Long Integers and Polynomial Evaluation with Estrin's Scheme

Bodrato, Marco; Zanoni, Alberto

doi:10.1109/synasc.2011.17

Cited by 8 publications

(11 citation statements)

References 15 publications

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“…To see this, notice that the output of the evaluation is of bitsize O(d(τ + L)). If we use the divide and conquer approach [10,19], each evaluation costs OB(d(τ + L)) and the overall complexity is OB(d(τ + L) lg 2 (τ + L)) = OB(d(τ + L)). This bound is the same as the one supported by αDES.…”

Section: Remark 5 (Des)mentioning

confidence: 99%

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On the boolean complexity of real root refinement

Pan

Tsigaridas

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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We assume that a real square-free polynomial A has a degree d, a maximum coefficient bitsize τ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the Double Exponential Sieve algorithm (also called the Bisection of the Exponents), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of t = 2 −L . The algorithm has Boolean complexity OB(d 2 τ + dL). Our algorithms support the same complexity bound for the refinement of r roots, for any r ≤ d.

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Section: Remark 5 (Des)mentioning

confidence: 99%

“…Each evaluation costs OB(d(τ − lg w + L)), using the divide and conquer scheme [19,10]. Hence, the overall complexity is OB(d(τ − lg w + L) lg(d)) = OB(d(dτ + L)) = OB(d 2 τ + dL).…”

Section: Remark 8 (Bis)mentioning

confidence: 99%

On the boolean complexity of real root refinement

Pan

Tsigaridas

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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“…Estrin's method has since been adopted in many applications [5,24] for fast evaluation of polynomials. It works by reforming (1) as follows: If k is even: else if k is odd:…”

Section: Parallel Methodsmentioning

confidence: 99%

Square-rich fixed point polynomial evaluation on FPGAs

Xu¹,

Fahmy

McLoughlin

2014

Proceedings of the 2014 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays

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Polynomial evaluation is important across a wide range of application domains, so significant work has been done on accelerating its computation. The conventional algorithm, referred to as Horner's rule, involves the least number of steps but can lead to increased latency due to serial computation. Parallel evaluation algorithms such as Estrin's method have shorter latency than Horner's rule, but achieve this at the expense of large hardware overhead. This paper presents an efficient polynomial evaluation algorithm, which reforms the evaluation process to include an increased number of squaring steps. By using a squarer design that is more efficient than general multiplication, this can result in polynomial evaluation with a 57.9% latency reduction over Horner's rule and 14.6% over Estrin's method, while consuming less area than Horner's rule, when implemented on a Xilinx Virtex 6 FPGA. When applied in fixed point function evaluation, where precision requirements limit the rounding of operands, it still achieves a 52.4% performance gain compared to Horner's rule with only a 4% area overhead in evaluating 5 th degree polynomials.

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“…To see this, notice that the output of the evaluation is of bitsize O((τ + L) d). If we use the divide and conquer approach (Bodrato and Zanoni, 2011;Hart and Novocin, 2011), each evaluation…”

Section: Double Exponential Sievementioning

confidence: 99%

“…, using the divide and conquer scheme (Hart and Novocin, 2011;Bodrato and Zanoni, 2011). Hence, the overall complexity…”

Section: Remark 8 (Bis) a Single Bisection Using Only Exact Arithmetmentioning

confidence: 99%