4 April 1986 The Matrix Exponential Approach To Elementary Operations
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Abstract
In 1971, J.S. Walther generalized and unified J.E. Volder's coordinate rotation (CORDIC) algorithms. Using Walther's algorithms a few commonly used functions such as divide, multiply-and-accumulate, arctan, plane rotation, arctanh, hyperbolic rotation can be implemented on the same simple hardware (shifters and adders, elementary controller) and computed in approximately the same time. Can other useful functions be computed on the same hardware by further generalizing these algorithms? Our positive answer lies in a deeper understanding of Walther's unification: the key to the CORDIC algorithms is that all of them effect the multiplication of a vector by the exponential of a 2 X 2 matrix. The importance of this observation is readily demonstrated as it easily yields the convergence conditions for the CORDIC algorithms and an efficient way of extending the domain of convergence for the hyperbolic functions. A correspondence may be established between elementary functions such as square-root, √(x2+y) , inverse square-root or cubic root and exponentials of simple matrices. Whenever such a correspondence is found, a CORDIC-like algorithm for computing the function can be synthesized in a very straightforward manner. The algorithms thus derived have a simple structure and exhibit uniform convergence inside an adjustable, precisely defined, domain.
© (1986) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Jean-Marc Delosme, Jean-Marc Delosme, "The Matrix Exponential Approach To Elementary Operations", Proc. SPIE 0696, Advanced Algorithms and Architectures for Signal Processing I, (4 April 1986); doi: 10.1117/12.936892; https://doi.org/10.1117/12.936892
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