Tomsk Polytechnic University

NUMERICAL METHODS

Workbook

Tomsk 2002

UDС 519.6(075.8)

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.

Contents

UDС 519.6(075.8) 2

Reviewed by: V.G. Spitsyn, Professor of the Automation and Computer Technology Department, TPU, D.Sc. 2

CONTENTS 3

INTRODUCTION 5

SYLLABUS 6

RATING 9

TIMING 10

PRACTISE 11

COMPUTER LABS 36

THE LABORATORY TASKS 37

with a subroutine of your choice accurate to four significant digits. Test cases: x = 0.5, x = 1. This integral occurs frequently in statistics (e.g., the normal distribution function). Try some test cases to make sure that your answer is correct. Use the trapezoidal rule to determine this integral with different numbers of subintervals. 37

Variant 3. Numerically evaluate the following definite integral accurate to 4 significant digits. 37

Use the Simpson's rule. Try some test cases to make sure that your answer is correct. 37

Variant 4. Evaluate the integral from Var. 3 using Simple Monte Carlo approximation. 37

Write a program to solve a set of linear algebraic equations of the form A·x = b, where A is a matrix of an arbitrary size specified by the user, b is an input vector, and x is the solution. 37

37

Solve this system by Gaussian elimination with back substitution. Substitute your results into the original equations to check your answers. 37

Variant 2. Solve the linear system from Var. 1 using Jacobi iteration. Check your results. 38

Variant 3. Solve the linear system from Var. 1 using Gauss - Seidel iteration. Substitute your results back into the original equations to verify your solution. 38

38

Use the Cramer's rule to solve for the x's. Compute the determinants using Gaussian elimination. Substitute your results back into the original equations to verify your solution. 38

Variant 5. Solve the linear system from Var. 4. Find A^-1 using Gaussian elimination with back substitution and finally the solution x. Multiply the inverse by the original coefficient matrix and assess whether the result is close to the identity matrix. Substitute your results back into the original equations to verify your solution. 38

Variant 1. Find the real root of using the bisection method. 38

Variant 2. Find the real root of using the secant method. 38

Variant 3. Find the real root of using the false position method. 38

Variant 4. Find the real root of using the Newton-Raphson method. 38

Reviewed by: V.G. Spitsyn, Professor of the Automation and Computer Technology Department, TPU, D.Sc. 41

Introduction

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.

Coverage includes:

• 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.