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CHAPTER 1. TRANSCENDENT EQUATIONS
1.1. Root separation
1.2. Dichotomy method
1.3. Chord method
1.4. Newton's method (tangent method)
1.5. Secant method
1.6. Simple iteration method
CHAPTER 2. LINEAR ALGEBRA PROBLEMS
2.1. Gaussian method with selection of the main element for solving SLAE
2.2. Iterative methods for solving SLAEs
2.3. Calculation of determinants
2.4. Calculating the elements of an inverse matrix
2.5. Calculation of eigenvalues of matrices
CHAPTER 3. DEPENDENCY INTERPOLATION
3.2. Lagrange interpolation polynomial
3.3. Newton's interpolation polynomial
3.4. .Using interpolation to solve equations
3.5. Interpolation method for determining matrix eigenvalues
3.6. Spline interpolation
CHAPTER 4. Least SQUARE METHOD
4.1. General algorithm
4.2. Power basis
4.3. Basis in the form of classical orthogonal polynomials
4.4. Basis in the form of orthogonal polynomials of discrete variable function
4.5. Linear version of OLS
4.6. Differentiation when approximating dependencies by least squares method
CHAPTER 5. DEFINITION OF AN INTEGRAL
5.1. Classification of methods
5.2. Rectangle Methods
5.3. A posteriori error estimates according to Runge and Aitken
5.4. Trapezoid method
5.5. Simpson method
5.6. Calculation of integrals with a given accuracy
5.7. Application of splines for numerical integration
5.8. Methods for the highest algebraic accuracy
5.9. Improper integrals
5.10. Monte Carlo methods
CHAPTER 6. CAUCHY PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS
6.1. Types of problems for ordinary differential equations
6.2. Euler method
6.3. Second order Runge-Kutta methods
6.6. Adams method
6.7. Geer's method
CHAPTER 7. BOUNDARY PROBLEMS
7.1. Finite difference method for linear boundary value problems
7.2. Shooting method for boundary value problems
7.3. Boundary value eigenvalue problems for ordinary differential equations
7.4. Shooting method for eigenvalue problem
7.5. Finite difference method for eigenvalue problems
7.6. Boundary value problem for differential equation in partial derivatives
CHAPTER 8. UNCONDITIONAL OPTIMIZATION OF FUNCTIONS
8.1. Golden ratio method
LIST OF PROGRAMS
1.1. Tabular method of root separation
1.2. Dichotomy method
1.3. Chord method
1.4. Newton's method
1.5. Newton's method in the complex domain
1.6. Secant method
1.7. Simple iteration method
2.1. Gaussian method for SLAE
2.2. Seidel method for SLAE
2.3. Calculation of Gaussian determinants
2.4. Matrix inversion
2.5. Direct method for calculating matrix eigenvalues
2.6. Iterative method for calculating the largest eigenvalue
3.1. Interpolation by canonical polynomial
3.2. Lagrange polynomial and its derivatives
3.3. Newton's polynomial and its derivatives
3.4. Parabola method
3.5. Interpolation method for calculating matrix eigenvalues
3.6. Spline interpolation
4.1. MLS with power basis
4.2. Gram matrix with power basis
4.3. OLS with an arbitrary basis
4.4. LSM with orthogonal basis
4.5. Linear version of OLS
4.6. Calculation of derivatives
5.1. Middle rectangle method
5.2. Trapezoid method
5.3. Simpson method
5.4. Simpson method with error estimation
5.5. Spline quadrature
5.6. Gaussian method with two nodes
5.7. Gaussian method with six nodes
5.8. Hermite quadrature with five nodes
5.9. Monte Carlo method
6.1. Euler method
6.2. Second order Runge-Kutta method with mean derivative correction
6.3. Second order Runge-Kutta method with midpoint correction
6.4. Fourth order Runge-Kutta method
6.5. Runge-Kutta-Merson method
6.6. Adams method
6.7. Geer's method
7.1. Finite difference method for linear boundary value problem
7.2. Shooting method for linear boundary value problem
7.3. Shooting method for eigenvalue problem
7.4. Finite difference method for eigenvalue problems
7.5. Drichlet's problem for Laplace's equation
8.1. Golden ratio method
8.2. Coordinate descent method
8.3. Gradient descent method
Brief summary of the book
The basic methods and algorithms of computational mathematics are outlined. The features of their software implementation on personal computers are considered. Descriptions and listings of about 150 programs in BASIC, Fortran and Pascal are provided. Parallel texts of programs in three languages will be useful to readers who speak one of them for the practical development of the other two. For scientific, engineering and technical workers of various specialties; may be useful for university students studying programming.
