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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