Subroutine Dependencies

Algorithm 9.2 shows the dependencies of the various subroutines on each other. We have grouped the routines together based on four loosely defined roles:

• Objective function. These are the routines which implement the objective function and its derivatives. This group is the only group that a user interested only in the basic functionality need ever modify to adapt the template. Functions in this group are used by functions in the geometrized objective function and high-level algorithm groups.

• Geometric implementation. These routines implement the geometric features of the Stiefel manifold. This includes the projection of unconstrained vectors onto the constraint (i.e., tangent) space (tangent), line movement (move), and the connection term used for covariant differentiation (connection). These routines are independent of the objective function group, but are essential to the geometrized objective function group.

• Geometrized objective function. This group uses the routines of the geometric implementation to project the differential and second differential of $F(Y)$ onto the constraint (i.e., tangent) surface producing the geometrically covariant gradient (grad) and Hessian (dgrad). Additionally, two routines for inverting the covariant Hessian (invdgrad_CG and invdgrad_MINRES) are located here. This group provides the raw tools out of which one builds implementations in the high-level algorithm group. Lastly, functions which detect (partition) and remove (nosym) block-diagonal orthogonal symmetries are found in the group.

• High-level algorithm. This group implements the constrained versions of various optimization algorithms. It is here that search directions are selected, line minimizations are performed (using Fline, dFline, and gradline), and convergence criteria are defined.

Lastly, every function reads from a global SGParameters structure whose fields contain information about the manifold and the computation. In our examples we have used a separate global FParameters to store information related to the computation of $F(Y)$. The fields of SGParameters are set in sg_min.