The usual formulation of efficient division uses Newton iteration to compute an inverse in a related domain where multiplicative inverses exist. On one hand, Newton iteration allows quotients to be calculated using an efficient multiplication method. On the other hand, working in another domain is not always desirable and can lead to a library structure where arithmetic domains are interdependent. This paper uses the concept of a whole shifted inverse and modified Newton iteration to compute quotients efficiently without leaving the original domain. The iteration is generic to domains having a suitable shift operation, such as integers or polynomials with coefficients that do not necessarily commute.

翻译：通常的高效除法公式采用牛顿迭代法在相关域中计算逆元，该域中存在乘法逆元。一方面，牛顿迭代法允许使用高效乘法方法计算商数；另一方面，在其他域中运算并非总是理想的选择，这可能导致算术域相互依赖的库结构。本文利用完整移位逆元的概念以及改进的牛顿迭代法，在不脱离原始域的情况下高效计算商数。该迭代方法适用于具有适当移位运算的域，例如整数或系数不一定可交换的多项式。