HP MLIB User's Guide Vol. 2 7th Ed.

1008 HP MLIB User’s Guide
What you need to know to use these subprograms
For notational purposes, assume that the eigenvalues λ
i
are ordered
λ
1
≤λ
2
≤ ... ≤λ
n
. The results from subroutines in this section normally form a
contiguous set of eigenvalues, say, λ
i
, λ
i+1
,..., λ
j
. You must designate which
subset you want through the following kinds of constraints, which usually
relate to the application-specific meaning of the eigenvalues:
1. Eigenvalues can be restricted to lie in a finite or semi-infinite interval (find
only eigenvalues in [a,b]).
2. Eigenvalues can be described by count (find only P eigenvalues) together
with a location descriptor based either on the magnitude (either the
smallest/lowest or largest/highest) or on the proximity to a fixed value
(closest to c).
These two characterizations can be used together or separately. Examples of
standard cases include:
• Find the lowest 10 eigenvalues
• Find all eigenvalues in [0.0, 100.0]
• Find the 3 eigenvalues closest to 53.1
This structure also permits combinations such as:
• Find the lowest 15 eigenvalues in [20.1, 75.3]
• Find the 10 eigenvalues closest to 15.0 in [10.0, 20.0]
The terms lowest and highest have two standard and different meanings in
algebra:
• Size—for example, least magnitude
• Ordinal position—algebraically least
These subprograms follow a common engineering usage, where the
interpretation is magnitude, not order. Thus, lowest means closest to zero and
highest farthest from zero. The two interpretations are the same when all
eigenvalues are positive, but the reverse when all eigenvalues are negative.
The greatest confusion arises when the eigenvalues of interest may include
both positive and negative numbers, that is, when the interval specifications
include zero in the interior of the interval. To help clarify the interpretation of
lowest in an interval containing zero in the interior, printed output from the
package will describe this as nearest to zero.
When zero lies in the interior of the interval, the eigenvalues highest in
magnitude may be drawn from both the algebraically least and the
algebraically greatest. These eigenvalues could form two disjoint sets of
contiguous eigenvalues rather than one. To avoid problems associated with
this, the subprograms currently allow finding the highest eigenvalues only with