Averaging Matlab matrix - performance

Question

In the Matlab programs I use I often have to average within a matrix (interpolation). The most straightforward way is to add the matrix and a shifted one (avg). However you could do the same operation using matrix multiplication (avg2). I noticed a considerable speed increase in the case of using matrix multiplication in the case of large matrices.
Could anyone explain why Matlab is able to process this multiplication faster than adding the same matrix? Also what are the possible downsides of using avg2() in respect to avg()?
Difference in runtime was a factor ~6 for this case (n=500).
function [] = speed()
%Speed test for averaging a matrix
n = 500;
A = rand(n,n);
tic
for i=1:100
avg(A);
end
toc
tic
for i=1:100
avg2(A);
end
toc
end
function B = avg(A,k)
if nargin<2, k = 1; end
if size(A,1)==1, A = A'; end
if k<2, B = (A(2:end,:)+A(1:end-1,:))/2; else B = avg(A,k-1); end
if size(A,2)==1, B = B'; end
end
function B = avg2(A,k)
if nargin<2, k = 1; end
if size(A,1)==1, A = A'; end
if k<2,
m = size(A,1);
e = ones(m,1);
S = spdiags(e*[1 1],-1:0,m,m-1)'/2;
B = S*A; else B = avg2(A,k-1); end
if size(A,2)==1, B = B'; end
end

Answer 1

Im afraid I cant give you an answer to the inner workings of the functions you are using. However, as they seem overly complicated, I felt I should make you aware of an easier (and a bit faster) way of doing this averaging.
You can instead use conv2 with a kernel of [0.5;0.5]. I have extended your code below:
function [A, T1, T2 T3] = speed()
%Speed test for averaging a matrix
n = 900;
A = rand(n,n);
tic
for i=1:100
T1 = avg(A);
end
toc
tic
for i=1:100
T2 = avg2(A);
end
toc
tic
for i=1:100
T3 = conv2(A,[1;1]/2,'valid');
end
toc
if sum(sum(abs(T3-T2))) > 0
warning('Method 3 not equal the other methods')
end
end
function B = avg(A,k)
if nargin<2, k = 1; end
if size(A,1)==1, A = A'; end
if k<2, B = (A(2:end,:)+A(1:end-1,:))/2; else B = avg(A,k-1); end
if size(A,2)==1, B = B'; end
end
function B = avg2(A,k)
if nargin<2, k = 1; end
if size(A,1)==1, A = A'; end
if k<2,
m = size(A,1);
e = ones(m,1);
S = spdiags(e*[1 1],-1:0,m,m-1)'/2;
B = S*A; else B = avg2(A,k-1); end
if size(A,2)==1, B = B'; end
end
Results:
Elapsed time is 10.201399 seconds.
Elapsed time is 1.088003 seconds.
Elapsed time is 1.040471 seconds.
Apologies if you already knew this.