computes the radius for a set of points whose X and Y coordinates are given as arrays:

n = 100;
x = rand(n,1);
y = rand(n,1);
rad = x;
for i=1:n
    rad(i) = sqrt(x(i)*x(i) + y(i)*y(i))
end

Convert it into a vectorized code.

Solution

n = 100;
x = rand(n,1);
y = rand(n,1);
rad = sqrt(x.*x + y.*y);

Note that the sqrt function here is a vectorized function. Its syntax is the same whether the argument is a scalar or a vector.

Logical array operators

If we want to perform logical operations over the elements of a vector (or matrix), then we can use MATLAB’s comparison operators which accept vectors as input arguments.

To find which elements of an array are positive, we can write:

A = [0.2 -2.0 1.0 9.0 -11.0 -98.1 -0.1 2.1 7.7];
A > 0

ans =
1x9 logical array
  1  0  1  1  0  0  0  1  1

Similarly, we can perform element-wise logical operations by using vectorization in MATLAB.

For additional details on vectorization in MATLAB, we suggest the following material:

• https://uk.mathworks.com/help/matlab/matlab_prog/vectorization.html
• http://www-h.eng.cam.ac.uk/help/tpl/programs/Matlab/tricks.html

Limitations and disadvantage of vectorization

• Vectorized code poses issues in parallelising the code for distributed computing.
• Sometimes it is difficult to spot bugs when you miss the period operator (.), especially if you don’t know when and where to use the period operator.

Key Points

• Vectorization in MATLAB is about replacing loops with appropriate array operations.

• Vectorized code runs much faster than the code with loops.

• Vectorization is concise and makes the code easier to understand and debug.