Why vectorized calculations are faster than for loops?

Why it's faster in Matlab? Is it because better memory treating, or paralleling? If this is only due parallel computation, on single-core laptop it will be now difference between? Thanks<

Answer 1

[Jan]

At first: There is no evidence that vectorized code is faster in general.

If a build-in function can be applied to a complete array, a vectorization is much faster than a loop appraoch. When large temporary arrays are required, the benefits of the vectorization can be dominated by the expensive allocation of the memory, when it does not match into the processor cache.

A secondray effect of vectorizing is that the code looks more clear, at least as a rule of thumb. A trivial example:

% Loops:
A = rand(10);
B = rand(10);
C = zeros(size(A));
for i2 = 1:size(A, 2)
  for i1 = 1:size(A, 1)  % Columns in the inner loop
    C(i1, i2) = A(i1, i2) + B(i1, i2);
  end
end

% Vectorized:
C = A + B;

The 2nd method is faster concerning the runtime, but also for the programming and debug time. There is almost no chance to create a bug and it will be very easy to understand the code, when the program needs changes in the future.

[Image Analyst]

The comment should say "Rows are in the inner loop", not columns.

Interestingly, with R2014b I'm not seeing much difference, at least with this example, when the rows or columns are in the inner loop. Maybe the arrays are too small. But since MATLAB stores in column major order, it should be faster to iterate over the rows first (inner loop).

% Loops:
A = rand(1500);
B = rand(1500);
C = zeros(size(A));
tic
for col = 1:size(A, 2) % Columns are in the outer loop
  for row = 1:size(A, 1)  % Rows in the inner loop
    C(row, col) = A(row, col) + B(row, col);
  end
end
toc
tic
for row = 1:size(A, 1)  % Rows in the outer loop
  for col = 1:size(A, 2)  % Columns are in the inner loop
    C(row, col) = A(row, col) + B(row, col);
  end
end
toc
tic
% Vectorized:
C = A + B;
toc

Times:

Elapsed time is 1.713167 seconds.
Elapsed time is 1.711998 seconds.
Elapsed time is 0.004025 seconds.

[Mikhail]

I still didn't understand why it happens? Why build-in function are faster? For what (fundamental) reason it happens?

[Keldon Alleyne]

Multiplying matrices in loops is O(N^3), while the fastest algorithms using other methods are O(N^2.3) - O(N^2.8), which can easily explain the differences in performance.

On my laptop with Matlab 2018b I get:

Elapsed time is 0.112362 seconds.
Elapsed time is 0.214544 seconds.
Elapsed time is 0.007935 seconds.