Vectorization and Broadcasting

Analogy categoryAnalogies

Why Vectorize: Replacing Loops with Array Operations

Vectorization means expressing an operation as a single array-wide command instead of an explicit for-loop over individual elements; because MATLAB's interpreter has per-iteration overhead, a vectorized statement like y = sin(x) applied to an entire vector x is typically far faster than looping and calling sin(x(i)) one element at a time, in addition to being shorter and easier to read.

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Cricket analogy: Vectorizing sin(x) over a whole vector at once is like a stadium's electronic scoreboard updating every player's strike rate simultaneously after a match, rather than a scorer manually recalculating one player's figures at a time, over by over.

Implicit Expansion (Broadcasting)

Since MATLAB R2016b, arithmetic operators automatically broadcast, or implicitly expand, operands of compatible but different sizes: adding a 3x1 column vector to a 1x4 row vector produces a 3x4 matrix where every combination of elements is summed, without needing an explicit repmat call. Two dimensions are broadcast-compatible if they are equal or one of them is 1; a 3x1 and a 1x4 combine because each mismatched dimension has a 1 on one side.

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Cricket analogy: Broadcasting a 3x1 column of bowler economy rates against a 1x4 row of over-numbers to produce a 3x4 combined matrix mirrors instantly cross-tabulating every bowler's performance against every over of a match without manually pairing each combination.

Common Vectorization Idioms

Beyond simple elementwise math, vectorization extends to conditional logic (using logical indexing instead of if-inside-a-loop), accumulation (using sum, cumsum, prod instead of running totals in a loop), and functional application (arrayfun/cellfun for per-element function calls that can't be simply vectorized). bsxfun still works for broadcasting on older MATLAB versions but is now largely redundant since implicit expansion was introduced.

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Cricket analogy: Using cumsum on ball-by-ball runs to get a running total mirrors a live scoreboard's running score, updated instantly for the whole innings at once instead of a scorer adding one ball's runs at a time.

matlab

x = linspace(0, 2*pi, 1000);

% Vectorized (fast)
y = sin(x);

% Non-vectorized equivalent (slow, avoid)
y2 = zeros(size(x));
for k = 1:numel(x)
    y2(k) = sin(x(k));
end

% Implicit expansion / broadcasting (R2016b+)
col = [1; 2; 3];        % 3x1
row = [10 20 30 40];    % 1x4
M = col + row;           % 3x4 result, every pairwise sum

% Logical indexing instead of loop+if
data = [-2 5 -1 8 0 -3];
positives = data(data > 0);   % [5 8]

% Accumulation without a loop
running = cumsum(data);

Two array dimensions are broadcast-compatible if they are equal or one of them is 1. A 3x1 plus a 1x4 works, producing a 3x4 result; a 3x1 plus a 1x5 also works; but a 3x2 plus a 1x4 fails because neither the 3-vs-1 pairing nor the 2-vs-4 pairing satisfies the rule on both dimensions simultaneously.

Implicit expansion can silently produce a much larger matrix than intended if you accidentally add a column vector to a row vector instead of two same-shaped vectors — check sizes with size() when a result unexpectedly balloons to an NxM matrix instead of the elementwise Nx1 result you meant.

Vectorization replaces explicit for-loops with whole-array operations, which is faster and more concise in MATLAB.
Implicit expansion (broadcasting), standard since R2016b, automatically expands operands whose dimensions are equal or 1.
Logical indexing (data(data>0)) replaces if-inside-a-loop filtering.
cumsum, sum, and prod replace manual running-total loops.
arrayfun/cellfun handle per-element function calls that resist direct vectorization.
bsxfun is now largely redundant, superseded by automatic implicit expansion.
Unexpected broadcasting can silently create a much larger result than intended — verify with size().