Why does expm command work too slow?

Hi, I am trying to calculate exponential of a matrix using expm command, however computation speed is too slow, compared to exp. Is there any way to obtain faster response from expm? Thanks...

A_aug = rand(6,6);
B_aug = rand(6,1);
syms t
M2 = expm(A_aug * t) * B_aug

[Volcano]

It is interesting that, the code below works faster.

A_aug = eye(6,6);

B_aug = rand(6,1);

syms t

M2 = expm(A_aug * t) * B_aug

[Matt J]

It is interesting that, the code below works faster.

Not surprising. When A_aug is diagonal, expm() is the same as exp().

Answer 1

[Star Strider]

If you want to multiply it by scalar or vector ‘t’, create an anonymous function rather than doing it symbolically —

A_aug = rand(6,6);
B_aug = rand(6,1);

M2 = @(t) expm(A_aug .* t) * B_aug;

t = 0:5;

tic
for k = 1:numel(t)
    Result(:,k) = M2(t(k));
end
toc

Elapsed time is 0.064424 seconds.

format shortE
Result

Result = 6×6

1.0e+00 * 4.6155e-01 9.8985e+00 1.9054e+02 3.7164e+03 7.2586e+04 1.4178e+06 7.7791e-01 1.3078e+01 2.5333e+02 4.9457e+03 9.6600e+04 1.8869e+06 5.8940e-01 8.4603e+00 1.6211e+02 3.1618e+03 6.1750e+04 1.2061e+06 3.4496e-01 1.2376e+01 2.4938e+02 4.8821e+03 9.5378e+04 1.8630e+06 4.4668e-01 1.4365e+01 2.8701e+02 5.6118e+03 1.0962e+05 2.1412e+06 6.9777e-01 9.9196e+00 1.9076e+02 3.7249e+03 7.2760e+04 1.4212e+06

You did not post the version you are using, however this should work in all recent versions.

.

[Volcano]

I am using 2021a. I will use expm command in SIMULINK, so cannot define a t vector. t will be updated in process.

[Star Strider]

Defining it as a scalar then should work. I am not familiar with Simulink (I have it, though I have not used it in a while) however it should have the ability to evaluate ans simulate state space systems available in it, if that is what you are doing, similar to the Control System Toolbox lsim function.

Answer 2

[Steven Lord]

Do you absolutely need the result as a (potentially very long) symbolic expression in t, or is computing this matrix exponential-vector product numerically sufficient for your needs? If the latter, either use expm with a numeric value for t or (if you're using release R2023b or later) use expmv.

expmv can be especially useful if you could substitute specifying a vector of values for t into the numerical calculation instead of performing the calculation symbolically and then substituting numeric values, as the third input to expmv can be a vector. If the vector is equally spaced, it can use an algorithm that takes advantage of that uniformity.

[Volcano]

I should run the expm for different t values in a process. I will perform this task in SIMULINK, so it may be useful to obtain a parametric result of matrix-exponential. Yes, the numerical t value may make the calculation faster, however my doesn't let me use pre-defined t values.