Parallel Subtraction Research Articles

On the parallel addition and subtraction of operators on a Hilbert space

In this paper, we extend the operations of parallel addition A:B and parallel subtraction from the cone of bounded nonnegative self-adjoint operators to the linear bounded operators on a Hilbert space. Some conditions for the relations to be true are studied and the solution of the equation A:X = B is investigated. Moreover, some relationships between the parallel addition and subtraction of projections are obtained.

Geometric approach to the parallel sum of vectors and application to the vector $\varepsilon $ -algorithm

The paper aims at answering the following question, in the scalar as well in the vector case: What do the famous Aitken’s \(\Delta ^{2}\) and Wynn’s \(\varepsilon \)-algorithm exactly do with the terms of the input sequence? Inspecting the rules of these algorithms from a geometric point of view leads to change the question into another one: By what kind of geometric object can the parallel (or harmonic) sum be represented? Thus, the paper begins with geometric considerations on the parallel addition and the parallel subtraction of vectors, including equivalent definition, and properties derived from the new point of view. It is shown how the parallel sum of vectors is related to the Bezier parabola controlled by these vectors and to their interpolating equiangular spiral. In the second part, observing that Aitken’s \(\Delta ^{2}\) and Wynn’s \(\varepsilon \)-algorithm may be defined through hybrid sums, mixing standard and parallel sums, the consequences are drawn regarding the way one can define and analyze these algorithms. New explanatory rules are derived for the \(\varepsilon \)-algorithm, providing a better understanding of its basic step and of its cross-rule.

Nonlinear Material Based All-Optical Parallel Subtraction Scheme: an Implementation

Non-linear material based all-optical switching mechanism is utilized here to implement the all-optical paral- lel subtraction scheme. Optical tree architectures here convert analog optical signal to the corresponding digital one. A two bit all-optical binary parallel subtractor has been proposed and which may be elevated to a higher bit parallel subtractor in course using all-optical half-subtractor and a full-subtractor. These are constructed by a composite slab of linear medium (LM) and non-linear medium (NLM) and it is the building block of our proposed subtructor circuit. These circuits are all-optical and fully parallel in nature. It can also gear up to the highest ability of optical performance in high-speed all-optical computing system.

Parallel subtraction of nonnegative forms

The notions of parallel sum and parallel difference of two nonnegative forms were introduced and studied by Hassi, Sebestyen, and de Snoo in [13] and [14]. In this paper we consider the parallel subtraction with much circumstances. Criteria are established for the solvability of the equation \({\mathfrak {t}}:{\mathfrak {z}}={\mathfrak {s}}\) with an unknown \({\mathfrak {z}}\) when \({\mathfrak {t}}\) and \({\mathfrak {s}}\) are given. We identify \({{\mathfrak {s}}\div {\mathfrak {t}}}\) as the minimal solution, and characterize all the solutions under the assumption \({\mathfrak {t}}\geqq\lambda {\mathfrak {s}}\) where λ>1. The Galois correspondence induced by the map \({\mathfrak {t}}\mapsto {\mathfrak {s}}\div {\mathfrak {t}}\) is also studied. We show that if the equation is solvable, then there is a unique \(\varrho_{{\mathfrak {s}}}\)-closed solution, namely \({\mathfrak {s}}\div {\mathfrak {t}}\). Finally, we consider some extremal problems such as the extreme points of the interval \([{\mathfrak {t}},{\mathfrak {w}}] = \{{\mathfrak {z}}: {\mathfrak {t}}\leqq {\mathfrak {z}}\leqq {\mathfrak {w}}\}\), and the characterization of the minimal forms in terms of the parallel sum.

Demonstration of an optoelectronic interconnect architecture for a parallel modified signed‐digit adder and subtracter

A space-position-logic-encoding scheme is proposed and demonstrated, This encoding scheme not only makes the best use of the convenience of binary logic operation, but is also suitable for the trinary property of modified signed-digit (MSD) numbers. Based on the space-position-logic-encoding scheme, a fully parallel modified signed-digit adder and subtracter is built using optoelectronic switch technologies in conjunction with fiber-multistage 3-D optoelectronic interconnects, Thus an effective combination of a parallel algorithm and a parallel architecture is implemented. In addition, the performance of the optoelectronic switches used in this system is experimentally studied and verified. Both the 3-bit experimental model and the experimental results of a parallel addition and a parallel subtraction are provided and discussed. Finally, the speed ratio between the MSD adder and binary adders is discussed and the advantage of the MSD in operating speed is demonstrated. (C) 1996 Society of Photo-Optical instrumentation Engineers.

Parallel subtraction of Fourier power spectrum using holographic interferometry

We propose and demonstrate parallel subtraction of the Fourier power spectrum of two images with the use of real-time holographic interferometry. The subtraction, which exhibits shift invariance, is achieved by employing a Mach-Zehnder interferometer and a photorefractive crystal. The results are presented and discussed.

Characterization of parallel subtraction

Parallel subtraction is an operation defined on pairs of positive operators. In terms of electrical networks, one may pose the following problem: Given an electrical network, represented by a specified positive operator, determine the set of positive operators which when connected in parallel with the specified operator yield another prescribed operator. The set of solutions of this electrical network problem is shown to have a minimum. The minimum is termed "the parallel difference of the fixed operators," and the operation is termed "parallel subtraction." The parallel difference is used to obtain explicit error estimates for an iteration procedure which approximates the geometric mean of positive operators. This concept of the geometric mean reduces to the square root of the product of the operators if the operators commute. Finally, by using the geometric mean, an operator version of the Gaussian mean is presented.

Parallel Subtraction of Matrices

A new Hermitian semidefinite matrix operation is studied. This operation-called parallel subtraction-is developed from the theory of parallel addition. Since the theory of parallel addition is motivated by the analysis of interconnected electrical networks, parallel subraction may be interpreted in terms of the synthesis of electrical networks. The idea of subtraction is also extended to hybrid addition.

Fast economical binary divider

By eliminating the shift operation of a shift-and-subtract divider, array dividers speed up division. Since n+1 subtractors are required, there is a heavy equipment cost. However, since each parallel subtraction is data dependent, the subtractions are sequential and only one subtractor is operating at any one time. Here a divider is developed which speeds division by eliminating the shift cycle, but which only requires two parallel subtractors working sequentially.