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gggggAssume that an n × n dot matrix is stored in consecutive n words of a computer with one word being n bit. This paper discusses eight operations to rotate or transpose the dot matrix in horizontal/vertical! slant direction. As is well known, such operations form a group. This paper presents a unified description of the algorithms, and it is shown that all eight algorithms can be derived by suit- ably setting the constants in the unified algorithm. Then it is shown that the algorithms can be transformed to each other by simple transformation rules for the constants, and those transformation rules are shown to form a group. The foregoing algorithms for the dot matrix are not unique, and there can be different descriptions for the algorithm producing the same result. Consequently, there can be various versions of the eight algorithms as well as versions of transformation rules. This paper shows that one of the two groups formed by the transformation rules is isomorphic to the group of algorithms, but the other is not isomorphic. [ABSTRACT FROM AUTHOR]

The real-time processing of the recursive least-square algorithm is difficult to execute by the conventional Neumann-type processor because of its large computational complexity. On the other hand, it is known that the recursive least-square algorithm based on the UD decomposition, which is equivalent to the recursive least-square algorithm, can be realized by the systolic array proposed by Rung. The systolic array to execute this algorithm is difficult to realize since it requires a tremendous number of elements and corrections. In the construction of the dedicated hardware, in general, the word length of the processor affects the processing speed and the hardware area. From such a viewpoint, it is important in the construction of the systolic array to execute the recursive least-square algorithm based on the UD decomposition that the word length of the processor should be minimized. To evaluate the word length, an error analysis is required for the finite word length operation of the recursive least-square algorithm based on the UD decomposition. This paper presents the finite word length floating-point error analysis for the recursive least-square algorithm, based on the UD decomposition, and evaluates the operation error. Then the convergence of the algorithm and the number of updates of the algorithm are evaluated analytically. Finally, by a computer simulation, the validity of the theoretical analysis for the convergence and the number of updates is verified. [ABSTRACT FROM AUTHOR]