While reading an article about iterative methods for solving nonlinear equations I can't understand what is *exponentially fitted osculating straight line*. Could someone please briefly explain this term to me?



What is exponentially fitted osculating straight line?
it's jargon for a function of the form
$$y(x)=e^{a(x-x\_0)}[b(x-x\_0)+c]$$
where the constants $a,b,c$ are determined such that $y(x)$ [osculates](http://en.wikipedia.org/wiki/Osculating_curve) the function $f(x)$ you are fitting to at $x=x\_0$:


$$y(x\_0)=f(x\_0),\;\;y'(x\_0)=f'(x\_0),\;\;y''(x\_0)=f''(x\_0).$$


This gives


$$c=f(x\_0),\;\;b+ac=f'(x\_0),\;\;2ab+a^2 c=f''(x\_0)$$


for example, see this [paper](http://www.emis.de/journals/HOA/JAM/Volume2012/294086.pdf)


