MTH603 Handouts Set 2 by VUBuddy

Table of Contents · Lecture # · Topics · Page # · Lecture 1 · Lecture 2 · Lecture 3

Solution of Non Linear Equations (Bisection Method)

Lecture 4

Solution of Non Linear Equations (Regula-Falsi Method)

Lecture 5

Solution of Non Linear Equations (Method of Iteration)

Lecture 6

Solution of Non Linear Equations (Newton Raphson Method)

Lecture 7

Solution of Non Linear Equations (Secant Method)

Lecture 8

Muller's Method

Lecture 9

Solution of Linear System of Equations (Gaussian Elimination Method)

Lecture 10

Solution of Linear System of Equations(Gauss–Jordon Elimination Method)

Lecture 11

Solution of Linear System of Equations(Jacobi Method)

Lecture 12

Solution of Linear System of Equations(Gauss–Seidel Iteration Method)

Lecture 13

Solution of Linear System of Equations(Relaxation Method)

Lecture 14

Solution of Linear System of Equations(Matrix Inversion)

Lecture 15

Eigen Value Problems (Power Method)

Lecture 16

Eigen Value Problems (Jacobi’s Method)

Lecture 17

Eigen Value Problems (continued)

Lecture 18

Interpolation(Introduction and Difference Operators)

Lecture 19

Interpolation(Difference Operators Cont.)

Lecture 20

Interpolation( Operators Cont.)

Lecture 21

Interpolation Newton’s Forward difference Formula

Lecture 22

Newton’s Backward Difference Interpolation Formula

Lecture 23

Lagrange’s Interpolation formula

Lecture 24

Divided Differences

Lecture 25

Lagrange’s Interpolation formula, Divided Differences (Examples)

Lecture 26

Error Term in Interpolation Formula

Lecture 27

Differentiation Using Difference Operators

Lecture 28

Differentiation Using Difference Operators (continued)

Lecture 29

Differentiation Using Interpolation

Lecture 30

Richardson’s Extrapolation Method

Lecture 31

Numerical Differentiation and Integration

Lecture 32

Numerical Differentiation and Integration(Trapezoidal and Simpsons Rules)

Lecture 33

Numerical Differentiation and Integration(Trapezoidal and Simpsons Rules)Continued

Lecture 34

Numerical Differentiation and Integration(Rombergs Integration and Double

integration)Continued

Lecture 35

Ordinary Differential Equations (Taylo's series method)Euler Method

Lecture 36

Ordinary Differential Equations (Euler Method)

Lecture 37

Ordinary Differential Equations (Runge-Kutta Method)

Lecture 38

Ordinary Differential Equations (Runge-Kutta Method)Continued

Lecture 39

Ordinary Differential Equations(Adam-Moultan’s Predictor-Corrector Method)

Lecture 40

Ordinary Differential Equations(Adam-Moultan’s Predictor-Corrector Method)

Lecture 41

Examples of Differential Equations

Lecture 42

Examples of Numerical Differentiation

Lecture 43

An Introduction to MAPLE

Lecture 44

Algorithms for method of Solution of Non-linear Equations

Lecture 45

Non-linear Equations Course Contents

Solution of Non Linear Equations

Solution of Linear System of Equations

Approximation of Eigen Values

Interpolation and Polynomial Approximation

Numerical Differentiation

Numerical Integration

Numerical Solution of Ordinary Differential Equations

We begin this chapter with some of the basic concept of representation of numbers on

computers and errors introduced during computation. Problem solving using computers and the

steps involved are also discussed in brief.

Number (s) System (s)

In our daily life, we use numbers based on the decimal system. In this system, we use ten

symbols 0, 1,…,9 and the number 10 is called the base of the system.

Thus, when a base N is given, we need N different symbols 0, 1, 2, …,(N – 1) to represent an

arbitrary number.

The number systems commonly used in computers are

Base, N Number

2 Binary

8 Octal

10 Decimal

16 Hexadecimal

An arbitrary real number, a can be written as

m m m

m m m a a N a N a N a a N a N

In binary system, it has the form,

m m m

m m m a a a a a a a

The decimal number 1729 is represented and calculated

(1729)

While the decimal equivalent of binary number 10011001 is

(1.1953125)

Electronic computers use binary system whose base is 2. The two symbols used in this system

are 0 and 1, which are called binary digits or simply bits.

The internal representation of any data within a computer is in binary form. However, we prefer

data input and output of numerical results in decimal system. Within the computer, the

arithmetic is carried out in binary form.

Conversion of decimal number 47 into its binary equivalent

Sol. Numerically, computed solutions are subject to certain errors. It may be fruitful to

identify the error sources and their growth while classifying the errors in numerical

computation. These are

Inherent errors,

Local round-off errors · Local truncation errors · Inherent errors

It is that quantity of error which is present in the statement of the problem itself, before

finding its solution. It arises due to the simplified assumptions made in the mathematical

modeling of a problem. It can also arise when the data is obtained from certain physical

measurements of the parameters of the problem.

Local round-off errors

Every computer has a finite word length and therefore it is possible to store only a fixed

number of digits of a given input number. Since computers store information in binary

form, storing an exact decimal number in its binary form into the computer memory gives

an error. This error is computer dependent.

At the end of computation of a particular problem, the final results in the computer,

which is obviously in binary form, should be converted into decimal form-a form

understandable to the user-before their print out. Therefore, an additional error is

committed at this stage too.

This error is called local round-off error.

(0.7625) (0.110000110011)

If a particular computer system has a word length of 12 bits only, then the decimal

number 0.7625 is stored in the computer memory in binary form as 0.110000110011.

However, it is equivalent to 0.76245.

Thus, in storing the number 0.7625, we have committed an error equal to 0.00005, which

is the round-off error; inherent with the computer system considered.

Thus, we define the error as

Error = True value – Computed value

Absolute error, denoted by |Error|,

While, the relative error is defined as

Relative error Error

Truevalue

Local truncation error

It is generally easier to expand a function into a power series using Taylor series

expansion and evaluate it by retaining the first few terms. For example, we may

approximate the function f (x) = cos x by the series

cos ( 1)

2! 4! (2 )! n

n x x x

x

n

If we use only the first three terms to compute cos x for a given x, we get an approximate

Answer: Here, the error is due to truncating the series. Suppose, we retain the first n

terms, the truncation error (TE) is given by