The question "where do black-box operators come from?" is really two different questions:
We answer the first question with a quotation from [20]:
Large matrices ... usually arise indirectly in the discretization of differential or integral equations. One might say, that if m is large, it is probably an approximation to ∞. It follows that most large matrices of computational interest are simpler than their vast number of individual entries might suggest.
The operators used in BBTools generally have the following sources:
A Matlab matrix can be converted to a black-box operator using either bbmatrix or blackbox. Operators, which are shipped with BBTools, can be found in the function reference.
The real power of BBTools comes from its ability to combine the other operators efficiently.
[20] Lloyd Nicholas Trefethen and David Bau III. Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997. ISBN: 0-89871-487-7. (Book)