Radix Representation Research Articles

Polygonal radix representations of complex numbers

Complex numbers can be represented in positional notation using certain digit sets. In this paper, we present the polygonal representation which uses zero and the n-roots of unity as digits. We give conditions on the base in order that every complex number be representable in such a system. We totally characterize complete polygonal numeration systems in imaginary quadratic fields.

Multiplicative circulant networks topological properties and communication algorithms

In this paper we study an interconnection network topology based on the radix representation of integers and modulo arithmetic. The multiplicative circulant graph MC( r, k) consists of n = r k vertices. Two vertices x and y are connected by an edge if and only if ¦x − y¦= r i mod n for some integer i, 0 ⩽ i ⩽ k − 1. Multiplicative circulant graphs can be considered as a kind of “twisted” torus graphs. We prove that for even r > 2 the diameter of the MC( r, k) network is kr 2 − ⌊ K 2 ⌋ which is smaller than the diameter of the corresponding torus. We also determine the average distance in the MC( r, k) graphs. The multiplicative circulant graphs are vertex symmetric and have Hamiltonian cycles, one of them being 0, 1, …, n − 1. The MC( r, k) graph can be recursively decomposed into r MC( r, k − 1) graphs and the later Hamiltonian cycle. Hypercubes and complete binary trees are embedded in MC(2, k) graphs with dilation and expansion 1. A ( pk)-dimensional hypercube can be embedded into MC(2 p , k) graph with expansion 1 and dilation 2 p-1 . Multiplicative circulant graphs have simple optimal routing and broadcasting algorithms. These algorithms are deadlock free for r = 2 and r = 3. A simple parallel semigroup and prefix computation algorithm for the multiplicative circulant networks are described.

Nonnegative Radix Representations for the Orthant 𝑅ⁿ₊

Let A A be a nonnegative real matrix which is expanding, i.e. with all eigenvalues | λ | > 1 |\lambda | > 1 , and suppose that | det ( A ) | |\det (A)| is an integer. Let D {\mathcal D} consist of exactly | det ( A ) | |\det (A)| nonnegative vectors in R n \mathbb {R}^n . We classify all pairs ( A , D ) (A, {\mathcal D}) such that every x x in the orthant R + n \mathbb {R}^n_+ has at least one radix expansion in base A A using digits in D {\mathcal D} . The matrix A A must be a diagonal matrix times a permutation matrix. In addition A A must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set D \mathcal D can be diagonally scaled to lie in Z n \mathbb {Z}^n . The proofs generalize a method of Odlyzko, previously used to classify the one–dimensional case.

AUTOMATIC MAPS ON A SEMIRING WITH DIGITS

A general framework for substitutions and automaticity of maps from a ring (or even from a semiring) with radix representation to a finite set is given. Our definition contains the classical notion of automaticity as well as the case of multidimensional automatic sequences and the case of sequences indexed by rational integers, by Gaussian integers or by quadratic irrationals.

Integer division in residue number systems

This contribution to the ongoing discussion of division algorithm for residue number systems (RNS) is based on Newton iteration for computing the reciprocal. An extended RNS with twice the number of moduli provides the range required for multiplication and scaling. Separation of the algorithm description from its RNS implementation achieves a high level of modularity, and makes the complexity analysis more transparent. The number of iterations needed is logarithmic in the size of the quotient for a fixed start value. With preconditioning it becomes the logarithm of the input bit size. An implementation of the conversion to mixed radix representation is outlined in the appendix.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Replicating Tessellations

A theory of replicating tessellation of $\mathbb{R}^n $ is developed that simultaneously generalizes radix representation of integers and hexagonal addressing in computer science. The tiling aggregates tesselate Euclidean space so that the $( m + 1 )$st aggregate is, in turn, tiled by translates of the mth aggregate, for each m in exactly the same way. This induces a discrete hierarchical addressing systsem on $\mathbb{R}^n $. Necessary and sufficient conditions for the existence of replicating tessellations are given, and an efficient algorithm is provided to determine whether or not a replicating tessellation is induced. It is shown that the generalized balanced ternary is replicating in all dimensions. Each replicating tessellation yields an associated self-replicating tiling with the following properties: (1) a single tile T tesselates $\mathbb{R}^n $ periodically and (2) there is a linear map A, such that $A( T )$ is tiled by translates of T. The boundary of T is often a fractal curve.

On symmetric radix representation of Gaussian integers

Symmetric radix representation and symmetric mixed-radix representation of Gaussian integers play a significant role in the residue arithmetic ofZ[i]. In the following, known results concerning corresponding representations of integers are generalized. It is shown that for any modulusmeZ[i] with\(m\bar m > 1\), except form=1±i, 2, there exists a unique symmetricm-radix representation of Gaussian integers.

The Fractal Dimension of Sets Derived from Complex Bases

AbstractFor each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λn/log(n2 + 1), where λn is the positive root of λ3 - (2n - 1) λ2 - (n - 1) 2λ - (n2 + 1).

