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Abstract: As a time discretization scheme for an ordinary differential equation with a stiff linear term, there is a class of methods that utilize the exponential or related functions of the coefficient matrix of the linear term. To implement these methods, we must compute a set of matrix functions called “-function”, that includes the exponential itself, and it is important to compute these functions efficiently and accurately. In this paper, we consider the modified scaling and squaring method for the computation of -function. An algorithm based on Higham’s method is defined, and the bounding parameter appropriate for -function is determined from an analysis of the truncation error under the assumption of the exact arithmetic. We also consider the propagation of the rounding error in the squaring process, and show that the error of -function is expected to be less than or roughly equal to that of the matrix exponential. Several evaluations are performed for famous test matrices, and the result shows that when the matrix exponential is computed accurately, the other -functions can also be obtained with the same level of accuracy. [Copyright &y& Elsevier]

In this paper, we introduce a new algorithm for sign detection in a residue number system, which can be applied to any number of moduli set. This technique is faster than most competitive published works, based on MRC or CRT. The CRT requires large modulo M and is therefore costly to implement. The mix radix conversion (MRC) is a sequential process which causes long delay. The proposed algorithm is based on CRT 11. No big modulo M = Πi=1pin pi operations are needed for the CRT II. The modulo multipliers in the CRT II are bounded by size √M. The new sign detector uses less hardware than previous techniques. Using the new moduli set (2, 2n − 1, 2n + 2n−1 − 1, 2n+1 + 2n − 1), the implementation of a sign detector is further simplified. [Copyright &y& Elsevier]

A multiplier-free residue to binary converter architecture based on the Chinese remainder theorem II (CRT II) [1] is presented. The paper also includes a binary to residue converter. This is achieved by introducing a new moduli set (2, 2n − 1, 2n + 2n−1 − 1, 2n+1 + 2n − 1) for RNS application. The complexity of conversion has been greatly reduced using CRT II with the new moduli set. The proposed hardware architecture replaces the necessary multiplication by shift-left operations. A similar hardware architecture is presented for the binary to residue conversion. [Copyright &y& Elsevier]