1.0 INTRODUCTION

In practice, the applications of Mathematics ultimately results in numerical form. These results may be from the evolution of formulae, solutions of equations or inferences drawn from tabulated data.

Numerical analysis may be described as that branch of Mathematics which provides with convenient methods for obtaining numerical solutions to such problems.

Suppose for a certain experiment the heights of a set of university students are measured. The numbers representing the heights are only approximations, true to two or three decimal places. In general the data represent approximations. Sometimes, the process used to deal with the data is approximate. So, the error in a computed result may be due to errors in the data or errors in the methods or both.

Numerical analysis deals with methods of which errors in computation is reduced to a minimum. With the advent of computers the demand for numerical methods increased rapidly in the applications to engineering and scientific fields. Today numerical methods have become an indispensable tool.

1.1 ACCURACY OF NUMBERS

The numbers and e etc written in this form are exact numbers. If  is written as 1.732, then it is an approximate number. We can also write as 1.73205, which is a better approximation. However, we cannot write the exact value of by finite number of digits, because has infinite number of decimals. Thus we deal with two types of numbers exact and approximate.

1.1.1 Significant Figures

The digits used to express a number meaningfully are called significant figures or significant digits. For instance the digits are all significant digits in any number. But 0 may or may not be a significant figure. If 0 is used to fix the decimal point or to fill places of unknown or discarded digits, then 0 is not a significant figure.

In the number of 0.000567, the significant figures are 5, 6, 7. The zeros are not significant, because they are used to fix the decimal point. But in the number 4.5037 all the five digits are significant. Here 0 is also a significant digit.

In 0.5000, only 5 is the significant figure.

The numbers 0.0028, 0.000035 contain only two significant figures 2, 8, and 3, 5 respectively.

1.1.2 Rounding Off of Numbers

We know is a never ending decimal. To use such a number in a practical computation we must cut down to a usable form such as 1.7 or 1.73 or 1.732 etc. This process of cutting off the extra digits is called rounding off of numbers.

Usually rounding off a number is done according to the following rules.

“To round off a number to n significant figures, discard all the digits to the right of the n^th digit, and if the (n + 1)^th digit is

1. (i) less than 5, then leave the n^th digit unaltered
2. (ii) greater than 5, then increase the n^th digit by 1
3. (iii) equal to 5, then the n^th digit is unaltered if it is even and increase the n^th digit by 1 if it is odd”

A number rounded off according to this rule is said to be correct to n significant places.

For example, the following numbers are rounded off to five significant figures.

23.876345 becomes 23.876

0.876345 becomes 0.87634

4.782250 becomes 4.7822

1.823156 becomes 1.8232

76.69954 becomes 76.700

1.1.3 A Safe Rule

In numerical analysis, often, we have to perform a sequence of arithmetical operations on numbers. During the computation retain one more figure (or decimal place) than that given in the data and round off after the last operation has been performed. When this practice is followed, no attention is paid to rounding off rule.

1.2 ERRORS AND THEIR ANALYSIS

In numerical computations we always look for the accuracy of the result obtained.

The size of the error in the computed value is usually expressed in two ways:

1. absolute error and (2) relative error

∴

and it is denoted by E_r.

∴

Sometimes, we express as percentages which will enable us for comparison.

∴ and it is denoted by E_p.

⇒

Note:

1. Absolute error is in terms of the unit used, where as relative error and percentage error are pure numbers independent of the unit of measurement.
2. It is obvious that the absolute error is related to the number of decimal places, whereas relative error is related to the number of significant figures.
3. If a number is correct to n decimal places, then absolute error is

   If a number is correct to n significant figures, then the relative error is

   In other words, a number is an approximation to x to n significant figures if

1.2.1 Classification of Errors

In numerical analysis errors are classified into two major categories.

1. (i) Inherent error
2. Truncation error

Inherent error: It is the error that is present in the statement of the problem before its analysis and solution. The inherent errors arise due to the simplication assumptions that are used in the mathematical formulation of the problem or due to the errors in measurements of the parameters of the problem.

Truncation error: It is due to those errors caused by the method. For example:

If is approximated by the cubic and computed then the error in the result is due to truncating the series.

The remainder after the fourth term in the Maclaurin’s series of is the truncation error.

We shall state some theorems without proof.

Theorem 1.1

If a number is correct to n significant figures and if a₁ is the first significant figure, then the relative error is less than i.e.

Theorem 1.2

If x is a number having n decimal digits and if is got by truncating to k digits, then the absolute error is

Absolute error due to rounding off to k digits

Theorem 1.3

If the relative error of any number is then the number is correct to n significant figures.

WORKED EXAMPLES

Example 1

If the number 23.876 is correct to 5 significant figures, then find the relative error.

Solution

Given the number is 23.876.

