# QR Algorithm

Chapter

First Online:

## 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].De Hoog FR, Hutchinson MF (1987) An efficient method for calculating smoothing splines using orthogonal transformations. Numer Math 50:311-319MathSciNetMATHCrossRefGoogle Scholar
- [2].George A, Heath MT (1980) Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl 34:69-83MathSciNetMATHCrossRefGoogle Scholar
- [3].Cox MG (1981) The least squares solution of overdetermined linear equations having band or augmented band structure. IMA J Numer Anal 1:3-22MathSciNetMATHCrossRefGoogle Scholar
- [4].Elden L (1984) An algorithm for the regularization of ill-conditioned banded least squares problems. SIAM J Sci Stat Comput 5:237-254MathSciNetMATHCrossRefGoogle Scholar
- [5].Golub GH, Van Loan CF (1996) Matrix computations. The Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
- [6].Stewart GW (1998) Matrix algorithms - basic decompositions. SIAM, PhiladelphiaGoogle Scholar
- [7].Higham NJ (2002) Accuracy and stability of numerical algorithms. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
- [8].Malcolm MA, Palmer J (1974) A fast method for solving a class of tridiagonal linear systems. Commun. ACM 17:14-17MathSciNetMATHCrossRefGoogle Scholar

## Copyright information

© The Author(s) 2013