Vectorizing nested for loops

How do I vectorize these for loops?

I am looking for the fastest implementation.

Thank you!

My current implementation is

d = zeros(N,N);
for i = 1:N
   for j = i:N
       d(i,j) = vecnorm( Phi(:,i) - Phi(:,j) ); 
   end
end

d = d + d';

Answer 1

[Jos (10584)]

A = magic(5) 
d = squareform(pdist(transpose(A))) % transpose to obtain vecnorm between columns

pdist and squareform are part of the Statistics Toolbox

[Adam]

Excellent! This is about an order of magnitude faster than the other answer.

Answer 2

[Bjorn Gustavsson]

You find a nice contribution from J Kirk on the file exchange: Distmat

In my historic appreciation efforts there is this Acklamized function for distmat:

function d = distmat1(x, y)
% DISTMAT1 Distance matrix (one for-loop).
%
%   D = DISTMAT1(X, Y) returns the distance matrix with all distances
%   between the points represented by the rows of X and Y.
%
%   DISTMAT1(X) is equivalent to DISTMAT1(X, X), but the former computes the
%   distance matrix faster.
%
%   Distance is Euclidean.
%
%   The calculation is done with one for-loop.
%
%   See also DISTMAT0, DISTMAT2, DISTMAT3.

%   Author:      Peter J. A.
%   Time-stamp:  2000-07-10 20:20:27
%   E-mail:      [e-mail]
%   URL:         http:xxxxxxxxxxxxxxx

error(nargchk(1, 2, nargin));

if nargin == 1                       % DISTMAT1(X)
  
  if ndims(x) ~= 2
    error('Input must be a matrix.');
  end
  
  m = size(x, 1);
  d = zeros(m, m);                  % initialise output matrix
  
  for i = 1:m-1
    xi = x(i,:);
    diff = xi(ones(m-i,1),:) - x(i+1:m,:);    % difference
    dist = sqrt(sum(abs(diff).^2, 2));        % distance
    d(i+1:m,i) = dist;
    d(i,i+1:m) = dist.';
  end
  
else                                 % DISTMAT1(X, Y)
  
  if ndims(x) ~= 2 | ndims(y) ~= 2
    error('Input must be two matrices.');
  end
  
  [mx, nx] = size(x);
  [my, ny] = size(y);
  
  if nx ~= ny
    error('Both matrices must have the same number of columns.');
  end
  
  m = mx;                   % number of rows in distance matrix
  n = my;                   % number of columns in distance matrix
  p = nx;                   % dimension of each point
  d = zeros(m, n);          % initialise output matrix
  
  % The for-loop is applied on the shortest dimension, the other is
  % vectorized.
  if m < n
    idx = ones(1, n);
    for i = 1:m
      xi     = x(i,:);
      d(i,:) = sqrt(sum(abs(y - xi(idx,:)).^2, 2)).';
    end
  else
    idx = ones(1, m);
    for j = 1:n
      yj     = y(j,:);
      d(:,j) = sqrt(sum(abs(x - yj(idx,:)).^2, 2));
    end
  end

end

Maybe no longer optimal coding - but the originator of this code was a master at writing good vectorized terse code, back in the olden days before bxfun arrayfun and all the tools making the youth of today .....

HTH