Methods and Applications of Error-Free Computation

Gregory, Robert T.; Krishnamurthy, E. V.

doi:10.1007/978-1-4612-5242-9

Cited by 111 publications

(47 citation statements)

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“…by equation (14). If q 1 ≥ 4 and q 2 ≥ 4 then, as in the remark following Proposition 9.3, we have log 2 (φ 2 (a)) ≤ 1 + log 2 (λ 2 (a) − 2) and the result follows.…”

mentioning

confidence: 61%

“…We have relied heavily on the p-adic approximation theory of [47,32]. Related results appear in [14,28,34].…”

Section: Related Literaturementioning

confidence: 99%

“…[47]). There are a number of decoding algorithms available in the context of Hensel and arithmetic codes [14,28,34]. However, there seem to be no published proofs that any of these algorithms satisfy both of the optimality properties mentioned above.…”

Section: Rational Approximation Algorithmmentioning

confidence: 99%

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Feedback shift registers, 2-adic span, and combiners with memory

1997

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Feedback shift registers with carry operation (FCSR's) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR's) are presented, including a synthesis algorithm (analogous to the Berlekamp-Massey algorithm for LFSR's) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the Marsaglia-Zaman random number generator. Possible variations on the FCSR architecture are indicated at the end.

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“…by equation (14). If q 1 ≥ 4 and q 2 ≥ 4 then, as in the remark following Proposition 9.3, we have log 2 (φ 2 (a)) ≤ 1 + log 2 (λ 2 (a) − 2) and the result follows.…”

mentioning

confidence: 61%

“…We have relied heavily on the p-adic approximation theory of [47,32]. Related results appear in [14,28,34].…”

Section: Related Literaturementioning

confidence: 99%

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Feedback shift registers, 2-adic span, and combiners with memory

1997

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“…As already mentioned RNS appears to be a suitable number system for implementing certain numerical methods for error-free solving of a system of linear equations [5,6,17,18]. In this section, we describe the basic mathematical principles of method solving SLE in the RNS.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Design of a Residue Number System Based Linear System Solver in Hardware

BuăźEk

Kubalik

Lórencz

et al. 2016

J Sign Process Syst

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This paper is focused on error-free solution of dense linear systems using residual arithmetic in hardware. The designed Modular System uses hardware identical Residual Processors (RP)s for solving independent systems of linear congruences and combines their solutions into the solution of the given linear system. This approach uses the residue number system which is based on the Chinese remainder theorem. In order to efficiently exploit parallel processing and cooperation of the individual components, a hardware architecture of the Modular System with several RPs is designed. In order to verify the proposed architecture, a Xilinx FPGA with a MicroBlaze processor was used. Experimental results are obtained for an evaluation FPGA board with Virtex 6. Results from implementation serve for subsequent theoretical analysis of the system performance for various linear system sizes and further improvement of the system. The proposed system can be useful as a special hardware peripheral or a part of an embedded system for solving large nonsingular systems of linear equations with integer, rational or floating-point coefficients with arbitrary precision.

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“…The FCSR synthesis problem was essentially studied in the context of Hensel and arithmetic codes [71,113,128], although no analysis of the aforementioned optimality properties was done in this early work.…”

Section: Rational Approximationmentioning

confidence: 99%

Algebraic Shift Register Sequences

Goresky

Klapper

2012

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Pseudo-random sequences are essential ingredients of every modern digital communication system including cellular telephones, GPS, secure internet transactions and satellite imagery. Each application requires pseudo-random sequences with specific statistical properties. This book describes the design, mathematical analysis and implementation of pseudo-random sequences, particularly those generated by shift registers and related architectures such as feedback-with-carry shift registers. The earlier chapters may be used as a textbook in an advanced undergraduate mathematics course or a graduate electrical engineering course; the more advanced chapters provide a reference work for researchers in the field. Background material from algebra, beginning with elementary group theory, is provided in an appendix.

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Methods and Applications of Error-Free Computation

Cited by 111 publications

References 0 publications

Feedback shift registers, 2-adic span, and combiners with memory

Feedback shift registers, 2-adic span, and combiners with memory

Design of a Residue Number System Based Linear System Solver in Hardware

Algebraic Shift Register Sequences

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