Historical Note on the Newton-Raphson Method of Approximation

Cajori, Florian

doi:10.2307/2973939

Cited by 17 publications

(10 citation statements)

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“…e same situation is with the three-point methods [28]. e following types of the method are used in this paper: the first-order Householder method (Newton-Raphson [18,19]), the 2nd order (Halley [69,70] and Schröder [25]), and the 3rd order, as well as the threepoint methods [28]. All these methods require the calculation of the derivatives which is usually underlined as the most important shortcoming of the Householder and the three-point methods compared with the simple fixedpoint procedure in respect of the Colebrook equation.…”

Section: Iterative Methods Adopted For the Colebrook Equationmentioning

confidence: 99%

“…f ″(x), Equation 14f ‴(x), Equation 16x 0 � 10.34052343 λ 0 � 0.009352225155363 17, and the symbolic derivatives f ′(x) and f ″(x): Equations 11and (14). 14x 0 � 7.990256504 λ 0 � 0.015663210285978 14x 0 � 7.990256504 λ 0 � 0.015663210285978 2, and direct calculation of λ: Equation (18).…”

Section: Ree-point Methodsmentioning

confidence: 99%

“…(b) Householder's iterative methods. On the other hand, Newton's method (also known as the Newton-Raphson method [17][18][19]) needs few iterations less compared to the fixed-point method to reach the same level of accuracy. A shortcoming of Newton's method is that it additionally requires the first derivative of the Colebrook function (here we show analytical form of the first derivative including the symbolic form generated in MATLAB [20]).…”

Section: Introductionmentioning

confidence: 99%

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Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

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Section: Iterative Methods Adopted For the Colebrook Equationmentioning

confidence: 99%

Section: Ree-point Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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“…30 Moritz Cantor reviewed the approximation methods of Newton, Raphson, Halley and de Lagny, describing Raphson as 'an absolute admirer and imitator of Newton', whose approximation method 'greatly resembled that of Newton'. 31 Reviewing the situation in 1911, Florian Cajori concluded that the method ought properly to be called the 'Newton-Raphson method'; 32 however, no person in the seventeenth or eighteenth centuries adopted such a view. Cajori's grounds for referring to the method as 'the Newton-Raphson method' may have been his view that 'If r is the approximation already reached, then Newton uses a divisor which in our modern notation takes the form f'(r)'.…”

Section: T H E M Y T H Blossomsmentioning

confidence: 99%

Thomas Simpson and ‘Newton's method of approximation’: an enduring myth

Kollerstrom

1992

Brit. J. Hist. Sci

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A resurgence of interest has occurred in ‘Newton's method of approximation’ for deriving the roots of equations, as its repetitive and mechanical character permits ready computer use. If x = α is an approximate root of the equation f(x) = 0, then the method will in most cases give a better approximation aswhere f′(x) is the derivative of the function into which α has been substituted. Older books sometimes called it ‘the Newton–Raphson method’, although the method was invented essentially in the above form by Thomas Simpson, who published his account of the method in 1740. However, as if through a time-warp, this invention has migrated back in time and is now matter-of-factly placed by historians in Newton's De analysi of 1669. This paper will describe the steps of this curious historical transposition, and speculate as to its cause.

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“…Usually, these equations are solved in multiple iterations and hence require substantial processing and memory requirements [ 13 , 14 ]. One of the primary methods used for the solutions of nonlinear equations is Newton-Raphson method [ 15 ], which is widely known for its quadratic convergence features. However, the conventional load flow studies do not account for the presence of STATCOM device(s) in the system and hence the method must be redesigned for the STATCOM buses.…”

Section: Introductionmentioning

confidence: 99%

STATCOM Estimation Using Back-Propagation, PSO, Shuffled Frog Leap Algorithm, and Genetic Algorithm Based Neural Networks

Soodi

Vural

2018

Computational Intelligence and Neuroscience

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Different optimization techniques are used for the training and fine-tuning of feed forward neural networks, for the estimation of STATCOM voltages and reactive powers. In the first part, the paper presents the voltage regulation in IEEE buses using the Static Compensator (STATIC) and discusses efficient ways to solve the power systems featuring STATCOM by load flow equations. The load flow equations are solved using iterative algorithms such as Newton-Raphson method. In the second part, the paper focuses on the use of estimation techniques based on Artificial Neural Networks as an alternative to the iterative methods. Different training algorithms have been used for training the weights of Artificial Neural Networks; these methods include Back-Propagation, Particle Swarm Optimization, Shuffled Frog Leap Algorithm, and Genetic Algorithm. A performance analysis of each of these methods is done on the IEEE bus data to examine the efficiency of each algorithm. The results show that SFLA outperforms other techniques in training of ANN, seconded by PSO.

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Historical Note on the Newton-Raphson Method of Approximation

Cited by 17 publications

References 0 publications

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Thomas Simpson and ‘Newton's method of approximation’: an enduring myth

STATCOM Estimation Using Back-Propagation, PSO, Shuffled Frog Leap Algorithm, and Genetic Algorithm Based Neural Networks

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