Matrix product error accumulated during iteration

I have a problem when multiplying unitary matrix many times. Initially, a unitary(or symmetric in the following example) matrix is constructed as U. Then I multiply the matrix in an iteractive way and check the unitarity of the results by comparing U*U' with identity. They are close in the first few steps but deviate gradually. Any suggestions to reduce the error and always keep U unitary?

The following is the code.

clear all
M = rand(10);
M=0.5*(M+M');
U  = expm(-1i*M);
for t_i = 1:100
     U = U*U;
     norm = trace(U*U'-eye(10))
end

[Bjorn Gustavsson]

Why not let expm handle all of that and skip the iteration?

[zhao hongzheng]

I am acutually constructing a fibonacci sequence of a matrix, for instance

U_n = U_{n-2}*U_{n-1}, with the first two unitaries as U_1=expm(-1i*M1),U_2=expm(-1i*M2). Thus in each step, I need build matrix by iteration, which can not be simply use a single expm.

Answer 1

[Jan]

You can find the nearest unitary matrix in each iteration using an SVD:

for t_i = 1:100
     [S,~,D] = svd(U * U);
     U       = S * D';
     norm    = trace(U * U' - eye(10))
end

[zhao hongzheng]

Thank you so much! It's very helpful.

[Jan]

Does it solve your problem? Then please mark the answer as accepted.