A Modification of the Fast Inverse Square Root Algorithm

Walczyk, Cezary J.; Moroz, Leonid; Cieśliński, Jan L.

doi:10.20944/preprints201908.0045.v1

Cited by 5 publications

(2 citation statements)

References 28 publications

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“…After sample digestion and spike-sample equilibration, the Os fractions are typically extracted with an organic solvent and purified by microdistillation Waters 1996, Birck et al 1997), with the Re fractions being purified by anion exchange chromatography (Morgan et al 1991, Selby andCreaser 2001). The Re and Os isotope compositions of the purified Re and Os fractions from samples are determined by negative-thermal ionisation mass spectrometry (N-TIMS) and/or (multi-collector) inductively coupled plasma-mass spectrometry (MC-ICP-MS/ICP-MS) (Creaser et al 1991, V€ olkening et al 1991, Reisberg and Meisel 2002, Meisel et al 2003, Walczyk 2004. The MC-ICP-MS/ICP-MS permits faster isotope ratio determination and simpler elemental purification chemistry.…”

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A Matrix‐Matched Reference Material for Validating Petroleum Re‐Os Measurements

Liu

Selby

2017

Geostandard Geoanalytic Res

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This study presents two matrix‐matched reference materials developed for petroleum Re‐Os measurements. We present the Re and Os mass fractions and 187Re/188Os and 187Os/188Os values (ratio of the number of atoms of the isotopes) for repeatedly measured aliquots (ca. 120–150 mg test portions) of the NIST Research Material 8505 (RM 8505) crude oil, and its asphaltene and maltene fractions, and ~ 90 g of homogeneous asphaltene powder isolated from this oil. Measurements were performed using the Carius tube‐isotope dilution negative‐thermal ionisation mass spectrometry methodology. The RM 8505 crude oil contains 1.98 ± 0.07 ng g−1 Re and 25.0 ± 1.1 pg g−1 Os, with Re‐Os isotope amount ratios of 452 ± 6 for 187Re/188Os and 1.51 ± 0.01 for 187Os/188Os (n = 20, 95% conf.). The homogeneous asphaltene sample contains 16.52 ± 0.10 ng g−1 Re and 166.0 ± 0.9 pg g−1 total Os, and possesses isotope amount ratios of 574 ± 3 for 187Re/188Os and 1.64 ± 0.01 for 187Os/188Os (n = 24, 95% conf.). The intermediate precision of these data makes the RM 8505 whole oil and the (~ 90 g) homogenised asphaltene appropriate petroleum matrix‐matched reference materials for Re‐Os measurements. The asphaltene fraction of the oil is the main carrier of Re and Os of the RM 8505 whole oil, and caution is suggested in using asphaltene and maltene fractions of a single oil for Re‐Os geochronology.

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confidence: 99%

A Matrix‐Matched Reference Material for Validating Petroleum Re‐Os Measurements

Liu

Selby

2017

Geostandard Geoanalytic Res

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“…The seed value can be obtained via a variety of approaches, including the use of lookup tables 33,34 , polynomial approximation 35 , or a combination of both, called piecewise polynomials [36][37][38] . The seed can also be generated by faster means of bit-manipulation techniques 39 and the selection of a magic constant [40][41][42] . In addition, a nonstandard method by Schwarz and Flynn in the form of Partial Product Array (PPA) is available 43 .…”

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Novel seed generation and quadrature-based square rooting algorithms

Altamimi

Youssef

2022

Sci Rep

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The square root operation is indispensable in a myriad of computational science and engineering applications. Various computational techniques have been devised to approximate its value. In particular, convergence methods employed in this regard are highly affected by the initial approximation of the seed value. Research shows that the provision of an initial approximation with higher accuracy yields fewer additional iterations to calculate the square root. In this article, we propose two novel algorithms. The first one presents a seed generation technique that depends on bit manipulation and whose output is to be used as an initial value in the calculation of square roots. The second one describes a quadrature-based square rooting method that utilizes a rectangle as the plane figure for squaring. We provide error estimation of the former using the vertical parabola equation and employ a suitable lookup table, for the latter, to store needed cosine values. The seed generation approach produces a significant reduction in the number of iterations of up to 84.42% for selected convergence methods. The main advantages of our proposed square rooting algorithm lie in its high accuracy and in its requirement of just a single iteration. Our proposed algorithm also provides for lower computational latency, measured in the number of clock cycles, compared to Newton–Raphson’s and Bakhshali’s square rooting methods.

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Improving the Accuracy of the Fast Inverse Square Root by Modifying Newton–Raphson Corrections

Walczyk

Moroz

Cieśliński

2021

Entropy

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Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton–Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs.

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A Modification of the Fast Inverse Square Root Algorithm

Cited by 5 publications

References 28 publications

A Matrix‐Matched Reference Material for Validating Petroleum Re‐Os Measurements

A Matrix‐Matched Reference Material for Validating Petroleum Re‐Os Measurements

Novel seed generation and quadrature-based square rooting algorithms

Improving the Accuracy of the Fast Inverse Square Root by Modifying Newton–Raphson Corrections

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