curvefitting - sum of exponential function

I have data from experiment.

I want to find the best-fit curve for this and I expect that the result will be “∑A(k)*exp(-t/lamda(k))” A(k) and lamda(k) are coefficients

Since I don’t have “curvefitting tool”, <1>does it work for such function above?

If not, <2>how can I find the best coefficient with the least calculation time?

Answer 1

[Arturo Gonzalez]

Per this answer, you can do it with the following matlab code

https://math.stackexchange.com/a/3808325/605948

clear all;
clc;
% get data
dx = 0.001;
x = (dx:dx:1.5)';
y = -1 + 5*exp(0.5*x) + 4*exp(-3*x) + 2*exp(-2*x);
% calculate n integrals of y and n-1 powers of x
n = 3;
iy = zeros(length(x), n);
xp = zeros(length(x), n+1);
iy(:,1) = cumtrapz(x, y);
xp(:,1) = x;
for ii=2:1:n
iy(:, ii) = cumtrapz(x, iy(:, ii-1));
xp(:, ii) = xp(:, ii-1) .* x;
end
xp(:, n+1) = ones(size(x));
% get exponentials lambdas
Y = [iy, xp];
A = pinv(Y)*y;
Ahat = [A(1:n)'; [eye(n-1), zeros(n-1, 1)]];
lambdas = eig(Ahat);
lambdas
% get exponentials multipliers
X = [ones(size(x)), exp(lambdas'.*x)];
P = pinv(X)*y;
P
% show estimate
y_est = X*P;
figure();
plot(x, y); hold on;
plot(x, y_est, 'r--');