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#### Stability of .

The deflating orthogonal transformation construction shown in Algorithm 7.8 is clearly stable (i.e., componentwise relatively accurate representation of the transformation one would obtain in exact arithmetic). There is no question that the similarity transformation numerically preserves the eigenvalues of . However, there is a serious question about how well these transformations perform numerically in preserving Hessenberg form during purging. A modification to the basic algorithm is needed to assure that if , then is numerically Hessenberg (i.e., that the entries below the subdiagonal are all tiny relative to ).

The fact that is Hessenberg depends upon the term vanishing in the expression

However, on closer examination, we see that

where is the first component of . Therefore, || g || = 1| _1 |,

so there may be numerical difficulty when the first component of is small. To be specific, and thus in exact arithmetic. However, in finite precision, the computed . The error will be on the order of relative to , but

so this term may be quite large. It may be as large as order if . This is of serious concern and will occur in practice without the modification we now propose.

The remedy is to introduce a step-by-step acceptable perturbation and rescaling of the vector to simultaneously force the conditions

to hold with sufficient accuracy in finite precision. To accomplish this, we shall devise a scheme to achieve

numerically. As shown in [420], this modification is sufficient to establish numerically, relative to for .

The basic idea is as follows: If, at the th step, the computed quantity is not sufficiently small, then it is adjusted to be small enough by scaling the vector with a number and the component with a number . With this rescaling just prior to the computation of , we then have and , where . Certainly, should not be altered with this scaling and this is therefore required. This gives the following system of equations to determine and : if , (_j )^2 + (1 - _j^2)^2 &=& 1,
y_j^* H_j q_j + _j _j _j+1 &=& _M _j+1. If is on the order of , then the scaling may be absorbed into without alteration of and also without effecting the numerical orthogonality of . When is modified, it turns out that none of the previously computed , , need to be altered. After step , the vector is simply rescaled in subsequent steps, and the formulas defining , are invariant with respect to scaling of . For complete detail, one should consult  [420].

The code shown in Algorithm 7.9 implements this scheme to compute an acceptable .In practice, this transformation may be computed and applied in place to obtain and without storing . However, that implementation is quite subtle and the construction of is obscured by the details required to avoid creating additional storage. This code departs slightly from the above description since the vector is rescaled to have unit norm only after all of the columns to have been determined.

There are several implementation issues.

(3)
The perturbation shown here avoids problems with exact zero initial entries in the eigenvector . In theory, this should not happen when is unreduced but it may happen numerically when a diagonal entry of is small but not zero. There is a cleaner implementation possible that does not modify zero entries. This is the simplest (and crudest) correction.

(10)
As soon as is sufficiently large, there is no need for any further corrections. Deleting this if-clause reverts to the unmodified calculation of shown in Algorithm 7.8.

(16)
This shows one of several possibilities for modifying to achieve the desired goal of numerically tiny elements below the subdiagonal of . More sophisticated strategies would absorb as much of the scaling as possible into the diagonal element . Here, the branch that does scale instead of is designed so that .

Next: Locking and Purging in Up: Orthogonal Deflating Transformation Previous: Purging .   Contents   Index
Susan Blackford 2000-11-20