It is generally a difficult task to construct efficient implementations of numerical mathematical algorithms for execution on high-performance computer systems. The difficulty arises from the need to express an implementation in a form that reflects the nature of the computer system, rather than a form that reflects the computations performed by the algorithm.This thesis develops the method of program transformation to derive automatically efficient implementations of algorithms from high-level, machine-independent specifications.The primary system considered is the AMT DAP array processor, but sequential and vector systems are also considered.The transformational method is further extended to automatically tailor implementations to use sparse programming techniques.