Vectorization and Logical Arrays

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Memory Allocation

When you add an element to a MATLAB matrix beyond the current bounds of the matrix, it automatically expands. However, this process of allocating memory takes time. Always know the size of the data structure ahead of time.

function squares()
	for n=1:100000
		a(n) = n^2
	end
end

90 seconds

function squares()
	a = zeros(1,100000);
	for n=1:100000
		a(n) = n^2
	end
end

0.004 seconds

Looping vs Vectorizing

Math operators are not always more efficient than using a looping construct. However, some tricks are available if you cast your code as matrix operations that make your code more elegant, easier to read, and possibly faster.

Example of equivalent functions:

function n=flipcoins()
	n=0;
	for m = 1:1000000
	q = rand(1);
		if (q>0.5)
			n = n+1;
		end
	end
end

0.15s

function n=flipcoins()
	q=rand(1000000,1);
	n = sum(q>0.5);
end

0.09s

Logical Indexing

Consider a=[2 5 7; 8 9 1] subscripted indexing: a(1,3), a([2.1]), a(:,3) linear indexing a(4)

In logical indexing, the index is a logical matrix of the same size as the referenced matrix. The result is that the expression refers only to those elements in the referenced matrix that corresponds to the 1 (TRUE) vlaues in the indexing matrix.

When alone or on the right hand side, use logical indexing to provide a single column vector that contains all the values for which the indexing matrix is 1.

a=[5 3 2; 6 1 0];
q = logical([1 0 1; 1 0 0]);
a(q)

Results in a column vector of [5;6;2]

a=[5 3 2; 6 1 0];
r = a < 3;
a(r)

Results in a column vector of `[1;2;0]

You can also use it to change those values that satisfies r:

a(r)=-1

Results in [5 3 -1; 6 -1 -1]

function calcmatrix() %.09s
	X=zeros(2001); 
	a=1;  
	for m = -1:0.01:1
		b=1;  
		for n = -1:0.01:1;
			X(a,b) = m^2+n^2; 
			if X(a,b) > 1	
 				X(a,b)=1;
			end
		end
		a=a+1;
		end
end

It can be simplified as:

function calcmatrix() %0.007s
	[n,m]=meshgrid(-1:0.01:1,-1:0.01:1);
	X = m.^2+n.^2; X(X>1)=1;
end

Helper functions

all returns true if all elements are true any returns true if any elements are true

helpful for vectorizing code sum returns the sum along a dimension cumsum returns the cumulative sum along a specified dimension. usage: cumsum([MATRIX],DIMENSION) (1 is row, 2 is column) prod returns the product along a dimension cumprodreturns the cumulative product along a specified dimension prod return differences between adjacent elements. (approx. derivative)

sub2ind returns the index given the subscripts ind2sub returns the subscripts given the index permute permutes the dimensions of a matrix ipermute inverse permutes the dimensions shiftdim shift array dimensions
circshift shifts all elements in a specified direction reshape reshape the array

find returns the indices of nonzero elements (linear indexing) sort sort elements along a dimension unique returns the unique elements in an array meshgrid returns a mesh in x y order ndgrid returns a mesh in row column order repmat replicate and tile an array squeeze eliminate singleton dimensions