Vectorization: matrix array multiplication element wise one by one - matlab

Question

I have a matrix:
R = [0 -1;1 0];
array = 1:1:10;
Also x0 = [2;1]
How can I obtain another array in the most efficient way without loop?
array2 = [expm(1*R) expm(2*R) expm(3*R) .... expm(10*R)];
Then I want to obtain
array3 of dimension 2 by 10 such that:
array3 = [expm(1*R)*x0 expm(2*R)*x0 expm(3*R)*x0 .... expm(10*R)*x0];

Answer 1

From wikipedia:
If a matrix is diagonal its exponential can be obtained by exponentiating each entry on the main diagonal.
Given that a block diagonal matrix can be created from {1*R, 2*R,...} then its exponential can be obtained and reshaped to a [2 * n] and it can be multiplied by x0.
However its performance may be worse than for loop.
R = [0 -1;1 0];
array = 1:1:10;
x0 = [2;1]
n = numel(array);
result = reshape(expm(kron(spdiags(array.',0,n,n),R))*repmat(x0,n,1),2,[]);
For array of small size (less than 70 elements) full matrix is more efficient:
result = reshape(expm(kron(diag(array),R))*repmat(x0,n,1),2,[]);

Answer 2

Well I see that the matrix R that you have is 2x2. In case it is always 2x2, then you can use the following function (Wikipedia) to calculate the exponential:
function output = expm2d(A)
% Assuming t = 1 from Evaluation by Laurent series (https://en.wikipedia.org/wiki/Matrix_exponential#Evaluation_by_Laurent_series)
s = trace(A) / 2;
q = sqrt(-det(A - s*eye(size(A))));
output = exp(s) * ((cosh(q) - s * sinh(q) / q) * eye(size(A)) + (sinh(q) * A / q));
end
Using the excellent comparison function provided by thewaywewalk, I got the following results:
When using expm:
>> bench
ans =
0.0181 %// rahnema
0.1075 %// thewaywewalk arrayfun
0.1139 %// thewaywewalk accumarray
When using expm2d:
>> bench
ans =
0.0048 %// rahnema
0.0161 %// thewaywewalk arrayfun
0.0222 %// thewaywewalk accumarray
As you can see, using the function for 2d matrices leads to a 10x decrease in the runtime. Of course, this cannot be used when R is not 2x2.
Edit:
When using expm2d for A = 1:100:
>> bench
ans =
0.1379 %// rahnema
0.1415 %// thewaywewalk arrayfun
0.1756 %// thewaywewalk accumarray

Answer 3

I still don't know if I got your question right. Here are two solutions which are not fully vectorized, but fairly fast:
R = [0 -1;1 0];
A = 1:1:10;
x0 = [2;1];
%// option 1
temp = arrayfun(#(x) (expm(R*x)*x0).', A, 'uni', 0);
array3 = vertcat( temp{:} )
%// option 2
temp = accumarray( (1:numel(A)).', A(:), [], #(x) {(expm(R*x)*x0).'})
array3 = vertcat( temp{:} )
Benchmark
I haven't considered Leander's Answer as it doesn't calculate array3:
function [t] = bench()
R = [0 -1;1 0];
A = 1:1:10;
x0 = [2;1];
% functions to compare
fcns = {
#() compare1(A,R,x0);
#() compare2(A,R,x0);
#() compare3(A,R,x0);
};
% timeit
t = zeros(3,1);
for ii = 1:100;
t = t + cellfun(#timeit, fcns);
end
end
function array3 = compare1(A,R,x0) %rahnema1
n = numel(A);
array3 = reshape(expm(kron(diag(A),R))*repmat(x0,n,1),2,[])
end
function array3 = compare2(A,R,x0) %thewaywewalk 1
temp = arrayfun(#(x) (expm(R*x)*x0).', A, 'uni', 0);
array3 = vertcat( temp{:} )
end
function array3 = compare3(A,R,x0) %thewaywewalk 2
temp = accumarray( (1:numel(A)).', A(:), [], #(x) {(expm(R*x)*x0).'});
array3 = vertcat( temp{:} )
end
Results:
for A = 1:1:10;
0.1006 %// rahnema
0.2831 %// thewaywewalk arrayfun
0.3103 %// thewaywewalk accumarray
As kron gets really slow for large arrays, the benchmark results change for A = 1:1:100;:
4.0068 %// rahnema
1.8045 %// thewaywewalk arrayfun
2.4257 %// thewaywewalk accumarray