• Journal of the Korea Institute of Information and Communication Engineering
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• v.11 no.1
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• pp.46-55
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• 2007

The Newton-Raphson's algorithm for finding a floating point reciprocal and inverse square root calculates the result by performing a fixed number of multiplications. In this paper, an improved Newton-Raphson's algorithm is proposed, that performs multiplications a variable number. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation is derived from many reciprocal and inverse square tables with varying sizes. The superiority of this algorithm is proved by comparing this average number with the fixed number of multiplications of the conventional algorithm. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a reciprocal and inverse square root unit. Also, it can be used to construct optimized approximate tables. The results of this paper can be applied to many areas that utilize floating point numbers, such as digital signal processing, computer graphics, multimedia, scientific computing, etc.

• The KIPS Transactions:PartA
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• v.12A no.5 s.95
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• pp.413-420
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• 2005

The Newton-Raphson iterative algorithm for finding a floating point reciprocal square mot calculates it by performing a fixed number of multiplications. In this paper, a variable latency Newton-Raphson's reciprocal square root algorithm is proposed that performs multiplications a variable number of times until the error becomes smaller than a given value. To find the rediprocal square root of a floating point number F, the algorithm repeats the following operations: '$X_{i+1}=\frac{{X_i}(3-e_r-{FX_i}^2)}{2}$, $i\in{0,1,2,{\ldots}n-1}$' with the initial value is '$X_0=\frac{1}{\sqrt{F}}{\pm}e_0$'. The bits to the right of p fractional bits in intermediate multiplication results are truncated and this truncation error is less than '$e_r=2^{-p}$'. The value of p is 28 for the single precision floating point, and 58 for the double precision floating point. Let '$X_i=\frac{1}{\sqrt{F}}{\pm}e_i$, there is '$X_{i+1}=\frac{1}{\sqrt{F}}-e_{i+1}$, where '$e_{i+1}{<}\frac{3{\sqrt{F}}{{e_i}^2}}{2}{\mp}\frac{{Fe_i}^3}{2}+2e_r$'. If '$|\frac{\sqrt{3-e_r-{FX_i}^2}}{2}-1|<2^{\frac{\sqrt{-p}{2}}}$' is true, '$e_{i+1}<8e_r$' is less than the smallest number which is representable by floating point number. So, $X_{i+1}$ is approximate to '$\frac{1}{\sqrt{F}}$. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications Per an operation is derived from many reciprocal square root tables ($X_0=\frac{1}{\sqrt{F}}{\pm}e_0$) with varying sizes. The superiority of this algorithm is proved by comparing this average number with the fixed number of multiplications of the conventional algorithm. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a reciprocal square root unit. Also, it can be used to construct optimized approximate reciprocal square root tables. The results of this paper can be applied to many areas that utilize floating point numbers, such as digital signal processing, computer graphics, multimedia, scientific computing, etc.

• Proceedings of the IEEK Conference
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• 2005.11a
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• pp.715-718
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• 2005

Mobile devices is getting to include more functions according to the demand of digital convergence. Applications based on 3D graphic calculation such as 3D games and navigation are one of the functions. 3D graphic calculation requires heavy calculation. Therefore, we need dedicated 3D graphic hardware unit with high performance. 3D graphic calculation needs a lot of complicated floating-point arithmetic operation. However, most of current mobile 3D graphics processors do not have efficient architecture for mobile devices because they are based on those for conventional computer systems. In this paper, we propose arithmetic units for special functions of lighting operation of 3D graphics. Transcendental arithmetic units are designed using approximation of logarithm function. Special function units for lighting operation such as reciprocal, square root, reciprocal of square root, and power can be obtained. The proposed arithmetic unit has lower error rate and smaller silicon area than conventional arithmetic architecture.

• Journal of the Korean Society for Heat Treatment
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• v.4 no.3
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• pp.39-46
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• 1991

The effects of austenitizing temperatures and times and cooling rate on the characteristics of continuous air cooling have been investigated for 0.3%C-Mn steels microalloyed with vanadium. Transformation start temperatures have been found to be measured from temperature-time curve directly obtained with continuous air cooling and to decrease with increasing austenitizing temperature, cooling rate and Mn contents. The coarsening behavior of austenite grain size has been measured to abnormally grow at $1050^{\circ}C$ and rapidly grow at $1200^{\circ}C$. It has been found that the volume fraction of pearlite was linealy proportional to the reciprocal square root of austenite grain size. The hardness has been measured to increase with increasing cooling rate up to $250^{\circ}C/min.$ and to remain relatively unchanged in the range of $250{\sim}400^{\circ}C/min.$ showing that hardness valves for steel with a higher Mn content increase more than those for steel with a lower Mn content. The impact property has been found to decrease with increasing of austenite grain size but does not linealy change with the reciprocal square root of austenite grain size.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.33B no.6
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• pp.54-63
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• 1996

We propose a 3rd order blind equalizer that incorporates a new transform method using either square root operation ($\sqrt{x]$) or reciprocal operation (1/x) in order to transform symmetric distribution of PAM signals at the transmitter, to asymmetric one. At the receiver, either the square operation or the reciprocal operation is needed to recover the asymmetrically transformed signals to the original ones after eualization. The reslts of the computer simulation, using the new method are better than the existing transform method using natural logarithm operation by the maximum of 8 dB in MSE. In addition, as the skewness of the asymmetrically transformed distribution has small values, the performances are improved.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.9 no.1
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• pp.188-198
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• 2005

