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Calculation the integral of the exponential of a matrix

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I want to integrate of an exponential of a matrix ( E= exp (Mmat.*(t0-t))). First of all ‘expm’ built-in function in MATLAB has complexity of order N^3 ; while my matrix is a circulant matrix, so I can use a theorem instead that makes complexity of the order of NlogN;
The algorithm is :
( in this algorithm, E=exp(A) which A is a matrix )
By this algorithm I can easily calculate the exp of a matrix but my problem now is the calculation of the integral of this function I mean
load filem.mat
expMt=ifft(expFF);% the first coulmn of exp (mt)
for x=2:322
expMmat(:,x) = circshift(expMDelt,x-1,1);
I don’t know how can I define it as a function. I can use handle function or function defining in a separate file code but how can I do it. Also I can use integral built in finction in matlab.
Also I think I have an another option and calculate the amount of the function in each time and use the trapz function for integration but I think Integral is more suitable for this situation. I attach my code and the data. I really appreciate any help
thanks in advance
raha ahmadi
raha ahmadi on 21 Jul 2021
Dear Toresten thank you for your responce. My matrix is usually sparse and inv function is really inefficient. Altough I use the other algorithms but I get inappropriate answers With best regards

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Accepted Answer

David Goodmanson
David Goodmanson on 20 Jul 2021
Edited: David Goodmanson on 20 Jul 2021
Hi raha,
I think you can go with the following technique, which does integration, differentiation etc. at the eigenvalue level. I was casual about it and created the circulant matrices by concatentation which of course would have to be improved for large matrices.
t = 4;
m1 = 2*rand(5,1)-1
M = circ(m1)
expM = expm(M*t)
% calculate N = expm(M*t)
c1 = M(:,1);
lam = fft(c1);
lamfun = exp(lam*t);
i1 = ifft(lamfun);
N = circ(i1)
max(abs(N-expM),[],'all') % check should be small
% calculate D = d/dt expm(M*t)
c1 = M(:,1);
lam = fft(c1);
lamfun = (exp(lam*t)).*lam;
i1 = ifft(lamfun);
D = circ(i1)
max(abs(D - M*expM),[],'all')
max(abs(D - expM*M),[],'all')
% calculate I = Int expm(M*t) dt (indefinite integral)
c1 = M(:,1);
lam = fft(c1);
lamfun = (exp(lam*t))./lam;
i1 = ifft(lamfun);
I = circ(i1)
max(abs(I*M -expM),[],'all')
max(abs(M*I -expM),[],'all')
function M = circ(m1)
% create circulant matrix by circular shift of columns
n = size(m1,1);
M = m1;
for k = 1:n-1
M = [M circshift(m1,k,1)];
raha ahmadi
raha ahmadi on 22 Jul 2021
I 'm very happy to hear that. The reference for the algorithm is this < > I agree with you the algorithm is very helpful. I hope you the best and of course thanks again:)

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