MATLAB^{®} is optimized for operations involving matrices and vectors. The process of revising loopbased, scalaroriented code to use MATLAB matrix and vector operations is called vectorization. Vectorizing your code is worthwhile for several reasons:
Appearance: Vectorized mathematical code appears more like the mathematical expressions found in textbooks, making the code easier to understand.
Less Error Prone: Without loops, vectorized code is often shorter. Fewer lines of code mean fewer opportunities to introduce programming errors.
Performance: Vectorized code often runs much faster than the corresponding code containing loops.
This code computes the sine of 1,001 values ranging from 0 to 10:
i = 0; for t = 0:.01:10 i = i + 1; y(i) = sin(t); end
This is a vectorized version of the same code:
t = 0:.01:10; y = sin(t);
The second code sample usually executes faster than the first
and is a more efficient use of MATLAB. Test execution speed on
your system by creating scripts that contain the code shown, and then
use the tic
and toc
functions
to measure their execution time.
This code computes the cumulative sum of a vector at every fifth element:
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
Using vectorization, you can write a much more concise MATLAB process. This code shows one way to accomplish the task:
x = 1:10000; xsums = cumsum(x); y = xsums(5:5:length(x));
Many vectorizing techniques rely on flexible MATLAB indexing methods. Three basic types of indexing exist:
In subscripted indexing, the index values indicate their position
within the matrix. Thus, if A = 6:10
, then A([3
5])
denotes the third and fifth elements of vector A
:
A = 6:10; A([3 5])
ans = 8 10
Multidimensional arrays or matrices use multiple index parameters for subscripted indexing.
A = [11 12 13; 14 15 16; 17 18 19] A(2:3,2:3)
A = 11 12 13 14 15 16 17 18 19 ans = 15 16 18 19
In linear indexing, MATLAB assigns every element of a matrix a single index as if the entire matrix structure stretches out into one column vector.
A = [11 12 13; 14 15 16; 17 18 19]; A(6) A([3,1,8]) A([3;1;8])
ans = 18 ans = 17 11 16 ans = 17 11 16
In the previous example, the returned matrix elements preserve the shape specified by the index parameter. If the index parameter is a row vector, MATLAB returns the specified elements as a row vector.
Note:
Use the functions 
With logical indexing, the index parameter is a logical matrix
that is the same size as A
and contains only 0s
and 1s.
MATLAB selects elements of A
that contain
a 1
in the corresponding position of the logical
matrix:
A = [11 12 13; 14 15 16; 17 18 19]; A(logical([0 0 1; 0 1 0; 1 1 1]))
ans = 17 15 18 13 19
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;
Note:
Placing a period ( 
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
, for which the corresponding
elements of D
are nonnegative: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 compare two vectors of the same size, 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
Note:

Matrix operations act according to the rules of linear algebra. These operations are most useful in vectorization 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(x^{2}  y^{2})
To evaluate this function at every combination of points in
the x
and y
, you need to define
a grid of values:
x = 2:0.2:2; y = 1.5:0.2:1.5; [X,Y] = meshgrid(x,y); F = X.*exp(X.^2Y.^2);
meshgrid
,
you might need to write two for
loops to iterate
through vector combinations. The function ndgrid
also
creates number grids from vectors, but can construct grids beyond
three dimensions. meshgrid
can only construct
2D and 3D grids.In some cases, using matrix multiplication eliminates intermediate steps needed to create number grids:
x = 2:2; y = 1:0.5:1; x'*y
ans = 2.0000 1.0000 0 1.0000 2.0000 1.0000 0.5000 0 0.5000 1.0000 0 0 0 0 0 1.0000 0.5000 0 0.5000 1.0000 2.0000 1.0000 0 1.0000 2.0000
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 5by5 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
The bsxfun
function provides a way of combining
matrices of different dimensions. 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. Your first thought might be to compute the
simple difference, A  mean(A)
. However, MATLAB throws
an error if you try this code because the matrices are not the same
size. Instead, bsxfun
performs the operation
without explicitly reconstructing the input matrices so that they
are the same size.
A = [97 89 84; 95 82 92; 64 80 99;76 77 67;...
88 59 74; 78 66 87; 55 93 85];
dev = bsxfun(@minus,A,mean(A))
dev = 18 11 0 16 4 8 15 2 15 3 1 17 9 19 10 1 12 3 24 15 1
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.
A number of different ways exist for finding the redundant elements
of a vector. One way involves the function diff
.
After sorting the vector elements, equal adjacent elements produce
a zero entry when you use the diff
function on
that vector. Because diff(x)
produces a vector
that has one fewer element than x
, you must add
an element that is not equal to any other element in the set. NaN
always
satisfies this condition. Finally, you can use logical indexing to
choose the unique elements in the set:
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
unique
function:y=unique(x);
unique
function might provide more functionality
than is needed and slow down the execution of your code. Use the tic
and toc
functions
if you want to measure the performance of each code snippet.Rather than merely returning the set, or subset, of x
,
you can count the occurrences of an element in a vector. After the
vector sorts, you can use the find
function to
determine the indices of zero values in diff(x)
and
to show where the elements change value. The difference between subsequent
indices from the find
function indicates the
number of occurrences for a particular element:
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
find
function does not return
indices for NaN
elements. You can count the number
of NaN
and Inf
values using
the isnan
and isinf
functions.count_nans = sum(isnan(x(:))); count_infs = sum(isinf(x(:)));
Function  Description 

all  Test to determine if all elements are nonzero 
any  Test for any nonzeros 
cumsum  Find cumulative sum 
diff  Find differences and approximate derivatives 
find  find indices and values of nonzero elements 
ind2sub  Convert from linear index to subscripts 
ipermute  Inverse permute dimensions of a multidimensional array 
logical  Convert numeric values to logical 
meshgrid  Generate X and Y arrays
for 3D plots 
ndgrid  Generate arrays for multidimensional functions and interpolations 
permute  Rearrange dimensions of a multidimensional array 
prod  Find product of array elements 
repmat  Replicate and tile an array 
reshape  Change the shape of an array 
shiftdim  Shift array dimensions 
sort  Sort array elements in ascending or descending order 
squeeze  Remove singleton dimensions from an array 
sub2ind  Convert from subscripts to linear index 
sum  find the sum of array elements 