I've a table of data in the following format:

x--> 0.1 3.767 4.395 5.0223

y

0.1 | 1 1.5 2.0 2.0---->z

0.6764 | 1 2.0 2.2 3.599-->z

1.10146 | 2 2.2 2.5 3.686-->z

1.3855 | 2.5 2.618 2.673 2.718-->z

Now using the above table, i wish to find z at unknown point say (x1,y1). As far as i know, i've two options:

(1) Interpolation and extrapolation (2) Curve Fitting.

I mention my problems with each:

**(1) Interpolation and extrapolation**: I can do bilinear interpolation to some satisfaction but what if the interpolation point is out of range. That is I'm not sure about the best method to do extrapolation. Please suggest.

**(2)Curve Fitting:**I'm thinking of Polynomial curve fitting. I'm not sure about the degree of polynomial. Is there any criteria do choose the degree of polynomial? Now suppose I fix the degree to three, what is the best menthod to obtain the coefficients of the polynomial? Please help. The last question related to curve fitting is How do I know the error i'm making in getting z for an unknow point (x1, y1). Thanks in advance.