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##

Sample Problems and Their Differentials

This section serves as a sample guide on the manipulation
of objective functions of matrices with orthonormal columns.
We have found a few common tricks worth emphasizing.

Once one has a formula for the objective function , we define
the formula for implicitly by
,
where (or any curve for which ).
The reader may recall that
, so it functions just like the real
inner product for vectors,^{} and the implicit definition of is
actually the directional derivative interpretation of the
gradient of as an unconstrained function in a Euclidean space.

For most of the functions we have used in our examples, the easiest
way to obtain the formula for is to actually use the implicit
definition.

For example, if
one then has

Since the value of the trace is invariant under cyclic permutations
of products and transposes, we may rewrite this equation as

and, since is unrestricted, this implies that
.
The process we recommend is:

- try to write as a trace;
- compute
, where we let
;
- use trace identities to rewrite every term to have a in the front;
- strip off the leaving the .

As a check, we recommend using the finite difference `dF.m`

code
supplied in the subdirectory `finitediff`

to check derivations
before proceeding.

The software needs a function called `ddF.m`

, which returns
for
. The sort of second
derivative information required by the software is easier to derive
than the first. If one has an analytic expression for , then
one need only differentiate.

If, for some reason, the computation for `ddF.m`

costs much more than two evaluations of with `dF.m`

, the
reader may just consider employing the finite difference function for
`ddF.m`

found in `finitediff`

(or simply use `ddF.m`

as a check).

It is, however, strongly suggested that one use an analytic expression
for computing , as the finite difference code for it requires a
large number of function evaluations ().

**Subsections**

** Next:** The Procrustes Problem
** Up:** Nonlinear Eigenvalue Problems with
** Previous:** MATLAB Templates
** Contents**
** Index**
Susan Blackford
2000-11-20