AllCombinations

I am not sure what the difference between extrapolation and interpolation is. I think interpolation is the fit of a function through a finite set of points and extrapolation might be looking at what a function does beyond the data set? As for "seeming contrived," I am learning about polynomial interpolation and the instructor made a side note that there are an infinite number of functions and that "most if not all" could be interpolated, and that random function was the example of one such function. The points were my own made up numbers for the sake of asking my question. If I did not provide both data points and the structure of the function to be fit, I was sure to be accused of not providing enough information. I do find the idea of interpolation or curve-fitting or whatever it is called fascinating. I like the idea of there being actual methods for finding/approximating an infinite number of points between two points that are known. It is amazing that math can be manipulated that way. I mean, I know that math is a vast subject, but this is all new to me. Being pleasantly surprised by this concept, my immediate inquiry was... is there one single method for the interpolation of any number of points in any number of variables in any coordinate system and in any random format, such as the composition of polynomials, exponentials, and trigonometric functions? It's just a curiosity. Does there exist one "almighty" method, for lack of terminology? You said the subject is vast. How many methods are there? There must be hundreds, I take it, considering how many kinds of functions there are.

Ah, quite right. Tangent is periodic. So we could use c=ln(Pi). And that allows for a and b to be solved for. That was easier than I expected. I guess I was intimidated by the composition of transcendental functions. That is a satisfactory method for three points. If, supposing we had ten points or a thousand, are there easier methods for this kind of interpolation (if I am using the right word) than substitution and elimination? Yet, even as I ask that, it strikes me as silly to imagine there being one method specifically for the form of the one I gave you. I guess what I am trying to ask is, 1) is there a numerical method that specializes in transcendental curve-fitting, and 2) is there any such thing as a "universal" method of interpolation where one might specify the desired format and the data set and get out an appropriate function? Thanks.

studiot, I thought about trying it as a system of equations but the point through the origin is giving me trouble for c. I get: 0=tan(exp©) which leads to 0=exp© which doesn't solve for c because ln(0) is undefined. So if we ignore the zero solution we have, in trying elimination and substitution, a system of three unknowns with only two equations. Thanks.

Hi. I don't know much about interpolation or curve fitting and need help, please. Fit/approximate a function through the points (0,0), (7,11), (13,33). The type of function is of the form y=tan(exp(ax^2+bx+c)). Any help is appreciated. I hope I have given enough information. Thanks.