How to normalize a vector in MATLAB efficiently? Any related built-in function ?

Question

I normalize a vector V in MATLAB as following:

normalized_V = V/norm(V);

however, is it the most elegant (efficient) way to normalize a vector in MATLAB?

Answer 1

The only problem you would run into is if the norm of V is zero (or very close to it). This could give you Inf or NaN when you divide, along with a divide-by-zero warning. If you don't care about getting an Inf or NaN, you can just turn the warning on and off using WARNING:

oldState = warning('off','MATLAB:divideByZero');  % Return previous state then
                                                  %   turn off DBZ warning
uV = V/norm(V);
warning(oldState);  % Restore previous state

If you don't want any Inf or NaN values, you have to check the size of the norm first:

normV = norm(V);
if normV > 0,  % Or some other threshold, like EPS
  uV = V/normV;
else,
  uV = V;  % Do nothing since it's basically 0
end

If I need it in a program, I usually put the above code in my own function, usually called unit (since it basically turns a vector into a unit vector pointing in the same direction).

Answer 2

The original code you suggest is the best way.

Matlab is extremely good at vectorized operations such as this, at least for large vectors.

The built-in norm function is very fast. Here are some timing results:

V = rand(10000000,1);
% Run once
tic; V1=V/norm(V); toc           % result:  0.228273s
tic; V2=V/sqrt(sum(V.*V)); toc   % result:  0.325161s
tic; V1=V/norm(V); toc           % result:  0.218892s

V1 is calculated a second time here just to make sure there are no important cache penalties on the first call.

Timing information here was produced with R2008a x64 on Windows.

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EDIT:

Revised answer based on gnovice's suggestions (see comments). Matrix math (barely) wins:

clc; clear all;
V = rand(1024*1024*32,1);
N = 10;
tic; for i=1:N, V1 = V/norm(V);         end; toc % 6.3 s
tic; for i=1:N, V2 = V/sqrt(sum(V.*V)); end; toc % 9.3 s
tic; for i=1:N, V3 = V/sqrt(V'*V);      end; toc % 6.2 s ***
tic; for i=1:N, V4 = V/sqrt(sum(V.^2)); end; toc % 9.2 s
tic; for i=1:N, V1=V/norm(V);           end; toc % 6.4 s

IMHO, the difference between "norm(V)" and "sqrt(V'V)" is small enough that for most programs, it's best to go with the one that's more clear. To me, "norm(V)" is clearer and easier to read, but "sqrt(V'V)" is still idiomatic in Matlab.