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In my project work, I've had to use the LMA technique for some non linear fitting. I really want to understand what it's doing rather than just using it. I have a poor knowledge of non linear fitting so please bear with me!

When producing the line of best fit, it needs to follow the data points extremely closely, obviously. To do this it minimises the sum of the square of the residuals, correct? So when I'm reading this literature, why does it always talk about minimising the function? I understand the LMA is only able to find a local minimum and not a global one, and hence it's important to set sensible initial starting parameters.

But why does the method need to find the minimum? Is it that a local minimum that might not be the global minimum trick it into thinking it found it, and send the curve off in the wrong direction? (In my head I have the idea of an x^3 trend with an x^2 fit if that makes any sense).

When the literature is talking about minimisation, what is it referring to? The residuals or the curve itself?

Also, for bonus points, could anyone explain what a trust region is in simple terms?

Any insight at all would be greatly appreciated!

Thank you in advance.