Linearization of $A\exp(-x/t)+Bx+C$

Question

I have an experimental graph with the $y(x) = A\exp(-x/t)+Bx+C$ approximation.

I have tried to get a linearization form for it. For a simple function, such as $y(x) = A\exp(-x/t)$ is pretty obvious how to do that: $-t\ln(y(x)/A) = x$. So, if I plot g(x) = $-t\ln(y(x)/A)$ vs x, I get a linear function.

However, I cannot do that for $y(x)-C = A\exp(-x/t)+Bx$. I know there is the Lambert W function, but I don't know how to use it and plot the linear graph.

Answer 1

I agree with the judicious answer from Claude Leibovici. As far as I know there is no simple way to transform your non-linear equation to a linear equation (wrt some parameters to be optimised).

Nevertheless one can transform the non-linear equation to a linear equation involving an integral which can be numerically computed (approximately).

The method is shown below with your example of data (not exaclty because the data is obtained in scannig your graph. This isn't numerically accurate).

Don't be surprised that the numerical values of the parameters be slightly different from whose obtained thanks to non-linear regression. This is due to :

• The non accurate graphical scanning for the data,

• The regression of an integral equation instead of the equation itself,

• The numerical integration which is not very accurate because the too low number of points.

Dispite all, one cannot distinguish the curve from this method to the curve from non-linear regression.

REFERENCE

For more explanation about the method of linearisation thanks to integral or differential equations : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales .

Answer 2

You have $n$ dta points$(x_i,y_i)$ and you want to fit the model $$y=A e^{-\frac{x}{t}}+Bx+C$$ which is nonlinear.

Now, give $t$ a value : this makes the model totally linear and easy to solve.

So, for a given $t$, you even have the analytical expressions of $\big(A(t),B(t),C(t)\big)$.

Now, vary $t$ in order to find the minimum of the sum of the squares. If you do not access a nonlinear regression softwre, zoom more and more.

You could even use Newton method with numerical derivatives to solve $$\frac {d\,SSQ}{dt}=0$$ starting from an approximate solution ot $t_{\text{(min)}}$.

You are done.