scholarly journals Solving Separable Nonlinear Equations Using LU Factorization

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yun-Qiu Shen ◽  
Tjalling J. Ypma
Keyword(s):  
Nonlinear Equations ◽  
Iterative Process ◽  
Full Rank ◽  
Original System ◽  
System Of Equations ◽  
Lu Factorization ◽  
Numerical Implementation ◽  
Direct Solution ◽  
Differentiable Functions ◽  
The Matrix

Separable nonlinear equations have the form where the matrix and the vector are continuously differentiable functions of and . We assume that and has full rank. We present a numerical method to compute the solution for fully determined systems () and compatible overdetermined systems (). Our method reduces the original system to a smaller system of equations in alone. The iterative process to solve the smaller system only requires the LU factorization of one matrix per step, and the convergence is quadratic. Once has been obtained, is computed by direct solution of a linear system. Details of the numerical implementation are provided and several examples are presented.

A new analytic procedure to determine hypocentral parameters of local seismic events

1984 ◽  
Vol 74 (2) ◽  
pp. 655-667
Author(s):  
D. Caccamo ◽  
G. Neri
Keyword(s):  
Least Squares ◽  
Iterative Process ◽  
Classical Method ◽  
Earth Model ◽  
Seismic Events ◽  
System Of Equations ◽  
Analytic Procedure ◽  
The Earth ◽  
The Matrix ◽  
Hypocentral Parameters

Abstract A new procedure for locating local earthquakes is proposed. Essentially, this procedure consists in solving—by means of least-squares technique—a system of equations which is formally analogous to that of Geiger (Ax = b) but different from his in the values of the matrix A and vector b elements. This difference makes our procedure more reliable than Geiger's because it significantly reduces the cases of both iterative process divergence and low precision. Among the factors contributing to this progress the lesser possibility that the matrix A contains columns proportional (or nearly proportional) one to another is considered of particular influence. More than 20,000 hypocentral calculations have been performed on simulated shocks: significant differences in the number of good locations were revealed between our procedure and the classical method of Geiger (1910). Ours is more precise, particularly when few stations were used or networks with an unsatisfactory geometry. The earth model also influences the observed differences as it contributes to generating those analytical conditions which make the calculation convergence more problematic when applying Geiger's method. Further applications are currently carried out in order to verify the procedure features for velocity laws and station configuration different from those used in this study.

scholarly journals METHODS AND ALGORITHNS FOR INCREASING THE SPEED OF COMPU-TING PROCESSES FOR CALCULATING HYDRAULIC NETWORKS

2019 ◽  
Vol 7 (3) ◽  
pp. 55-61
Author(s):  
Elena Kitayceva
Keyword(s):  
Mathematical Modeling ◽  
Nonlinear Equations ◽  
Network Topology ◽  
Iterative Process ◽  
Flow Distribution ◽  
System Of Nonlinear Equations ◽  
System Of Equations ◽  
Hydraulic Network ◽  
Hydraulic Networks ◽  
The Mathematical Model

The article is devoted to mathematical modeling of flow distribution in hydraulic net-works. Calculations of hydraulic networks are carried out at the stage of their design and operation. The results of numerical simulation are used to control the operation of the hy-draulic network in real time. The mathematical model of the distribution of flows in the hydraulic network is a system of nonlinear equations. The nodal pressures method used to solve the system of equations numerically is the n-dimensional Newton method. To ensure stable and fast convergence of the iterative process, it is proposed to use the initial approx-imation taking into account the network topology and parameters of its objects, use the lower relaxation factor and optimize the structure of the Maxwell matrix. The algorithms presented in the paper allow one to significantly reduce the dimension of the system of nonlinear equations being solved.

scholarly journals A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots

2021 ◽  
Vol 5 (1) ◽  
pp. 25
Author(s):  
Víctor Galilea ◽  
José M. Gutiérrez
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
Iterative Process ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Schröder’S Method

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials.

Research of stamping of unequal channel bars. Part 3. Force parameters and shape change of the billet when extruding channels. 1. Kinematic and stress states of the billet

2021 ◽  
pp. 65-71
Author(s):  
A.L. Vorontsov
Keyword(s):  
Plastic Flow ◽  
Complete System ◽  
Shape Change ◽  
Plane Deformation ◽  
System Of Equations ◽  
Flow Rates ◽  
Die Forging ◽  
The Matrix ◽  
Stress States ◽  
Theory Of Plastic Flow

On the basis of the complete system of equations of the theory of plastic flow, the kinematic and stress states of the billet are determined when the channels are extruded under conditions of plane deformation of the misaligned position of the punch and the matrix. Keywords: die forging, extrusion, misaligned position, punch, matrix, plane deformation, plastic flow rates, stresses. [email protected]

Research of stamping of unequal channel bars. Part 3. Force parameters and shape change of the billet when extruding channels. 2. Determination of the extrusion force, maximum pressure on the matrix wall and the heights of the walls, taking into account the elastic deformation jf the matrix

