The coefficient matrix A (1.9) in the normal equations (1.7) will be ill-conditioned for small λ, causing the number of correct digits in the computed spline to be small. To try to compensate for this problem, one can reformulate spline smoothing as a basic least-squares problem and solve it using a QR factorization. De Hoog and Hutchinson , building on earlier work [2, 3, 4] on general banded least-squares problems, presented a QR algorithm for spline smoothing. In this chapter we will evaluate the condition number of the coefficient matrix, present a faster and more compact QR algorithm, and determine whether this alternative is preferable to solving the normal equations.
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