This paper considers learning a dynamical system using a recurrent neural network (RNN) with hidden units. Such an RNN does not produce a dynamical system on the visible state space unless a mapping from the visible state space to the hidden state space is successfully specified. We propose an affine neural dynamical system (A-NDS) as a dynamical system that an RNN can actually produce on the visible state space to approximate a target dynamical system. An n-dimensional A-NDS is parametrically represented by a suitable pair of an RNN with n visible units and r hidden units, and an affine mapping from the n-dimensional space to the r-dimensional space. However, this parametric representation has redundancy. We construct a unique parametric representation of an A-NDS with the aim of building efficient learning algorithms of a dynamical system using an RNN.

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THis report compares the behaviors of three multiplicative algorithms for floating-point divide operation. They are the Newton-Raphson method, Goldschmidt's algorithm, and another method that simply evaluates the Taylor series expansion of a reciprocal. Goldschmidt's algorithm is based on the same series but differs from the alternative method in a manner of evaluating the series. The behaviors of the three methods are compared using two kinds of models for each method : a performance model, which describes latency, and an accuracy model, which describes the upper bound for the error of the quotient. Particular emphasis is placed on development of the accuracy models. Validity of the accuracy models were empirically verified with numerical tests. It is shown that, with a practical choice of the number of iterations, k, the magnitude of the relative error is bounded by 3×2^<-p> (2k+1)×2^<-p> and (k+1)×2^<-p> where p is the size of the mantissa represented in bits, for the Newton-Raphson method, Goldschmidt's algorithm, and the alternative method, respectively (results on a floating-point unit with a multiplyadd-fused configuration). For the Newton-Raphson method, any number of additional iterations further reduce the bound to (8/3)×2^<-p>. The performance models indicate that, on a pipelined floating-point unit, Goldschmidt's algorithm and the alternative method are equally faster than the Newton-Raphson method. As a result, in general, the alternative method is promising in the case of k≤3, a very realistic value.

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The purpose of this paper is to introduce a new method to compute the zeros of polynomials. A lot of methods are available for the determination of the zeros of polynomials^[6]. The feature of our method is in using the errors of numerical integrals of the logarithmic derivative of the polynomial. Our method is explained in two algorithms. The fundamental algorithm finds the zero which is the nearest to the initial value as our theory indicates. We propose the applied algorithm as a practical one, which converges more quickly than dose the fundamental algorithm.

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