2021 ◽  
pp. 63-69
Author(s):  
A.L. Vorontsov
Keyword(s):  
Plastic Flow ◽  
Elastic Deformation ◽  
Shape Change ◽  
Plane Deformation ◽  
Maximum Pressure ◽  
Extrusion Force ◽  
System Of Equations ◽  
The Matrix ◽  
Theory Of Plastic Flow

On the basis of the system of equations of the theory of plastic flow, the forces, the maximum pressure on the wall of the matrix and the heights of the obtained walls when extruding channels are determined, taking into account the elastic deformation of the matrix. Keywords: die forging, extrusion, misalignment, punch, matrix, plane deformation, stresses. [email protected]

scholarly journals Numerical Implementation of the Fictitious Domain Method for Elliptic Equations

2014 ◽  
Vol 60 (3) ◽  
pp. 219-223 ◽  
Author(s):  
Almas N. Temirbekov ◽  
Waldemar Wójcik
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Iterative Process ◽  
Computational Algorithm ◽  
Fictitious Domain ◽  
Special Method ◽  
Numerical Implementation ◽  
Fictitious Domain Method ◽  
Varying Coefficients ◽  
Domain Method

Abstract In this paper, we consider an elliptic equation with strongly varying coefficients. Interest in the study of these equations is connected with the fact that this type of equation is obtained when using the fictitious domain method. In this paper, we propose a special method for the numerical solution of elliptic equations with strongly varying coefficients. A theorem is proved for the rate of convergence of the iterative process developed. A computational algorithm and numerical calculations are developed to illustrate the effectiveness of the proposed method.

Singular analysis of stereoscopic serif

2021 ◽  
Vol 966 (12) ◽  
pp. 21-30
Author(s):  
E.G. Voronin
Keyword(s):  
System Of Equations ◽  
Stereo Images ◽  
Geometric Conditions ◽  
Basis Plane ◽  
The Matrix ◽  
Spatial Coordinates ◽  
Singular Numbers ◽  
Singular Analysis

The article is devoted to the development of a methodological apparatus for evaluating the influence of geometric survey conditions on the ratio of errors in the planned and altitude components of determining the terrain points’ spatial coordinates. For this purpose, an approach based on the singular analyzing the matrix of the system of equations for intersecting lines in the basis plane is used. It is shown that singular numbers characterize the ellipse of coordinate definition errors in plane and height. They are calculated for the most typical cases of stereo surveying. It is noted that the geometric conditions of stereo photography are mainly determined by the serif angle. By interpolating the results of calculating singular numbers, the formulas characterizing the direct and inverse dependence of the ratio of the stereo-cut in plan and height on the serif angle errors’ are obtained. The author considers the practical issues of using the developed methodological apparatus to justify the solutions related to the assessment of the ratio between the errors in determining the plane coordinates and the heights of terrain points from stereo images.

scholarly journals A Note on Full-Rank Factorization of Matrix

2019 ◽  
Vol 15 (2) ◽  
pp. 152-154
Author(s):  
Gyan Bahadur Thapa ◽  
J. López-Bonilla ◽  
R. López-Vázquez
Keyword(s):  
Singular Value Decomposition ◽  
Full Rank ◽  
Singular Value ◽  
The Matrix ◽  
Rank Factorization ◽  
Value Decomposition

We exhibit that the Singular Value Decomposition of a matrix Anxm implies a natural full-rank factorization of the matrix.

scholarly journals A New Model for Fast Analysis of Leveling Process

2012 ◽  
Vol 586 ◽  
pp. 389-393 ◽  
Author(s):  
Jindřich Petruška ◽  
Tomas Návrat ◽  
František Šebek
Keyword(s):  
Finite Element Method ◽  
Iterative Process ◽  
Direct Problem ◽  
Direct Solution ◽  
The Finite Element Method ◽  
Fast Analysis ◽  
Plastic Bending ◽  
Optimal Setting ◽  
Reliable Solution ◽  
User Friendly

The paper deals with numerical analysis of the process of roller leveling of long products. The problem of multiple elastic plastic bending is solved by a program in Matlab, which is based on the Finite Element Method. The aim is to provide a simple, user friendly tool, capable of quick and reliable analysis of the process without the necessity to work with large multipurpose FEM packages. Direct solution of the problem is formulated, starting from the roll intermeshing and product geometry. Roller loading, product deflection, curvature and plastification are the output parameters of the program. With fast and reliable solution of the direct problem, optimal setting of the leveling process will be sought in an iterative process.

scholarly journals On exponential decay for linear porous-thermo-elasticity system

2012 ◽  
Vol 45 (4) ◽  
Author(s):  
Przemysław Głowiński ◽  
Andrzej Łada
Keyword(s):  
Exponential Decay ◽  
Original System ◽  
System Of Equations ◽  
Elasticity System

AbstractWe study the problem of exponential decaying for solutions of porous-thermoelasticity system, when timeIn the considerations we apply the idea of compact decoupling for the system of equations. The exponential decaying property is proved first for the corresponding decoupled system, which is simpler to handle, then the property is derived for the original system.

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