This thesis describes a flexible framework for abstract numerical algorithms which treats algorithms as objects and makes them reusable, composable, and modifiable. These algorithm objects are implemented using the Rice Vector Library (RVL) interface, decoupling the algorithmic code from the details of linear algebra and calculus in Hilbert Space. I made many improvements to the RVL design, including abstract return types for reductions. These improvements allowed me to demonstrate the breadth of this design by incorporating semantically similar objects from other packages which had significant syntatic differences to the RVL objects. By adapting other libraries, I gain access to a variety of tools, including parallel linear algebra implementations. The benefits of the algorithm framework can be seen when abstract numerical algorithms are coupled with parallel simulators without needing to modify either the algorithm or the simulator.