NAG Library Chapter Introduction
F05 – Orthogonalization
1 Scope of the Chapter
This chapter is concerned with the orthogonalization of vectors in a finite dimensional space.
2 Background to the Problems
Let be a set of linearly independent vectors in -dimensional space; .
We wish to construct a set of
- – the vectors form an orthonormal set; that is,
for , and ;
- – each is linearly dependent on the set .
2.1 Gram–Schmidt Orthogonalization
The classical Gram–Schmidt orthogonalization process is described in many textbooks; see for example Chapter 5 of Golub and Van Loan (1996)
It constructs the orthonormal set progressively. Suppose it has computed orthonormal vectors
which orthogonalise the first
. It then uses
In finite precision computation, this process can result in a set of vectors
which are far from being orthogonal. This is caused by
being small compared with
. If this situation is detected, it can be remedied by reorthogonalising the computed
, that is, repeating the process with the computed
. See Danial et al. (1976)
2.2 Householder Orthogonalization
An alternative approach to orthogonalising a set of vectors is based on the
factorization (see the F08 Chapter Introduction
), which is usually performed by Householder's method. See Chapter 5 of Golub and Van Loan (1996)
matrix whose columns are the
vectors to be orthogonalised. The
upper triangular matrix and
matrix, whose columns are the required orthonormal set.
Moreover, for any such that , the first columns of are an orthonormal basis for the first columns of .
Householder's method requires twice as much work as the Gram–Schmidt method, provided that no re-orthogonalization is required in the latter. However, it has satisfactory numerical properties and yields vectors which are close to orthogonality even when the original vectors are close to being linearly dependent.
3 Recommendations on Choice and Use of Available Routines
The single routine in this chapter, F05AAF
, uses the Gram–Schmidt method, with re-orthogonalization to ensure that the computed vectors are close to being exactly orthogonal. This method is only available for real vectors.
To apply Householder's method, you must use routines in Chapter F08
The example programs for F08AEF (DGEQRF)
or F08ASF (ZGEQRF)
illustrate the necessary calls to these routines.
4 Routines Withdrawn or Scheduled for Withdrawal
Danial J W, Gragg W B, Kaufman L and Stewart G W (1976) Reorthogonalization and stable algorithms for updating the Gram–Schmidt factorization Math. Comput. 30 772–795
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore