• 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.

• Journal of Korea Society of Industrial Information Systems
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• v.18 no.6
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• pp.31-37
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• 2013

Modern graphic processors, multimedia processors and audio processors mostly use floating-point number. High-level language such as C and Java use both single precision and double precision floating-point number. In this paper, an algorithm which computes the reciprocal of double precision floating-point number using a 32 bit multiplier is proposed. It divides the mantissa of double precision floating-point number to upper part and lower part, and calculates the reciprocal of the upper part with Newton-Raphson algorithm. And it computes the reciprocal of double precision floating-point number with calculated upper part reciprocal as the initial value. Since the number of multiplications performed by the proposed algorithm is dependent on the mantissa of floating-point number, the average number of multiplications per an operation is derived from some reciprocal tables with varying sizes.

• 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.

• The KIPS Transactions:PartA
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• v.12A no.2 s.92
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• pp.95-102
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• 2005

The Newton-Raphson iterative algorithm for finding a floating point reciprocal which is widely used for a floating point division, calculates the reciprocal by performing a fixed number of multiplications. In this paper, a variable latency Newton-Raphson's reciprocal algorithm is proposed that performs multiplications a variable number of times until the error becomes smaller than a given value. To find the reciprocal of a floating point number F, the algorithm repeats the following operations: '$'X_{i+1}=X=X_i*(2-e_r-F*X_i),\;i\in\{0,\;1,\;2,...n-1\}'$ with the initial value $'X_0=\frac{1}{F}{\pm}e_0'$. 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 27 for the single precision floating point, and 57 for the double precision floating point. Let $'X_i=\frac{1}{F}+e_i{'}$, these is $'X_{i+1}=\frac{1}{F}-e_{i+1},\;where\;{'}e_{i+1}, is less than the smallest number which is representable by floating point number. So, $X_{i+1}$ is approximate to $'\frac{1}{F}{'}$. 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 tables $(X_0=\frac{1}{F}{\pm}e_0)$ 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 unit. Also, it can be used to construct optimized approximate reciprocal 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.

• IEMEK Journal of Embedded Systems and Applications
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• v.9 no.5
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• pp.285-292
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• 2014

The commonly used Newton-Raphson's floating-point number divider algorithm performs two multiplications in one iteration. In this paper, a tentative K'th Newton-Raphson's floating-point number divider algorithm which performs K times multiplications in one iteration is proposed. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation in single precision and double precision divider is derived from many reciprocal tables with varying sizes. In addition, an error correction algorithm, which consists of one multiplication and a decision, to get exact result in divider is proposed. 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 floating point number divider unit. Also, it can be used to construct optimized approximate reciprocal tables.

• IEMEK Journal of Embedded Systems and Applications
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• v.13 no.1
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• pp.45-51
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• 2018

In this paper, a tentative Kth order Newton-Raphson's floating point number Nth root algorithm for K order convergence rate in one iteration is proposed by applying Taylor series to the Newton-Raphson root algorithm. Using the proposed algorithm, $F^{-1/N}$ and $F^{-(N-1)/N}$ can be computed from iterative multiplications without division. It also predicts the error of the algorithm iteration and iterates only until the predicted error becomes smaller than the specified value. 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 floating point number Nth root unit.

• Proceedings of the KIEE Conference
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• 1997.07c
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• pp.1094-1096
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• 1997

이제까지 전력계통의 상태를 알기 위한 조류계산은 전력방정식에 대한 직접적인 풀이로써 이루어졌다. 그런데 만일 자코비안 행열이 singular나 유사 singular가 된다면 그 전력 방정식의 해를 구할 수가 없게 된다. singular나 유사 singular가 되는 자코비안 행열을 가지는 전력방정식을 풀기 위하여 보조 방정식을 추가하여 미분불능인 점을 미분가능으로 변환하는 continuous method가 있다. 그런데 continuous method에서 보조 방정식은 원함수의 성질에 따라 달라지므로 어떤 일반적인 형태를 가지지 않는다. 따라서 본 논문에서는 전력방정식을 제약조건으로 가지고 부하 모선의 전력이 수용가의 요구량과 거의 일치되도록 하는 것을 목적 함수로 하는 최적화 문제로 조류계산을 변환하여 풀이하였다. 이러한 최적화 기법을 이용함으로써 continuous method에서 보조방정식이 원함수에 따라 계속적으로 변하는 문제를 개선하였다.