Numerical Methods I

2021 ◽  
Author(s):  
Boris Obsieger
Keyword(s):  
Numerical Methods ◽  
Computer Program ◽  
Iterative Methods ◽  
Error Propagation ◽  
Random Variables ◽  
Regression Analyses ◽  
Practical Application ◽  
Numeral Systems ◽  
Roots Of Polynomials ◽  
Digital Computers

Textbook of several universities. 2nd edition. The color edition is also available at Glasstree Bookstore. It is recommended for students. The series of books Numerical Methods is written primarily for students at technical universities, but also as a useful handbook for engineers, PhD students and scientists. This volume introduces the reader into numeral systems and representation of numbers in digital computers. Possibly the most important part of this book are descriptions of differences between constant and random variables, related types of errors and error propagation. These topics are supplemented with various types of regression analyses. Finally, direct and iterative methods for finding roots of polynomials are explained. Practical application is supported by 77 examples and 13 algorithms. For reasons of simplicity, algorithms are written in pseudo-code, so they can easily be included in any computer program.

Numerical Methods I

2021 ◽  
Author(s):  
Boris Obsieger
Keyword(s):  
Numerical Methods ◽  
Direct Methods ◽  
Historical Background ◽  
Random Errors ◽  
Practical Application ◽  
Fourth Degree ◽  
Numerical Errors ◽  
The Third ◽  
Numeral Systems ◽  
The Given

Textbook, established at several universities. Second edition. *** Written primarily for students at technical studies. Valuable handbook for engineers, PhD students and scientists. *** Published in several variants. *** Seven chapters. In the first chapter, a historical background and basic properties of various numeral systems, as well as conversion of numbers from one system to another are briefly explained. In the second chapter, numbers in digital computers, namely integers and floating point numbers are described. This helps the reader to choose precision and range limits of stored numbers. The third chapter explains constant variables and related numerical errors, including error propagation and algorithm instability. The fourth and fifth chapters explain random variables and related random errors, uncertainty, confidence level, as well as propagation of random errors. Various types of regression analyses of experimental data are described in the sixth chapter. Direct methods for finding roots of the third and fourth degree polynomials are described in the seventh chapter, followed by general iterative methods for polynomials of any degree. *** Why the presented topics are so important? Simply, they are common to all numerical methods. *** Practical application is supported by 84 examples and 17 algorithms. For reasons of simplicity, algorithms are written in pseudo-code, so they can easily be implemented in any computer program. Finally, the given text with 98 figures and 52 tables represents a valuable background for understanding, applying and developing various numerical analyses.

scholarly journals A Neural Network-Based Approach for Approximating Arbitrary Roots of Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 317
Author(s):  
Diogo Freitas ◽  
Luiz Guerreiro Lopes ◽  
Fernando Morgado-Dias
Keyword(s):  
Iterative Methods ◽  
Fundamental Problem ◽  
Iterative Algorithms ◽  
Rounding Errors ◽  
Suggested Approach ◽  
Simultaneous Iterative Methods ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Comparative Results ◽  
The One

Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.

scholarly journals Application of quasiconformal mapping methods to the investigation of the explosive processes impact on the environment

2019 ◽  
pp. 17-18
Author(s):  
Kateryna Mykolaiivna Malash ◽  
Andrii Yaroslavovych Bomba
Keyword(s):  
Numerical Methods ◽  
Mathematical Models ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Practical Application

The mathematical models used to study explosive processes are given. A class of problems investigating the influence of explosive processes on the environment by the quasiconformal mappings numerical methods are outlined and their practical application are described

scholarly journals Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Computer Program ◽  
Order Statistics ◽  
Stock Exchange ◽  
Random Variables ◽  
Integral Representations ◽  
Symmetric Distribution ◽  
Random Vectors ◽  
Dependent Random Variables ◽  
Exact Distributions

AbstractIntegral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.

The Application Research of Rough Set on Customer Risk

2010 ◽  
Vol 20-23 ◽  
pp. 731-734
Author(s):  
Hong Jun Guan
Keyword(s):  
Risk Management ◽  
Supply Chain ◽  
Supply Chain Management ◽  
Computer Program ◽  
Rough Set ◽  
Relational Theory ◽  
Application Research ◽  
Practical Application ◽  
Chain Management

As an important role of supply chain, management impact deeply on the efficiency of the supply chain. This article first introduced the relational theory of rough set, then described the application on the customer risk management based on the analysis of a large number of historical transactions, gave the realization of computer program, and detailed the practical application. It has greatly proved the effective of the theory.

Some simple properties of sums of random variables having long-range dependence

1989 ◽  
Vol 424 (1867) ◽  
pp. 255-262 ◽  
Keyword(s):  
Numerical Methods ◽  
Long Range ◽  
Random Variables ◽  
Higher Order ◽  
Long Range Dependence ◽  
Finite Variance ◽  
Sums Of Random Variables ◽  
Tensor Methods ◽  
Non Gaussian ◽  
Special Case

The higher-order moments and cumulants of sums of a special case of random variables having long-range dependence are investigated. Tensor methods are used to simplify the calculations. The limiting form of the second and third cumulants as the number of variables added becomes large is studied by analytical and numerical methods. The implications are discussed for the existence of non-gaussian limits of sums of random quantities of finite variance and long-range dependence.

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