Personal computers (PCs) are widely being introduced into science and technology, education, management activities, technological processes, etc. The effectiveness of using PCs is primarily associated with software, both with the availability of ready-made packages of system and general programs, and with the user’s ability to adapt them to solve specific problems.
Mathematical modeling of processes and phenomena in various fields of science and technology is one of the main ways to obtain new knowledge and technological solutions. To implement mathematical modeling a researcher, regardless of his specialty, must know a certain minimum set of computational mathematics algorithms, as well as master the methods of their software implementation on a personal computer. Such knowledge and skills are also necessary when using ready-made software packages, otherwise planning a computational experiment and interpreting its results will be difficult.
Currently, there is extensive literature on computational methods and programming in algorithmic languages. However, a relatively small number of publications combine these two areas.
Of the books on computational mathematics with universal content, intended for persons who are not specialists in this field, we note, in which the accessibility of the presentation is combined with sufficient rigor and practical orientation of the algorithms presented. Its popularity among scientists and engineers is evident in numerous references to it in scientific publications related to computational experiments in mathematical modeling in various fields of science and technology. IN last years A number of books have been published, which present a wide range of methods and algorithms, as well as works in which individual sections of computational mathematics are given in more depth.
Among the books that combine the presentation of computational algorithms with their implementation in the BASIC language, we note, and in the Fortran language -. The author is not aware of similar works with programs in the Pascal language, where methods of computational mathematics would be systematically presented.
When working on a PC, the programming languages BASIC, Fortran and Pascal are widely used, each of which has certain advantages and disadvantages.
Thus, BASIC is characterized by weak structure, relatively slow speed of execution of programs of computational algorithms, the ability side effects due to the "overlapping" of variables in subroutines. But at the same time, BASIC programs are distinguished by readability and visibility, brevity and the presence of an interactive mode, the convenience of directly making additions and corrections without the use of editor programs and recompilation of the program. Such features allow you to use BASIC to implement relatively simple algorithms, as well as when checking and debugging individual fragments of complex algorithms and programs.
Fortran is characterized by insufficient structure, the presence of many archaisms preserved from the times of the first computers, uncontrolled declarations and the introduction of new default variables. But at the same time, a wealth of experience in using the language has been accumulated and extensive software packages have been created for solving applied problems, system mathematical software and, in particular, optimizing compilers for using Fortran on different computers have been developed. Scientists and engineers are attracted to Fortran because of its ease of working with complex variables and functions.
In teaching programming and practicing the use of personal computers, the Pascal language is now widely used due to its structure, clear and unambiguous grammar, and ease of working with file structures. However, some cumbersomeness of writing programs due to the need to describe all the objects used, insufficient development of problem-solving mathematical software, and the lack of optimizing compilers on some PCs are obstacles in solving problems of mathematical modeling in the Pascal language.
Due to the specified features of programming languages in different stages For solving applied problems, it can be advantageous to use different languages or combine them at one stage when programming parts of one problem. Since each language has its own set of tools for software implementation of algorithms, “literal” translation of programs from one language to another is not always possible. The same algorithm must be written in each programming language using its own visual arts. Here a situation arises similar to translating a text from one natural language to another.
In this book, classical methods of computational mathematics are illustrated with parallel programs in BASIC, FORTRAN and Pascal. In total there are about 150 completed programs. The programs were compiled so that they were easy to read and modernize, and used as a basis for development software systems. Without any special difficulties, programs can be adapted to other types of PCs. In programs, where possible without compromising readability and simplicity, the number of variables and operators used is minimized, and the text of each section provides summary computational method and the problem used for the example, the information necessary to transfer the method algorithm to the program is given, and a generalized block diagram of the program is considered. More detailed descriptions are given for programs in the BASIC language, where attention is drawn to the “pitfalls” and the logic of using certain constructions is explained. In explanations of programs in Fortran and Pascal, attention is drawn only to distinctive features from BASIC programs.
A reader who knows one of these programming languages will be able to practically master the other two with the help of this book.
