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Numerical Methods

Jeffrey R. Chasnov

Adapted for:

Numerical Methods for EngineersClick to view a promotional video The Hong Kong University of Science and Technology

Department of Mathematics

Clear Water Bay, Kowloon

Hong KongCopyright

c○2012 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Preface

What follows were my lecture notes for Math 3311:Introduction to Numerical Meth- ods, taught at the Hong Kong University of Science and Technology. Math 3311, with two lecture hours per week, was primarily for non-mathematics majors and was required by several engineering departments. I also have some free online courses on Coursera. A lot of time and effort has gone into their production, and the video lectures for these courses are of high quality. You can click on the links below to explore these courses. If you want to learn differential equations, have a look at

Differential Equations for Engineers

If your interests are matrices and elementary linear algebra, try

Matrix Algebra for Engineers

If you want to learn vector calculus (also known as multivariable calculus, or calcu- lus three), you can sign up for

Vector Calculus for Engineers

And if your interest is numerical methods, have a go at

Numerical Methods for Engineers

JeffreyR. Chasnov

Hong Kong

February 2021

iii

1 IEEE Arithmetic

1.1 Definitions

1.2 Numbers with a decimal or binary point

1.3 Examples of binary numbers

1.4 Hex numbers

1.5 4-bit unsigned integers as hex numbers

1.6 IEEE single precision format:

1.7 Special numbers

1.8 Examples of computer numbers

1.9 Inexact numbers

1.9.1 Find smallest positive integer that is not exact in single precision

1.10 Machine epsilon

1.11 IEEE double precision format

1.12 Roundoff error example

2 Root Finding

2.1 Bisection Method

2.2 Newton"s Method

2.3 Secant Method

2.3.1 Estimatep2=1.41421356 using Newton"s Method. . . . . . . 8

2.3.2 Example of fractals using Newton"s Method

2.4 Order of convergence

2.4.1 Newton"s Method

2.4.2 Secant Method

3 Systems of equations

3.1 Gaussian Elimination

3.2LUdecomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Partial pivoting

3.4 Operation counts

3.5 System of nonlinear equations

4 Least-squares approximation

4.1 Fitting a straight line

4.2 Fitting to a linear combination of functions

5 Interpolation

5.1 Polynomial interpolation

5.1.1 Vandermonde polynomial

5.1.2 Lagrange polynomial

5.1.3 Newton polynomial

5.2 Piecewise linear interpolation

5.3 Cubic spline interpolation

5.4 Multidimensional interpolation

33
v

6 Integration

6.1 Elementary formulas

6.1.1 Midpoint rule

6.1.2 Trapezoidal rule

6.1.3 Simpson"s rule

6.2 Composite rules

6.2.1 Trapezoidal rule

6.2.2 Simpson"s rule

6.3 Local versus global error

6.4 Adaptive integration

7 Ordinary differential equations

7.1 Examples of analytical solutions

7.1.1 Initial value problem

7.1.2 Boundary value problems

7.1.3 Eigenvalue problem

7.2 Numerical methods: initial value problem

7.2.1 Euler method

7.2.2 Modified Euler method

7.2.3 Second-order Runge-Kutta methods

7.2.4 Higher-order Runge-Kutta methods

7.2.5 Adaptive Runge-Kutta Methods

7.2.6 System of differential equations

7.3 Numerical methods: boundary value problem

7.3.1 Finite difference method

7.3.2 Shooting method

7.4 Numerical methods: eigenvalue problem

7.4.1 Finite difference method

7.4.2 Shooting method

53
vi CONTENTS

Chapter 1 IEEE Arithmetic

1.1

Definitions

Bit = 0 or 1

Byte = 8 bits

Word = Reals: 4 bytes (single precision)

8 bytes (double precision)

= Integers: 1, 2, 4, or 8 byte signed

1, 2, 4, or 8 byte unsigned

1.2

Numbers with a decimal or binar ypoint

Decimal: 10

3102101100101102103104

Binary: 2

322212021222324

1.3

Examples of binar ynumbers

Decimal Binary

1 1 2 10 3 11 4 100

0.5 0.1

1.5 1.1

1.4

Hex numbers

1.5

4-bit unsigned integers as hex numbers

Decimal Binary Hex

1 0001 1

2 0010 2

3 0011 3

10 1010 a

15 1111 f

1.6. IEEE SINGLE PRECISION FORMAT:

1.6

IEEE single precision for mat:

quotesdbs_dbs14.pdfusesText_20

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