The Goldschmidt iterative algorithm for finding a floating point square root calculated it by performing a fixed number of multiplications. In this paper, a variable latency Goldschmidt's square root algorithm is proposed, that performs multiplications a variable number of times until the error becomes smaller than a given value. To find the square root of a floating point number F, the algorithm repeats the following operations: $R_i=\frac{3-e_r-X_i}{2},\;X_{i+1}=X_i{\times}R^2_i,\;Y_{i+1}=Y_i{\times}R_i,\;i{\in}\{{0,1,2,{\ldots},n-1} }}'$with the initial value is $'\;X_0=Y_0=T^2{\times}F,\;T=\frac{1}{\sqrt {F}}+e_t\;'$. The bits to the right of p fractional bits in intermediate multiplication results are truncated, and this truncation error is less than $'e_r=2^{-p}'$. The value of p is 28 for the single precision floating point, and 58 for the doubel precision floating point. Let $'X_i=1{\pm}e_i'$, there is $'\;X_{i+1}=1-e_{i+1},\;where\;'\;e_{i+1}<\frac{3e^2_i}{4}{\mp}\frac{e^3_i}{4}+4e_{r}'$. If '|X_i-1|<2^{\frac{-p+2}{2}}\;'$ is true, $'\;e_{i+1}<8e_r\;'$ is less than the smallest number which is representable by floating point number. So, $\sqrt{F}$ is approximate to $'\;\frac{Y_{i+1}}{T}\;'$. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation is derived from many reciprocal square root tables ($T=\frac{1}{\sqrt{F}}+e_i$) with varying sizes. The superiority of this algorithm is proved by comparing this average number with the fixed number of multiplications of the conventional algorithm. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a square root unit. Also, it can be used to construct optimized approximate reciprocal square root tables. The results of this paper can be applied to many areas that utilize floating point numbers, such as digital signal processing, computer graphics, multimedia, scientific computing, etc.

• Textile Coloration and Finishing
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• v.9 no.5
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• pp.1-9
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• 1997

In the single yarn spinning process by the ring spinning system, the finer the fineness of yarn and the lower the twist coefficient, the lower the breaking strength and breaking elongation. The change of yarn specific volume to yarn number agreed with Peirce's formula in the range of Ne 50 to 70, but above that range the values of the experiment are higher than that of the formula. The change of diameter of yarn to the reciprocal of the square root of yarn number agreed with Peirce's formula in the range of under 0.14, but above that value the values of the experiment are higher than that of the formula. In breaking strength variation according to twist constant of single yarn, as the twist coefficient increased, breaking strength increased. At 5.8∼6.0 of twist coefficient the maximum breaking strength was shown, but above that value breaking strength decreased. Breaking elongation also showed a similar tendency. But at 6.0∼6.5 of twist coefficient the maximum breaking elongation was shown. Also spinning tension increased as twist coefficient increased. Twist coefficient, breaking strength and breaking elongation according to the number of coils stayed almost the same. Yarn spinning tension according to the number of coils at the maximum of diameter was the lowest value. The speed of the traveller at the maximum of diameter was the highest value.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.17 no.8
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• pp.1891-1898
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• 2013

To efficiently execute 3D graphic APIs, such as OpenGL and Direct3D, special purpose arithmetic unit(SFU) which supports floating-point sine, cosine, reciprocal, inverse square root, base-two exponential, and logarithmic operations is designed. The SFU uses second order minimax approximation method and lookup table method to satisfy both error less than 2 ulp(unit in the last place) and high speed operation. The designed circuit has about 2.3-ns delay time under 65nm CMOS standard cell library and consists of about 23,300 gates. Due to its maximum performance of 400 MFLOPS and high accuracy, it can be efficiently applicable to mobile 3D graphics application.

• Journal of Pharmaceutical Investigation
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• v.33 no.3
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• pp.245-253
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• 2003

Dissolution profile comparsions can be done by virtue of the similarity factor $(f_2)$. It is a logarithmic reciprocal square root transformation of the sum of squared error of % dissolution differences between two profiles at several time points. It gives information on the degree of similarity between the two profiles: An $f_2$ value between 50 and 100 suggests the similarity/equivalence of the two dissolution curves being compared. The objective of this report was to provide a careful examination on the $f_2$ metrics in detail. It was shown that $f_2$ values exceeded 50, when relative differences in % dissolved between two products were less than 15% at all time points. The similarity factor value was also found to be greater than 50, in cases when absolute % dissolution differences were below 10% at all time points. Interestingly, the $f_2$ value was changed by the number of the time points selected for calculation. In particular, $f_2$ tended to have higher values, when the $f_2$ metrics used a large number of time points in which % dissolved reached plateau. Finally, since the similarity factor was a sample statistics, it was impossible to infer type I/II errors and sampling error. Despite certain limitations inherited in the $f_2$ metrics, it was easy and convenient to evaluate how similar the two dissolution profiles were.