Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms

Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms.

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An Iterative Hybrid Algorithm for Roots of Non-Linear Equations

Eng ◽

10.3390/eng2010007 ◽

2021 ◽

Vol 2 (1) ◽

pp. 80-98

Author(s):

Chaman Lal Sabharwal

Keyword(s):

Hybrid Algorithm ◽

Analytic Solution ◽

Linear Equations ◽

Root Finding ◽

Engineering Sciences ◽

Non Linear ◽

False Position ◽

Newton Raphson ◽

And Performance ◽

Non Linear Equations

Finding the roots of non-linear and transcendental equations is an important problem in engineering sciences. In general, such problems do not have an analytic solution; the researchers resort to numerical techniques for exploring. We design and implement a three-way hybrid algorithm that is a blend of the Newton–Raphson algorithm and a two-way blended algorithm (blend of two methods, Bisection and False Position). The hybrid algorithm is a new single pass iterative approach. The method takes advantage of the best in three algorithms in each iteration to estimate an approximate value closer to the root. We show that the new algorithm outperforms the Bisection, Regula Falsi, Newton–Raphson, quadrature based, undetermined coefficients based, and decomposition-based algorithms. The new hybrid root finding algorithm is guaranteed to converge. The experimental results and empirical evidence show that the complexity of the hybrid algorithm is far less than that of other algorithms. Several functions cited in the literature are used as benchmarks to compare and confirm the simplicity, efficiency, and performance of the proposed method.

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Extension of the finite-difference Newton- Raphson algorithm to the simultaneous optimization of trajectories and associated parameters

10.2514/6.1968-115 ◽

1968 ◽

Cited By ~ 2

Author(s):

C. VAN DINE

Keyword(s):

Finite Difference ◽

Simultaneous Optimization ◽

Newton Raphson ◽

Raphson Algorithm

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A Class of Decision Processes Showing Policy-Improvement/Newton–Raphson Equivalence

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800001261 ◽

1989 ◽

Vol 3 (3) ◽

pp. 397-403 ◽

Cited By ~ 1

Author(s):

P. Whittle

Keyword(s):

Optimality Condition ◽

Regularity Condition ◽

Decision Processes ◽

Policy Improvement ◽

Improvement Algorithm ◽

Newton Raphson ◽

Raphson Algorithm

A condition expressed in Eq. (7) is given which, with one simplifying regularity condition, ensures that the policy-improvement algorithm is equivalent to application of the Newton–Raphson algorithm to an optimality condition. It is shown that this condition covers the two known cases of such equivalence, and another example is noted. The condition is believed to be necessary to within transformations of the problem, but this has not been proved.

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A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods

Mathematics ◽

10.3390/math9111306 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1306

Author(s):

Elsayed Badr ◽

Sultan Almotairi ◽

Abdallah El Ghamry

Keyword(s):

Comparative Study ◽

Numerical Results ◽

Traditional Methods ◽

Running Time ◽

Root Finding ◽

Average Running Time ◽

False Position ◽

Newton Raphson ◽

Number Of Iterations

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time.

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An efficient algorithm for estimating the parameters of superimposed exponential signals in multiplicative and additive noise

International Journal of Applied Mathematics and Computer Science ◽

10.2478/amcs-2013-0010 ◽

2013 ◽

Vol 23 (1) ◽

pp. 117-129 ◽

Cited By ~ 2

Author(s):

Jiawen Bian ◽

Huiming Peng ◽

Jing Xing ◽

Zhihui Liu ◽

Hongwei Li

Keyword(s):

Parameter Estimation ◽

Convergence Rate ◽

Efficient Algorithm ◽

Numerical Experiments ◽

Additive Noise ◽

Mean Squared Errors ◽

Least Squares Estimators ◽

The Mean ◽

Newton Raphson ◽

Raphson Algorithm

This paper considers parameter estimation of superimposed exponential signals in multiplicative and additive noise which are all independent and identically distributed. A modified Newton-Raphson algorithm is used to estimate the frequencies of the considered model, which is further used to estimate other linear parameters. It is proved that the modified Newton- Raphson algorithm is robust and the corresponding estimators of frequencies attain the same convergence rate with Least Squares Estimators (LSEs) under the same noise conditions, but it outperforms LSEs in terms of the mean squared errors. Finally, the effectiveness of the algorithm is verified by some numerical experiments.

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Generalized Newton–Raphson algorithm for high dimensional LASSO regression

Statistics and Its Interface ◽

10.4310/20-sii643 ◽

2021 ◽

Vol 14 (3) ◽

pp. 339-350

Author(s):

Yueyong Shi ◽

Jian Huang ◽

Yuling Jiao ◽

Yicheng Kang ◽

Hu Zhang

Keyword(s):

High Dimensional ◽

Lasso Regression ◽

Newton Raphson ◽

Generalized Newton ◽

Raphson Algorithm

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Numerical methods for kinetic equations

Acta Numerica ◽

10.1017/s0962492914000063 ◽

2014 ◽

Vol 23 ◽

pp. 369-520 ◽

Cited By ~ 123

Author(s):

G. Dimarco ◽

L. Pareschi

Keyword(s):

Computational Complexity ◽

Numerical Methods ◽

State Of The Art ◽

Kinetic Equations ◽

Numerical Techniques ◽

Asymptotic Preserving ◽

Hybrid Schemes ◽

Velocity Models ◽

Current State ◽

Number Of Particles

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

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