Efficient method for solving a system of nonlinear equations

Waseem, Muhammad; Noor, Muhammad Aslam; Noor, Khalida İnayat

doi:10.1016/j.amc.2015.11.061

Cited by 18 publications

(22 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These methods were obtained by using Taylor's expansion of the function f(x) or some decomposition techniques applied to another nonlinear equation equivalent (or approximate) to the original f(x) = 0. In the works of Darvishi et al, 34 Waseema et al, 35 and Traub, 36 these techniques were applied to develop a two-step Newton method for solving a system of nonlinear equations, which is an extension of the approach. 33 Besides, a normalized Newton method was proposed by Tapia et al 30 for the general system of nonlinear equations whose solution x satisfies ||x|| = 1.…”

Section: A Mnnm For Eigenpairsmentioning

confidence: 99%

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

Zhao

Zheng

Liang

et al. 2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

This paper is concerned with computing -eigenpairs of symmetric tensors.We first show that computing -eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex-valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method. K E Y W O R D S-eigenpairs, US-eigenpairs, cubical convergence, modified normalized Newton method, Newton correction method, symmetric tensors INTRODUCTIONThe eigenvalue problems of tensors have attracted much attention in recent years because of their applications in high-order Markov chains, 1 magnetic resonance imaging, 2 quantum entanglement, 3,4 automatical control, 5 the best rank-one approximations in statistical data analysis, 6-10 learning binary latent variable models, 11 and so on. Unlike matrices, there are various definitions of eigenvalues for tensors, including -eigenvalues, -eigenvalues, and -eigenvalues; see, for example, other works 12-18 and the references therein. In this paper, we are mainly concerned with the calculation of -eigenvalues and corresponding -eigenvectors. This kind of eigenvalues and eigenvectors has practical applications, for example, the smallest -eigenvalue can reflect the stability of a nonlinear autonomous system in automatic control, and the principal -eigenvector can depict the orientations of nerve fibers in the voxel of white matter of human brain. 5,19 Computing -eigenpairs of a third or higher order tensor is equivalent to finding nontrivial solutions of a system of inhomogeneous polynomial equations in several variables. This implies that the task in the computation of -eigenpairs of a tensor will be much more difficult when its order and dimension are very large. For this reason, some iterative methods have been developed to compute one or more -eigenpairs of the tensors with special structure, such as nonnegative tensors and symmetric tensors.Recently, for nonnegative tensors, Guo et al. 20 proposed a modified Newton iteration to compute some nonnegative -eigenpairs and showed that this method has local quadratic convergence under appropriate assumptions. Kuo et al. 21 presented a homotopy continuation method for the same purpose. For symmetric tensors, Cui, Dai and Numer Linear Algebra Appl. 2020;27:e2284.wileyonlinelibrary.com/journal/nla

show abstract

Section: A Mnnm For Eigenpairsmentioning

confidence: 99%

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

Zhao

Zheng

Liang

et al. 2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…The authors in [6] proposed an effective algorithm called Algorithm 2.4. They compared their algorithm against a large number of algorithms and showed its superiority to the others.…”

Section: Benchmark I3mentioning

confidence: 99%

“…They compared their algorithm against a large number of algorithms and showed its superiority to the others. The comparison is based on the following two problems [6]. E 0 2 1 3 3 1 0 2 i 3 2 i 2 i 1 3 i 2 0 i 3 2 0 i 2 i 2 1 3 i 2 3 i 2 1 3 i 2 i 2 2 i 2 i i 2 i 2 3 i 2 i 3…”

Section: Benchmark I3mentioning

confidence: 99%

Probability-Directed Problem Optimization Technique for Solving Systems of Linear and Non-Linear Equations

Al-Muhammed¹

2019

Computer Science &Amp; Information Technology (CS &Amp; IT )

View full text Add to dashboard Cite

Although many methods have been proposed for solving linear or nonlinear systems of equations, there is always a pressing need for more effective and efficient methods. Good methods should produce solutions with high precision and speed. This paper proposed an innovative method for solving systems of linear and nonlinear equations. This method transforms the problem into an optimization problem and uses a probability guided search technique for solving this optimization problem, which is the solution for the system of equations. The transformation results in an aggregate violation function and a criterion function. The aggregation violation function is composed of the constraints that represent the equations and whose satisfaction is a solution for the system of equations. The criterion function intelligently guides the search for the solution to the aggregate violation function by determining when the constraints must be checked; thereby avoiding unnecessary, timeintensive checks for the constraints. Experiments conducted with our prototype implementation showed that our method is effective in finding solutions with high precision and efficient in terms of CPU time.

show abstract

“…The algorithm was then applied by Kardan and Akbarzadeh 5 to solve FKP of parallel robots. Moreover, Waseem et al 8 constructed a sequence of high-order iterative methods for nonlinear equations system through a new decomposition technique mainly due to Daftardar-Gejji and Jafari 14 and proved that the methods were higher-order convergent and had better efficiency index. This decomposition of nonlinear operator N(X) is quite different from the Adomian decomposition, and it can be decomposed as…”

Section: High-order Iterative Methodsmentioning

confidence: 99%

“…In this article, by combining a new method based on signal decomposition and reconstruction with high-order iterative method 7,8 together with the use of a new decomposition technique, a new hybrid strategy is proposed for solving the FKP of parallel robots. An effective way to increase the convergence speed of numerical algorithms is to provide them with more accurate initial guesses.…”

Section: Introductionmentioning

confidence: 99%

Efficient hybrid method for forward kinematics analysis of parallel robots based on signal decomposition and reconstruction

Wang

et al. 2017

Advances in Mechanical Engineering

View full text Add to dashboard Cite

This article combines a new method based on signal decomposition and reconstruction with a fifth-order numerical algorithm and proposes an efficient hybrid method for solving forward kinematics problem of parallel manipulators. In this hybrid method, new method can first generate an approximate solution of the forward kinematics problem, which will be taken as initial guess for the fifth-order numerical technique. The answer with desired level of accuracy is then obtained. The superposition principle is proposed stating that each rod's displacement required to drive the mobile platform with 6-degree-of-freedom coupled motion is approximately a sum of the rod's displacement required to drive it with each of six single basic motions including roll q 1 , pitch q 2 , yaw q 3 , surge q 4 , sway q 5 , and heave q 6 . By solving the equations established in accordance with the superposition principle, the approximate solution is also gained. Superposition principle is more applicable to the robots with small rotational workspace. The solution is directly solved under superposition principle. Even though the rotational workspace is expanded, the solution still meets the required accuracy of initial value of high-order iterative technique for obtaining its exact solution. The proposed method is then applied to a test bench for bogie parameters and a Stewart platform. Simulation results demonstrate that maximum absolute error of displacement along X-, Y-, and Z-axis is 0.174 mm and its relative error is about 3.4&. Using this method will reduce about 34.1% of the required number of iterations to solve the forward kinematics problem compared with improved hybrid method and up to 49.8% compared with the Newton-Raphson method.

show abstract

Efficient method for solving a system of nonlinear equations

Cited by 18 publications

References 31 publications

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

Probability-Directed Problem Optimization Technique for Solving Systems of Linear and Non-Linear Equations

Efficient hybrid method for forward kinematics analysis of parallel robots based on signal decomposition and reconstruction

Contact Info

Product

Resources

About