MTH603 Handouts Set 3

Numerical Analysis

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

Introduction

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.

Errors in Computations

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 TE (2 2)! n x n

The TE is independent of the computer used. If we wish to compute cos x for accurate with five significant digits, the question is, how many terms in the expansion are to be included? In this situation

.5 10 (2 2)! n x n Taking logarithm on both sides, we get (2 2)log log[(2 2)!] log 5 6log 10 0.699 5.3 n x n

or log[(2 2)!] (2 2)log 5.3 n n x We can observe that, the above inequality is satisfied for n = 7. Hence, seven terms in the expansion are required to get the value of cos x, with the prescribed accuracy The truncation error is given by

TE 16! x