I know how to perform a linear regression on a set of points. That is, I know how to fit a polynomial of my choice, to a given data set, (in the LSE sense). However, what I do not know, is how to force my solution to go through some particular points of my choosing. I have seen this be done before, but I cannot remember what the procedure was called, let alone how it was done.
As a very simple and concrete example, let us say that I have 100 points scattered on the x-y plane, and I choose to fit a whatever-order polynomial through them. I know how to perform this linear regression very well. However, let us say that I want to 'force' my solution, to go through, say, three of my data points at x-coordinates $x=3$, $x=19$, and $x=89$, (and their corresponding y-coordinates of course).
What is this general procedure called, how is it done, and are there any particular pitfalls I need to be aware of?
I would like to add, that I am looking for a concrete way to do this. I have written a program that actually does the linear regression in one of two ways, inverting the covariance matrix directly, or through gradient descent. What I am asking is, how, exactly, step by step, do I modify what I did, such that I force the polynomial solution to go through specific points?