Dynamic memory allocation

Matlab supports dynamic memory allocation. It is important not to misuse this feature. Test the following code segment and explain your results:

n = 1000;
p = 500;
x = randn(n, 1);

tic
  X = [];
  for k = 1:p
    X(:, k) = x;
  end
toc

tic
  X = zeros(n, p);
  for k = 1:p
    X(:, k) = x;
  end
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tic
  X = [];
  for k = p:-1:1
    X(:, k) = x;
  end
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In the following example you can see that this is important when dealing with sparse matrices as well. A typical example, from Finite Elements, is when you have to assemble the element stiffness matrices to form the global stiffness matrix. The second part simulates a situation where we know the sparsity structure from the beginning, but where the elements have to be computed using the loops. If we know the elements as well, we can use the third form.

n = 30000;
m = 150;
tic
  X = sparse(n, n);
  for j = 1:m:n       % one can try loop interchange as well
    for k = 1:m:n
      X(j, k) = 1;
    end
  end
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% Suppose we know the sparsity structure
[rows, cols] = find(X);

tic
  X = sparse(rows, cols, ones(size(rows)));
  for j = 1:m:n
    for k = 1:m:n
      X(j, k) = 2;  % but not the elements
    end
  end
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tic
  X = sparse(rows, cols, ones(size(rows)));
toc