# Euler expansion of double exponential?

1. Feb 5, 2008

### Clifford

Simple question,

I have used the euler expansion to estimate a variable that grows as a single exponential.
adapt = Amax * exp(-tau*X);

In excerpted form:

for (i=1;i<npts; i++)
{
adapt = adapt[i-1] + (Amax -adapt[i-1]) * dt / tau;
}

where dt is the step size and tau is the 'time constant.'

Now, however, I think that the data would be better fit with a double exponential.

adapt = a(1) * exp(-tau1*X) + a(3) * exp(-tau2*X);

I am unsure how to expand this analogously to the single exponential.
thanks!

Clifford

2. Feb 7, 2008

### trambolin

Assuming that a(1) and a(3) are some incremental values, you can define your system as an autonomous system as, $$\dot{x} = Ax$$ where $$A$$ is a $$3 \times 3$$ matrix and $$x \in \mathbb{R}^3$$, then expand the matrix exponential. And take the first state.