# Least Squares Data Fitting

- What does the least-squares solution minimize?
- For a given model and given data points, can you form the system \({\bf A} {\bf x} \cong {\bf b}\) for a least squares problem?
- For a small problem, given some data points and a model, can you determine the least squares solution?
- In general, what can we say about the value of the residual for the least squares solution?
- What are the differences between least squares data fitting and interpolation?
- Given the SVD of a matrix \({\bf A}\), how can we use the SVD to compute the residual of the least squares solution?
- Given the SVD of a matrix \({\bf A}\), how can we use the SVD to compute the least squares solution? Be able to do this for a small problem.
- Given an already computed SVD of a matrix \({\bf A}\), what is the cost of using the SVD to solve a least squares problem \(A {\bf x} \cong {\bf b}\)?
- Why would you use the SVD instead of normal equations to find the solution to \({\bf A} {\bf x} \cong {\bf b}\)?
- Which costs less: solving a least squares problem via the normal equations or solving a least squares problem using the SVD?
- What is the difference between a linear and a nonlinear least squares problem? What sort of model makes it a nonlinear problem? For data points \((t_i, y_i)\), is fitting \(y = a \cos(t) + b\) where \(a\) and \(b\) are the coefficients we are trying to determine a linear or nonlinear least squares problem?