The first chapter discusses methods and algorithms for separating and refining the roots of transcendental equations with parameters. As examples, equations containing special functions of mathematical physics are used, including Bessel functions, elliptic integrals, the logarithmic derivative of the y-function, Fresnel integrals, and the probability integral. Routines for calculating these functions can be used as independent routines separately from routines for solving equations. The first chapter shows how to implement calculations with complex variables on different languages programming.
The second chapter discusses exact and iterative methods for solving systems of linear algebraic equations, calculation of determinants, inverse matrices, finding the eigenvalues of matrices.
The third chapter provides algorithms and programs for interpolation by polynomials and splines. Practical methods for numerical differentiation of approximating functions, the use of interpolation to solve equations and calculate eigenvalues of matrices are considered.
The fourth chapter outlines various options for the method least squares, used for processing experimental data, smoothing and differentiating dependencies, and reducing the amount of numerical information. Programs for the method with a power basis, a basis in the form of classical orthogonal polynomials and polynomials of a discrete variable, and a linear version of the method are given.
The fifth chapter contains a description of the most common methods of calculation definite integrals and programs that implement interpolation methods, methods of the highest algebraic accuracy and statistical tests are given.
The sixth chapter discusses algorithms for solving the Cauchy problem for a system of ordinary differential equations. Programs of Runge-Kutta methods of different orders are given, among which there is a version of the method with automatic selection of the integration step. Among the multipoint methods, the Adams and Geer methods of the forecast-correction type were selected.
The seventh chapter is devoted to methods for solving boundary value problems for ordinary differential equations and partial differential equations. Programs for shooting and finite difference methods are proposed for boundary value and eigenvalue problems. As an example of problems of the last class, we consider the problem of propagation electromagnetic waves in a coaxial waveguide structure.
In the eighth chapter, programs for elementary methods for unconditional minimization of functions of one and many variables are developed.
This book is intended for scientific, engineering and technical workers who are not specialists in the field of programming and computational mathematics, who want to pose and solve applied problems using a PC. The author does not claim completeness of coverage and depth of presentation of the selected methods; the material considered should be considered an introduction to the vast world of computational mathematics.
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Tomsk: MP "RASKO", 1991. - 272 p.
The basic methods and algorithms of computational mathematics are outlined. The features of their software implementation on personal computers are considered. Detailed descriptions and listings of about 150 programs in BASIC, Fortran and Pascal are provided. Parallel texts of programs in three languages will be useful to readers who speak one of them for the practical development of the other two.
_Transcendental equations.
Root separation.
Dichotomy method.
Chord method
Newton's method (tangent method).
Secant method.
Simple iteration method.
_Problems of linear algebra.
Gaycca method with selection of the main element.
Iterative methods for solving SLAEs.
Calculation of determinants.
Calculation of elements of the inverse matrix.
Calculation of eigenvalues of matrices.
_Dependency interpolation.
Interpolation by canonical polynomial.
Lagrange interpolation polynomial.
Newton's interpolation polynomial.
Using interpolation to solve equations.
Interpolation method for determining the eigenvalues of a matrix.
Spline interpolation.
_Least square method.
General algorithm.
Power basis.
Basis in the form of classical orthogonal polynomials.
Basis in the form of orthogonal polynomials of a discrete variable function.
Linear version of OLS.
Differentiation in approximation of least squares dependences.
_Definite integrals.
Classification of methods.
Rectangle methods.
Posterior error estimates according to Runge and Aitken.
Trapezoid method.
Simpson's method.
Calculation of integrals with a given accuracy.
Application of splines for numerical integration.
Methods of the highest algebraic accuracy.
Improper integrals.
Monte Carlo methods.
_Cauchy problem for ordinary differential equations.
Types of problems for ordinary differential equations.
Euler's method.
Second order Runge-Kutta methods.
Fourth order Runge-Kutta method.
Runge-Kutta-Merson method.
Adams method.
Geer's method.
_Boundary problems.
Finite difference method for linear boundary value problems.
Shooting method for boundary problems.
Boundary value eigenvalue problems for ordinary differential equations.
Shooting method for eigenvalue problem.
Finite difference method for eigenvalue problems.
Boundary value problem for a partial differential equation.
_Unconditional optimization of functions.
Golden section method.
Coordinate descent method.
The gradient descent method. You may be interested in a book with a similar structure:
Pao Y.C. Engineering Analysis: Interactive Methods and Programs with FORTRAN,
QuickBASIC, MATLAB, and Mathematica