AllCombinations

Thanks for all of your input. I appreciate the help. I think I have a clearer understanding of the difference between interpolation and optimization now.

Very nice. So, it would seem, for any three points a circle can be found to fit through them, unless they are colinear in which case the circle would have an infinite radius. So could interpolation be described as a form of optimization given a set of restraints? For example, an optimization through n points with the additional condition/restraint that the form must be a polynomial/rational/exponential/etc. Or perhaps I have it backwards and we should look at it as a form type, for instance a sine wave, that must fulfill the additional condition that it pass through a set of "special" points, special in that they are not what the sine wave would pass through on their own. Is that right, or is interpolation nothing to do with optimization?

So I Googled Hamilton - Lagrange and somehow ended up reading information on the Euler-Lagrange method. I watched https://www.youtube.com/watch?v=08vJyA-XD3Q which shows that the optimal path between two points is a straight line. My immediate question on this is how do we find the shortest path through, say, three points? I do not mean two linear equations that go from point a to point b and then from point b to point c, but rather how might one go about finding an optimal path? Does it become too complicated? Or is it a matter of there being many solutions? Right now I am not thinking about gravitational fields or air resistance, but only in pure terms as in interpolation. I welcome any input, insight, or suggeted reading.

Constant fighting and stupidity is why I try to avoid most forums. I agree, this is a nice one. The calculus of variations? I will certainly look in to it. To "determine the best curve through data points according to some pre established criterion" sounds like exactly what I have in mind. I like the beauty of pure mathematics but I also lean toward application. That the physical world speaks in mathematics is one of the things I like most about mathematics. Have a good night.

I understood your explanation, though I did not understand how your triangular grid relates to interpolation. I also understand the concept of a differential equation but I do not know what a finite element grid is, at least not by name. I don't know if that answered your question or not. Apart from taking a few local classes I am almost entirely self-taught by buying books and reading them, and from online courseware by Khan Academy, MIT, etc. on youtube. Teaching myself is something I can do. The hardest part is knowing what subjects to study and in what order. For instance, I just taught myself linear algebra and now I am just starting books on tensor analysis (the book by Bishop and Goldberg and another one by Grinfeld who also has a lecture series on Youtube) but I could just have easily tried tensors first only to realize I needed linear algebra as a prerequisite. I was lucky in this case. I didn't always have a particular goal in mind because small schools are the only ones I have ever had available to me and they do not teach anything further along than multivariable calculus and first-order ordinary differential equations. Now that I have explored on my own I find the subject of interpolation to be particularlly fascinating... I just don't know what to study or what direction to even head in. I mentioned numerical analysis but I don't really know if that is the right direction? I don't spend much time on forums but I thought it was a good time to first ask if there is one single method for fitting most functions before I further explored the appropriate subjects (whatever they turn out to be) or if the only way to learn how to fit a wide range of functions would be to study interpolation and curve-fitting in more depth. If I strike you as someone who doesn't really seem to know what they are doing, you are correct. That is why I was seeking advice. In summary, I understand calculus, (basic) differential equations, and linear algebra, I am now in the first few pages of a book on tensors, and I am interested in learning either about a general method for fitting a lot of different kinds of functions (algebraic and transcendental, anyway) or in seeking out a wide range of these methods, or both.

studiot: That is really fascinating. Thank you for taking the time to do all of that. You explained it really well. In your opinion, what subject or subjects should I study in order to learn more about interpolation and its various associated methods? Numerical analysis? Any other information is welcome. Thanks again for giving your time to share all of that. And no worries about the "contrived" remark. I wondered but I understand now. overtone: That is interesting. Interesting too that math has applications in those subjects, though I guess it makes sense when I consider topics like population, predator-prey models, and the spread of disease.

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.