Compute weighted summation of matrix power (matrix polynomial) in Matlab - matlab

Question

Given an nxn matrix A_k and a nx1 vector x, is there any smart way to compute
using Matlab? x_i are the elements of the vector x, therefore J is a sum of matrices. So far I have used a for loop, but I was wondering if there was a smarter way.

Answer 1

Short answer: you can use the builtin matlab function polyvalm for matrix polynomial evaluation as follows:
x = x(end:-1:1); % flip the order of the elements
x(end+1) = 0; % append 0
J = polyvalm(x, A);
Long answer: Matlab uses a loop internally. So, you didn't gain that much or you perform even worse if you optimise your own implementation (see my calcJ_loopOptimised function):
% construct random input
n = 100;
A = rand(n);
x = rand(n, 1);
% calculate the result using different methods
Jbuiltin = calcJ_builtin(A, x);
Jloop = calcJ_loop(A, x);
JloopOptimised = calcJ_loopOptimised(A, x);
% check if the functions are mathematically equivalent (should be in the order of `eps`)
relativeError1 = max(max(abs(Jbuiltin - Jloop)))/max(max(Jbuiltin))
relativeError2 = max(max(abs(Jloop - JloopOptimised)))/max(max(Jloop))
% measure the execution time
t_loopOptimised = timeit(#() calcJ_loopOptimised(A, x))
t_builtin = timeit(#() calcJ_builtin(A, x))
t_loop = timeit(#() calcJ_loop(A, x))
% check if builtin function is faster
builtinFaster = t_builtin < t_loopOptimised
% calculate J using Matlab builtin function
function J = calcJ_builtin(A, x)
x = x(end:-1:1);
x(end+1) = 0;
J = polyvalm(x, A);
end
% naive loop implementation
function J = calcJ_loop(A, x)
n = size(A, 1);
J = zeros(n,n);
for i=1:n
J = J + A^i * x(i);
end
end
% optimised loop implementation (cache result of matrix power)
function J = calcJ_loopOptimised(A, x)
n = size(A, 1);
J = zeros(n,n);
A_ = eye(n);
for i=1:n
A_ = A_*A;
J = J + A_ * x(i);
end
end
For n=100, I get the following:
t_loopOptimised = 0.0077
t_builtin = 0.0084
t_loop = 0.0295
For n=5, I get the following:
t_loopOptimised = 7.4425e-06
t_builtin = 4.7399e-05
t_loop = 1.0496e-04
Note that my timings fluctuates somewhat between different runs, but the optimised loop is almost always faster (up to 6x for small n) than the builtin function.