Using Vectorization

MATLAB^® is optimized for operations involving matrices and vectors. The process of revising loop-based, scalar-oriented code to use MATLAB matrix and vector operations is called vectorization. Vectorizing your code is worthwhile for several reasons:

i = 0;
for t = 0:.01:10
    i = i + 1;
    y(i) = sin(t);
end

t = 0:.01:10;
y = sin(t);

x = 1:10000;
ylength = (length(x) - mod(length(x),5))/5;
y(1:ylength) = 0;
for n= 5:5:length(x)
    y(n/5) = sum(x(1:n));
end 

x = 1:10000;
xsums = cumsum(x);
y = xsums(5:5:length(x)); 

Array Operations

Array operators perform the same operation for all elements in the data set. These types of operations are useful for repetitive calculations. For example, suppose you collect the volume (V) of various cones by recording their diameter (D) and height (H). If you collect the information for just one cone, you can calculate the volume for that single cone:

V = 1/12*pi*(D^2)*H;

Now, collect information on 10,000 cones. The vectors D and H each contain 10,000 elements, and you want to calculate 10,000 volumes. In most programming languages, you need to set up a loop similar to this MATLAB code:

for n = 1:10000
   V(n) = 1/12*pi*(D(n)^2)*H(n);
end

With MATLAB, you can perform the calculation for each element of a vector with similar syntax as the scalar case:

% Vectorized Calculation
V = 1/12*pi*(D.^2).*H;

Array operators also enable you to combine matrices of different dimensions. This automatic expansion of size-1 dimensions is useful for vectorizing grid creation, matrix and vector operations, and more.

Suppose that matrix A represents test scores, the rows of which denote different classes. You want to calculate the difference between the average score and individual scores for each class. Using a loop, the operation looks like:

A = [97 89 84; 95 82 92; 64 80 99;76 77 67;...
 88 59 74; 78 66 87; 55 93 85];

mA = mean(A);
B = zeros(size(A));
for n = 1:size(A,2)
    B(:,n) = A(:,n) - mA(n);
end

A more direct way to do this is with A - mean(A), which avoids the need of a loop and is significantly faster.

devA = A - mean(A)

devA =

    18    11     0
    16     4     8
   -15     2    15
    -3    -1   -17
     9   -19   -10
    -1   -12     3
   -24    15     1

Even though A is a 7-by-3 matrix and mean(A) is a 1-by-3 vector, MATLAB implicitly expands the vector as if it had the same size as the matrix, and the operation executes as a normal element-wise minus operation.

The size requirement for the operands is that for each dimension, the arrays must either have the same size or one of them is 1. If this requirement is met, then dimensions where one of the arrays has size 1 are expanded to be the same size as the corresponding dimension in the other array. For more information, see Compatible Array Sizes for Basic Operations.

Another area where implicit expansion is useful for vectorization is if you are working with multidimensional data. Suppose you want to evaluate a function, F, of two variables, x and y.

F(x,y) = x*exp(-x2 - y2)

To evaluate this function at every combination of points in the x and y vectors, you need to define a grid of values. For this task you should avoid using loops to iterate through the point combinations. Instead, if one of the vectors is a column and the other is a row, then MATLAB automatically constructs the grid when the vectors are used with an array operator, such as x+y or x-y. In this example, x is a 21-by-1 vector and y is a 1-by-16 vector, so the operation produces a 21-by-16 matrix by expanding the second dimension of x and the first dimension of y.

x = (-2:0.2:2)'; % 21-by-1
y = -1.5:0.2:1.5; % 1-by-16
F = x.*exp(-x.^2-y.^2); % 21-by-16

In cases where you want to explicitly create the grids, you can use the meshgrid and ndgrid functions.

Logical Array Operations

A logical extension of the bulk processing of arrays is to vectorize comparisons and decision making. MATLAB comparison operators accept vector inputs and return vector outputs.

For example, suppose while collecting data from 10,000 cones, you record several negative values for the diameter. You can determine which values in a vector are valid with the >= operator:

D = [-0.2 1.0 1.5 3.0 -1.0 4.2 3.14];
D >= 0

ans =

     0     1     1     1     0     1     1

Vgood = V(D >= 0);

MATLAB allows you to perform a logical AND or OR on the elements of an entire vector with the functions all and any, respectively. You can throw a warning if all values of D are below zero:

if all(D < 0)
   warning('All values of diameter are negative.')
   return
end

MATLAB can also compare two vectors with compatible sizes, allowing you to impose further restrictions. This code finds all the values where V is nonnegative and D is greater than H:

V((V >= 0) & (D > H))

To aid comparison, MATLAB contains special values to denote overflow, underflow, and undefined operators, such as Inf and NaN. Logical operators isinf and isnan exist to help perform logical tests for these special values. For example, it is often useful to exclude NaN values from computations:

x = [2 -1 0 3 NaN 2 NaN 11 4 Inf];
xvalid = x(~isnan(x))

xvalid =

     2    -1     0     3     2    11     4   Inf

Matrix Operations

When vectorizing code, you often need to construct a matrix with a particular size or structure. Techniques exist for creating uniform matrices. For instance, you might need a 5-by-5 matrix of equal elements:

A = ones(5,5)*10;

v = 1:5;
A = repmat(v,3,1)

A =

     1     2     3     4     5
     1     2     3     4     5
     1     2     3     4     5

The function repmat possesses flexibility in building matrices from smaller matrices or vectors. repmat creates matrices by repeating an input matrix:

A = repmat(1:3,5,2)
B = repmat([1 2; 3 4],2,2)

A =

     1     2     3     1     2     3
     1     2     3     1     2     3
     1     2     3     1     2     3
     1     2     3     1     2     3
     1     2     3     1     2     3


B =

     1     2     1     2
     3     4     3     4
     1     2     1     2
     3     4     3     4

Ordering, Setting, and Counting Operations

In many applications, calculations done on an element of a vector depend on other elements in the same vector. For example, a vector, x, might represent a set. How to iterate through a set without a for or while loop is not obvious. The process becomes much clearer and the syntax less cumbersome when you use vectorized code.

x = [2 1 2 2 3 1 3 2 1 3];
x = sort(x);
difference  = diff([x,NaN]);
y = x(difference~=0)

y =

     1     2     3

y=unique(x);

x = [2 1 2 2 3 1 3 2 1 3];
x = sort(x);
difference  = diff([x,max(x)+1]);
count = diff(find([1,difference]))
y = x(find(difference))

count =

     3     4     3


y =

     1     2     3

count_nans = sum(isnan(x(:)));
count_infs = sum(isinf(x(:)));

Functions Commonly Used in Vectorization