Computation of the Pressure Distribution in Hydrodynamic Bearings Using Newton’s Method

2004 ◽  
Vol 126 (2) ◽  
pp. 404-407 ◽  
Author(s):  
Lars Johansson and ◽  
Ha˚kan Wettergren
Keyword(s):  
Pressure Distribution ◽  
Newton Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reynolds Equation ◽  
Usual Sense ◽  
System Of Equations ◽  
Equilibrium Equations ◽  
Hydrodynamic Bearings

In this paper an algorithm is developed where Reynolds’ equation, equilibrium equations and non-negativity of pressure are formulated as a system of equations, which are not differentiable in the usual sense. This system is then solved using Pang’s Newton method for B-differentiable equations.

scholarly journals A Geometric Newton Method for Oja's Vector Field

2009 ◽  
Vol 21 (5) ◽  
pp. 1415-1433 ◽  
Author(s):  
P.-A. Absil ◽  
M. Ishteva ◽  
L. De Lathauwer ◽  
S. Van Huffel
Keyword(s):  
Vector Field ◽  
Newton Method ◽  
Newton’S Method ◽  
Matrix Equation ◽  
Orthogonal Group ◽  
Newton's Method ◽  
Newton Algorithm ◽  
The Matrix ◽  
Geometric Techniques ◽  
The Way

Newton's method for solving the matrix equation [Formula: see text] runs up against the fact that its zeros are not isolated. This is due to a symmetry of F by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a “geometric” Newton algorithm that finds the zeros of F. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method.

Formal proofs for theoretical properties of Newton's method

2011 ◽  
Vol 21 (4) ◽  
pp. 683-714 ◽  
Author(s):  
IOANA PAŞCA
Keyword(s):  
Multivariate Analysis ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Algorithms ◽  
System Of Equations ◽  
Proof Assistant ◽  
Formal Proofs ◽  
Original Result ◽  
Accurate Model ◽  
Formal Study

We discuss a formal development for the certification of Newton's method. We address several issues encountered in the formal study of numerical algorithms: developing the necessary libraries for our proofs, adapting paper proofs to suit the features of a proof assistant and designing new proofs based on the existing ones to deal with optimisations of the method. We start from Kantorovitch's Theorem, which gives the convergence of Newton's method in the case of a system of equations. To formalise this proof inside the proof assistant Coq, we first need to code the necessary concepts from multivariate analysis. We also prove that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input and that of the result. This proof is based on Kantorovitch's Theorem, but is an original result. An algorithm including rounding is a more accurate model for computations with Newton's method in practice.

Comparison of Iterative Methods for the Solution of Compressible-Fluid Reynolds Equation

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Nenzi Wang ◽  
Shih-Hung Chang ◽  
Hua-Chih Huang
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Compressible Fluid ◽  
Newton's Method ◽  
Reynolds Equation ◽  
Multigrid Method ◽  
Iterative Solution ◽  
Sor Method ◽  
Solution Methods ◽  
Iterative Solution Methods

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

scholarly journals A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ababu Teklemariam Tiruneh ◽  
W. N. Ndlela ◽  
S. J. Nkambule
Keyword(s):  
Newton Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Computational Effort ◽  
Secant Methods ◽  
Iterative Solutions ◽  
Solutions Of Equations ◽  
Modified Formula

An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.

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