Is least square fit using Pivoted QR decomposition a single pass or multipass fitting procedure?

I was refering, https://www.mathworks.com/help/curvefit/least-squares-fitting.html blog by matlab which tells that "the backslash operator uses QR decomposition with pivoting, which is a very stable algorithm numerically".

My question is, whether surface fitting using Pivoted QR decomposition is a single pass fitting or multipass fitting procedure? If it is trying to minimize the least square then how it is changing the coefficient values for every iteration?

Thanks

Answer 1

[John D'Errico]

Backslash does not use an iterative scheme, if that is what you are asking. As long as your model is linear in the parameters, a linear algebraic solution using pivoted QR is direct. No iterations at all. Some tools like lsqr are iterative. But not backslash.

You are thinking about linear least squares as if it is nonlinear least squares, where an iterative scheme is needed. Again, a model can have higher order terms in it and still be a linear least squares problem, as long as the model is linear in the coefficients.

Are you looking for a proof that it is possible to solve a (linear) least squares estimation using simple linear algebra, i.e., a QR? (Not hard to do, but not on topic for Answers.)

[Jay]

Thanks. Yes that is what I am looking for. I was wondering how it is getting optimized coefficient values in single pass?