实数和复数记数系统中乘法和除法的在线算法
On-line algorithms for multiplication and division in real and complex numeration systems
AI总结：
本文推广了Trivedi-Ercegovac的在线乘除法算法至任意实/复数基，定义OL性质并给出充分条件，证明满足条件时算法可行且线性时间，附三个实例。
摘要：
位置记数系统由基和数字集合给出。基是实数或复数 $β$，满足 $|β|>1$，数字集合 $A$ 是包含 0 的有限数字集合。因此，一个数可以被视为有限或无限的数字字符串。在线算法以串行方式逐块处理输入。由 Trivedi 和 Ercegovac 引入的在线算术是一种计算模式，其中操作数和结果以数字串行方式流过算术单元，从最高位数字开始。\n 在本文中，我们首先针对 $β$ 为任意实数或复数、数字为实数或复数的情况，推广了 Trivedi 和 Ercegovac 的乘法和除法在线算法。然后我们定义了所谓的 OL 性质，并证明如果 $(β, A)$ 具有 OL 性质，则通过 Trivedi-Ercegovac 算法可以实现在线乘法和除法。对于实数基 $β$ 和连续整数的数字集合 $A$，当 $\n# A > |β|$ 时，系统 $(β, A)$ 具有 OL 性质。对于复数基 $β$ 和连续整数的对称数字集合 $A$，当 $\n# A > β\overlineβ + |β+ \overlineβ|$ 时，系统 $(β, A)$ 具有 OL 性质。假设在系统 $(β, A)$ 中加法和减法可以并行实现，并且可以对分母进行预处理，则我们的乘法和除法在线算法具有线性时间复杂度。文中详细介绍了三个例子：基 $β=\frac{3+\sqrt{5}}{2}$，数字集合 $A=\{-1,0,1\}$；基 $β=2i$，数字集合 $A = \{-2,-1, 0,1,2\}$；以及基 $β= -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + ω$，其中 $ω= \exp{\frac{2iπ}{3}}$，数字集合 $A = \{0, \pm 1, \pm ω, \pm ω^2 \}$。
英文摘要：
A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $β$ such that $|β|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $β$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(β, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $β$ and a digit set $A$ of contiguous integers, the system $(β, A)$ has the OL Property if $\# A > |β|$. For a complex base $β$ and symmetric digit set $A$ of contiguous integers, the system $(β, A)$ has the OL Property if $\# A > β\overlineβ + |β+ \overlineβ|$. Provided that addition and subtraction are realizable in parallel in the system $(β, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $β=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $β=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $β= -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + ω$, where $ω= \exp{\frac{2iπ}{3}}$, with digits $A = \{0, \pm 1, \pm ω, \pm ω^2 \}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
