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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?

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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 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

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