Numerical Methods. Workbook. Tomsk: TPU Press, 2002, 41 pp.
Author: J.J. Katsman, Associate Professor, Ph.D.
Reviewed by: V.G. Spitsyn, Professor of the Automation and Computer Technology Department, TPU, D.Sc.
COMPUTER LABS. 36
THE LABORATORY TASKS. 37
Numerical methods are techniques (tools) by which mathematical problems are formulated so that they can be solved with a set of arithmetic operations. This workbook brings together some cornerstones of numerical methods. The material in this workbook is presented in the same order and with the same section numbers as in the textbook.
· Error and accuracy of calculations;
· Numerical integration: Newton - Cotes formulas, Monte Carlo integration;
· Linear algebra applications: Gaussian elimination with backsubstitution, Jacobi iteration, and more;
· Roots of equations: bracketing methods and method for nonlinear systems of equations;
· Ordinary differential equations: Euler's method, multistep methods, and more;
· Interpolation and extrapolation of functions: Newton and Lagrange polynomials, inverse interpolation, and spline interpolation.
Using a "learn by example" approach, this exploration of the fundamental numerical methods covers both modern and older, well-established techniques that are well-suited to the digital-computer solution of problems in many areas of science and engineering. A variety of examples are used to illustrate these applications to science and engineering problems.
Number of Credits:3 (about 150 hours of study)
Number of Weeks:16 (Full-time, distance learning)
The aim of the course is to provide a solid foundation in both applied and computational mathematical science with areas of application. The course shifts the emphasis away from rote symbol manipulation, and places more emphasis on numerical methods, with the goal of preparing you to make better use of calculus as a practical problem-solving tool.
TextThe textbook covers the course to be studied during one semester. It is intended for students who specialize mainly in Computer Science to help in understanding the fundamental principles of graphical and numerical methods. The book was written using many tools provided by the Open Source Community.
LabsFor one session each week you will meet in the Calculus (Engineering) Computer Lab. In your labs, you will work individually on computer assignments and projects, using the computer programming languages (Pascal, C++). Much of your work will be done in the university computer labs, but some of you may choose to do some of your work on your personal computers. As you work, you should discuss your results with your instructor and group mates.
You will do your lab work on PC, which allows you to integrate programs and results of calculations with fully word-processed text to produce a polished report.
HomeworkEach week, you will be assigned some problems to write up and hand in. These assignments will be graded and returned. In addition, you will be provided with a list of practice problems to do, even though they will not be handed in and graded. Exam problems will sometimes be modeled on these practice problems.
Once a week you will meet with your instructor in a classroom (without computers) to go over problems related to the material covered the previous week.
Contents of the Course
1. Error and Accuracy of Calculations
The errors associated with both calculations and measurements can be characterized with regard to their accuracy and precision. Numerical errors arise from the use of approximations to represent exact mathematical operations and quantities. These include truncation errors, which result when approximations are used to represent exact mathematical procedures, and round-off errors, which result when numbers having limited significant figures are used to represent exact numbers.
2. Numerical Integration
The Newton - Cotes formulas are the most common numerical integration schemes. Closed and open forms of the Newton - Cotes formulas are available. Some of the most widely used formulas are offered in detail: midpoint, trapezoidal and Simpson's rules. Here we have presented the error estimates for each rule. Simple Monte Carlo integration methods are discussed in detail.
3. Solution of Linear Algebraic Equations
Linear sets can be routinely solved by direct and iterative numerical methods: Cramers rule, Gaussian elimination with back substitution, Jacobi iteration, Gauss-Siedel iteration. Compute determinant of a matrix. Compute the inverse matrix.
4. Solution Methods of Non-Linear Equations
This chapter is devoted to bracketing methods for finding roots. Graphical methods are used to provide visual insight into the root location. Here we have presented some simple methods: bisection, secant and false position. Newton-Raphson method using derivative is discussed also. Newton-Raphson method is for finding roots of nonlinear systems of equations.
5. Integration of Ordinary Differential Equations
Solution of the Cauchy problem. We shall consider some major types of practical numerical methods for solving ODE¢s: Euler method, Runge-Kutta methods, and predictor-corrector methods.
6. Interpolation and Extrapolation of Functions
Interpolation and extrapolation schemes must model the function, between or beyond the known points. Polynomial interpolation: Gregory-Newton interpolating polynomial, Newton's interpolatory divided-difference formula, Lagrange interpolating polynomial. Inverse interpolation. Spline functions: linear splines, quadratic splines and cubic splines.