# QR Algorithm

• Howard L. Weinert
Chapter
Part of the SpringerBriefs in Computer Science book series (BRIEFSCOMPUTER)

## Abstract

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 [1], 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.

## Keywords

Condition Number Coefficient Matrix Normal Equation Cholesky Factor Spline Smoothing
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1].
De Hoog FR, Hutchinson MF (1987) An efficient method for calculating smoothing splines using orthogonal transformations. Numer Math 50:311-319
2. [2].
George A, Heath MT (1980) Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl 34:69-83
3. [3].
Cox MG (1981) The least squares solution of overdetermined linear equations having band or augmented band structure. IMA J Numer Anal 1:3-22
4. [4].
Elden L (1984) An algorithm for the regularization of ill-conditioned banded least squares problems. SIAM J Sci Stat Comput 5:237-254
5. [5].
Golub GH, Van Loan CF (1996) Matrix computations. The Johns Hopkins University Press, Baltimore
6. [6].
7. [7].
Higham NJ (2002) Accuracy and stability of numerical algorithms. SIAM, Philadelphia
8. [8].
Malcolm MA, Palmer J (1974) A fast method for solving a class of tridiagonal linear systems. Commun. ACM 17:14-17