Basic digit sets for radix representation

Abstract : The use of a negative base did not appear until the 1950s when several authors independently introduced the concept. Complement representation also became much discussed in this period as an alternative to sign magnitude for designing the arithmetic unit of a computer. The arithmetic of numbers represented in positional notation has a firm foundation derived from the theory of polynomial arithmetic that readily allows these extensions to negative bases and/or negative digit values, complement representation, and digit values in excess of the base. Our primary concern in this paper is the characterization and computation of those integral valued base and digit set pairs that provide complete and unique finite radix representation of the integers.

Conversion schemes in residue code

In this paper, an implementation scheme involving decoders and residue adders has been suggested to convert input data in fixed radix representation to residue representation. A method dealing with the reverse process is also demonstrated. A comparison has been made with the methods known hitherto.

Radix representations of quadratic fields

Let ϱ be an algebraic integer in a quadratic number field whose minimum polynomial is x 2 + p 1 + p 0. Then all the elements of the ring |Z [ϱ] can be written uniquely in the base ϱ as Σ k m =0 a kπ k , where 0 ⩽ a k < | p 0|, if and only if p 0 ⩾ 2 and −1 ⩽ p 1 ⩽ p 0.

On Minimum Weight Binary Representation of Integers and Continued Fractions with Application to Computer Arithmetic

We develop the concept of minimum weight binary continued fraction representation of a rational number as an extension of minimum weight binary radix representation of an integer. The relation of these representations to the attainment of optimum efficiency in the shift and add or subtract model of binary computer arithmetic is discussed.

Weak radix representation and cyclic codes over euclidean domains

Here we show that to some extent the two theories cyclic arithmetic coding and cyclic polynomial coding can be subsumed under a “unified” theory based on a more or less arbitrary Euclidean domain. In particular, we show that Hamming weight (for cyclic polynomial codes) and arithmetic weight (for cyclic arithmetic codes) are special cases of a more general notion of weight defined for Euclidean rings. of fundamental importance in this development is a general-ization of the notion of radix representation for integers which we call a weak radix representation. For several specific Euclidean rings we prove the existence of certain unique canonical weak radix representations for all ring elements. We show, for example, that every Gaussian integer has a unique representation of the form (1) each aj is zero or a unit, and whenever aj≠0 , then aj+1 = aj+2 = 0 This is analogous to a known result of Reitwiesner that each rational integer has a unique representation of the form (1) where r - 2, each aj. is 0 or ± 1...

A recursive radix conversion formula and its application to multiplication and division

A recursive formula for number conversion from one radix representation to another radix representation is presented. This formula differs from the existing ones in two major aspects. First, it utilizes a digit shift technique which provides faster accumulation of higher significant digits in the final result. Second, it is suitable for parallel computation, so that the length of time necessary for number conversion can be shortened. Thus the longer the digit number, the more appreciation in conversion time-saving will result. Applications of the recursive formula are studied in multiplication and division for negative radix numbers as well as for positive radix numbers. The multiplication and division presented here are especially useful for computations of η word precision, since successive bit-carrying propagation to the most significant digit hardly ever occurs.

Fourier Transform Computers Using CORDIC Iterations

The CORDIC iteration is applied to several Fourier transform algorithms. The number of operations is found as a function of transform method and radix representation. Using these representations, several hardware configurations are examined for cost, speed, and complexity tradeoffs. A new, especially attractive FFT computer architecture is presented as an example of the utility of this technique. Compensated and modified CORDIC algorithms are also developed.

On arithmetic weight for a general radix representation of integers (Corresp.)

In this correspondence we define a "nonadjacent form" for integers in an arbitrary radix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r &gt; 1</tex> . This form is proved to be unique, and the arithmetic weight of an integer is shown to be equal to the number of nonzero terms in the form. Two algorithms are presented for the computation of this form. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r = 2</tex> , our form coincides with the well-known modified binary nonadjacent form.

Complementary Two-Way Algorithms for Negative Radix Conversions

This paper describes two sets of algorithms in positive radix arithmetic for conversions between positive and negative integral radix representation of numbers. Each set consists of algorithms for conversions in either direction; these algorithms are mutually complementary in the sense they involve inverse operations depending upon the direction of conversion. The first set of algorithms for conversion of numbers from positive to negative radix (negative to positive radix) proceeds serially from the least significant end of the number and involves complementation and addition (subtraction) of unity on single-digit numbers. The second set of algorithms for conversion of numbers from positive to negative radix (negative to positive radix) proceeds in parallel starting from the full number (the most significant end of the number) and involves complementation and right (left) shift operations. The applications of these algorithms to integers, mixed integer-fractions, floating-point numbers, and for real-time conversions are given.

On Radix Representation and the Euclidean Algorithm

On Radix Representation and the Euclidean Algorithm

On Radix Representation and the Euclidean Algorithm

On Radix Representation and the Euclidean Algorithm

Conversion of Modular Numbers to their Mixed Radix Representation by Matrix Formula

Conversion of Modular Numbers to their Mixed Radix Representation by Matrix Formula