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This paper describes a unifying framework for nonsingular data transformations. It shows that a wide class of existing transformations may be expressed in this framework, allowing compound transformations to be performed in one step. Validity conditions for such transformations are developed as is the form of the transformed program and data. Constructive algorithms to generate data transformations for different applications are described and applied to example programs. It is shown that they can have a significant impact on program performance and may be used in situations where traditional loop transformations are inappropriate.