An efficient and stable algorithm for solving least squares problems \[ {min }_f \{ {{||Kf - g||}_2^2 + μ ^2 {||Lf||}_2^2 } \}, \] where the matrix K is ill conditioned, and K and L have band structure, is studied. The algorithm, which is a variant of an algorithm given by George and Heath, is based on $QR$-decomposition by plane rotations. The amount of work depends linearly on the dimension. It is shown that the algorithm does not introduce any unnecessary fill-in. The efficiency and stability of the new algorithm are compared to those of the algorithm based on normal equations. The algorithm can also be used for computing smoothing splines.
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Lars Eldén (1984) studied this question.