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## Main Question or Discussion Point

I have a simple problem. I have a set of 3D data points and I want to fit a line through them using linear least squares. I understand the basic approach required: set up two matrices such that Ax = b, then make it a square matrix A^t*Ax = A^t*b, then solve for x using a Cholesky decomposition. (I'm working in C++ using the Eigen 2.0 matrix library.)

The problem is that I'm not sure how to set up A and b to begin with. This would be simple if the data points were in 2D: b would be a column vector of all the ys, and A would be an Nx2 matrix with [x, 1] on each row. Then the unknowns would be x = [m, b] for the normal y = mx+b line.

But with 3D points, it's less clear to me how I have to arrange things.

Also, any suggestions for how to make the operation more numerically stable (such as subtracting off the mean from each data point, I know that one at least) would be welcome.

The problem is that I'm not sure how to set up A and b to begin with. This would be simple if the data points were in 2D: b would be a column vector of all the ys, and A would be an Nx2 matrix with [x, 1] on each row. Then the unknowns would be x = [m, b] for the normal y = mx+b line.

But with 3D points, it's less clear to me how I have to arrange things.

Also, any suggestions for how to make the operation more numerically stable (such as subtracting off the mean from each data point, I know that one at least) would be welcome.