Solvers as operators

This proposal details an interface hierarchy for objects whose major function is the mapping of a finite dimensional vector space of dimension m to another vector space of dimension n. This includes many important objects in a solver library, including matrices, their transposes and inverses, solvers, preconditioners, iterative methods, and nonlinear maps. A unifying framework for finite dimensional operators and solvers is proposed which utilizes the composition operation from the operator algebra to achieve great functionality while reducing the size of the interface and complexity of the class structure. A second composition operation is introduced to handle the composition of approximate solution techniques, and related to several common preconditioning techniques