scholarly journals A note on continuation methods for the solution of nonlinear equations

1977 ◽  
Vol 20 (2) ◽  
pp. 157-164 ◽  
Author(s):  
James P. Abbott ◽  
Richard P. Brent
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Continuation Method ◽  
Continuation Methods ◽  
Variable Order ◽  
Rapid Convergence

AbstractIn this note we present a variable order continuation method for the solution of nonlinear equations when only a poor estimate of a solution is known. The method changes continuously from one which improves the global convergence characteristics to one which attains rapid convergence to a solution and proves to be more efficient than methods previously presented in [2].

scholarly journals An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations

2020 ◽  
Author(s):  
Jamilu Sabi'u ◽  
Abdullah Shah
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Numerical Results ◽  
Search Direction ◽  
Projection Technique ◽  
Type Algorithm ◽  
Gradient Type

In this article, we proposed two Conjugate Gradient (CG) parameters using the modified Dai-{L}iao condition and the descent three-term CG search direction. Both parameters are incorporated with the projection technique for solving large-scale monotone nonlinear equations. Using the Lipschitz and monotone assumptions, the global convergence of methods has been proved. Finally, numerical results are provided to illustrate the robustness of the proposed methods.

Size-Dependent Geometrically Nonlinear Forced Vibration Analysis of Functionally Graded First-Order Shear Deformable Microplates

2016 ◽  
Vol 32 (5) ◽  
pp. 539-554 ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
A. Shahabodini
Keyword(s):  
Size Effect ◽  
Nonlinear Equations ◽  
Forced Vibration ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Continuation Method ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Governing Equations ◽  
First Order

AbstractIn this paper, a non-classical plate model capturing the size effect is developed to study the forced vibration of functionally graded (FG) microplates subjected to a harmonic excitation transverse force. To this, the modified couple stress theory (MCST) is incorporated into the first-order shear deformation plate theory (FSDPT) to account for the size effect through one length scale parameter, only. Strong form of nonlinear governing equations and associated boundary conditions are obtained using Hamilton's principle. The solution process is implemented on two domains. The generalized differential quadrature (GDQ) method is first employed to discretize the governing equations on the space domain. A Galerkin-based scheme is then applied to extract a reduced set of the nonlinear equations of Duffing-type. On the second domain, through a time differentiation matrix operator, the set of ordinary differential equations are transformed into the discrete form on time domain. Eventually, a system of the parameterized nonlinear equations is acquired and solved via the pseudo-arc length continuation method. The frequency response curve of the microplate is sketched and the effects of various material and geometrical parameters on it are evaluated.

scholarly journals A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Menglong Su ◽  
Yufeng Shang ◽  
Wenzhuang Zhu
Keyword(s):  
Fixed Point ◽  
Global Convergence ◽  
Continuation Method ◽  
Equality Constraints ◽  
Fixed Point Problems ◽  
Homotopy Continuation ◽  
Parameter Perturbation ◽  
Numerical Examples ◽  
Homotopy Continuation Method ◽  
Nonconvex Sets

In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.

scholarly journals An Improved Secant Algorithm of Variable Order to Solve Nonlinear Equations Based on the Disassociation of Numerical Approximations and Iterative Progression

2020 ◽  
Vol 12 (6) ◽  
pp. 50
Author(s):  
Christian Vanhille
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Secant Method ◽  
Variable Order ◽  
Discretization Step ◽  
Newton Raphson ◽  
Analytic Expression ◽  
The One ◽  
Derivatives Of ◽  
Raphson Method

We propose an iterative method to evaluate the roots of nonlinear equations. This Secant-based technique approximates the derivatives of the function numerically through a constant discretization step h disassociated from the iterative progression. The algorithm is developed, implemented, and tested. Its order of convergence is found to be h-dependent. The results obtained corroborate the theoretical deductions and evidence its excellent behavior. For infinitesimal h-values, the algorithm accelerates the convergence of the Secant method to order 2 (the one of the Newton-Raphson method) with no need for analytic expression of derivatives (the advantage of the Secant method).

scholarly journals A method with inertial extrapolation step for convex constrained monotone equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdulkarim Hassan Ibrahim ◽  
Poom Kumam ◽  
Auwal Bala Abubakar ◽  
Jamilu Abubakar
Keyword(s):  
Global Convergence ◽  
Theoretical Analysis ◽  
Nonlinear Equations ◽  
Large Scale ◽  
Descent Direction ◽  
Monotone Equations ◽  
Derivative Free ◽  
Speed Up ◽  
Sufficient Descent ◽  
Inertial Extrapolation

AbstractIn recent times, various algorithms have been incorporated with the inertial extrapolation step to speed up the convergence of the sequence generated by these algorithms. As far as we know, very few results exist regarding algorithms of the inertial derivative-free projection method for solving convex constrained monotone nonlinear equations. In this article, the convergence analysis of a derivative-free iterative algorithm (Liu and Feng in Numer. Algorithms 82(1):245–262, 2019) with an inertial extrapolation step for solving large scale convex constrained monotone nonlinear equations is studied. The proposed method generates a sufficient descent direction at each iteration. Under some mild assumptions, the global convergence of the sequence generated by the proposed method is established. Furthermore, some experimental results are presented to support the theoretical analysis of the proposed method.

scholarly journals The Pareto Tracer for General Inequality Constrained Multi-Objective Optimization Problems

2020 ◽  
Vol 25 (4) ◽  
pp. 80
Author(s):  
Fernanda Beltrán ◽  
Oliver Cuate ◽  
Oliver Schütze
Keyword(s):  
Optimization Problems ◽  
Continuation Method ◽  
Optimization Methods ◽  
New Method ◽  
Continuation Methods ◽  
Benchmark Problems ◽  
Multi Objective Optimization ◽  
Financial Applications ◽  
General Inequality ◽  
Multi Objective

Problems where several incommensurable objectives have to be optimized concurrently arise in many engineering and financial applications. Continuation methods for the treatment of such multi-objective optimization methods (MOPs) are very efficient if all objectives are continuous since in that case one can expect that the solution set forms at least locally a manifold. Recently, the Pareto Tracer (PT) has been proposed, which is such a multi-objective continuation method. While the method works reliably for MOPs with box and equality constraints, no strategy has been proposed yet to adequately treat general inequalities, which we address in this work. We formulate the extension of the PT and present numerical results on some selected benchmark problems. The results indicate that the new method can indeed handle general MOPs, which greatly enhances its applicability.

Four-Bar Path Generation Synthesis by a Continuation Method

1989 ◽  
Author(s):  
T. Subbian ◽  
D. R. Flugrad
Keyword(s):  
Nonlinear Equations ◽  
Continuation Method ◽  
System Of Nonlinear Equations ◽  
Path Generation ◽  
Four Bar Linkage

Abstract A different approach for the synthesis of four bar planar path generating mechanisms is presented. A continuation method is used to solve the system of nonlinear equations derived for the path generating problem. A brief description of the method is provided followed by the development of equations representing the four bar linkage. The implementation of the method for five position path generation is discussed in detail and the solutions for two examples are presented.

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