∴ the first significant figure is 2 and the number of significant digits is 5, n = 5, a₁ = 2

∴ the relative error

Example 2

Consider the number 52.43, which is correct to four significant figures. Find E_a and E_r. Also find the percentage error.

Solution

Given the number is 52.43. The first significant number is 5 and the number of significant figures is 4

∴

∴ by theorem 1 the relative error

⇒

The n^th place is

∴ the absolute error = 0.005

Percentage error = 0.0002 × 100 = 0.02

Example 3

Find the absolute error and relative error in takinπ

Solution

Given

∴

Example 4

Round off the numbers 784652 and 78.4625 to four significant digits and compute E_a, E_r, E_p.

Solution

1. (i) Given x = 784652

   Rounding off to 4 significant figures, we get x₁ = 784600

   

2. (ii) Given x = 78.4625

   Rounding off to 4 significant figures, we get x₁ = 78.4600

   

Example 5

A person measured a length as 3450 cm, where as its actual length is 3445 cm and another length as 145 cm where as its actual length is 140 cm Compare their absolute and relative errors.

Solution

1. (i) Given x = 3445, x₁ = 3450

   

2. (ii) Given x = 140, x₁ = 145

   

Though the absolute errors are same for both measurements, their relative errors differ more than 24 times.

1.3 A GENERAL FORMULA FOR ERROR

Let be a function of several variables and be differentiable. Let Δx₁, Δx₂,…,Δx_n be the errors in the variables so that the error in y is Δy.

We have to find the error relation.

We know that the total differential

∴ the error relation is

The relative error of y is

∴ The maximum value of

Note: Suppose then

The differential relation is

∴ the error relation is

The relative error in y is

Maximum relative error is

WORKED EXAMPLES

Example 6

Let Find the maximum error and relative error in y if x₁ = x₂ = x₃ = 1 and

Solution

Given

∴

∴ the error relation is

∴ the maximum error relation Δy is

But

At the point

and given that

When

∴ maximum error is

= 0.06 + 0.3 + 0.12 = 0.21

Maximum relative error

The general error formula can be used to find error in fundamental operations of arithmetic.

Example 7

Find the sum of the numbers 681.32, 521.7, 94.853, 5.9271, 0.0034, each being correct to its last digit. Also find the absolute error.

Solution

Here 52.7 has one decimal place

The absolute error =

This number is the one with greatest absolute

So, we round off all the numbers to two decimals.

∴ the round off numbers are 681.32, 94.85, 5.93, 0.00

So, their sum S = 681.32 + 521.7 + 94.85 + 5.93 + 0.00 = 1303.8

E_a = sum of absolute errors in each number

Rounding off error is 0.01.

∴ total error in S is 0.07 + 0.01 = 0.08

∴ S = 1303.8 ± 0.08

Example 8

Evaluate the number correct to 4 significant digits and find its absolute and relative error.

Solution

Evaluate the numbers to 4 significant digits using calculator ,

∴

We know that the absolute error in a number correct to 3 decimal places is

∴ the absolute error in the sum of the 3 numbers is

This shows that the sum is correct to 3 significant figures only,

∴

Then

1.4 ERROR IN SERIES APPROXIMATION

Power series expansion of a function is a very useful technique in theory and applications. The general method for expanding functions into power series is by means of Taylor’s formula.

1. (i) Taylor’s formula for about under valid conditions is

   

2. (ii) If then the expansion about the origin is

   

This is called Maclaurin’s formula.

The remainder after n terms is denoted by R_n and

If as then the series converges.

If we approximate by the first n terms, then the maximum error committed in the truncation is R_n.

Conversely, if the accuracy required is given in advance, the we can find the number of terms n to be taken.

1.4.1 Error in Some Important Series

1. Logarithmic series

   

   where or then the error committed in truncating is less than the first term neglected

   ∴

2. Binomial series

   

   where

   If then

   If then

3. The exponential series

   

   Error R_n is maximum when

   If maximum. relative

WORKED EXAMPLES

Example 9

Compute truncating after the third term. Find the error.

Solution

where

Put

∴

and

which is correct to 7 decimal places, because in the first significant value is 4 occurring in the 8^th place.

Note: Using calculator

Example 10

Find the number of terms n to be taken in the expansion of correct to 7 significant figures, when x = 1.

Solution

The Maclaurin’s series for

Maximum relative

Maximum relative error, when is

For 7 significant figure accuracy, we must have, by theorem 2

This means we have to take 11 terms of the series for 7 significant figure accuracy.

Exercises 1.1

1. Round of the following numbers correctly to four significant figure. 23.7642, 53266, 0.070037, 0.0052725.
2. Find the sum of the number which are correct to the number of significant figure given 142.6, 26.23, 0.23425, 220.44, 3.42.

Answers 1.1

1. (1) 23.76, 53270 = 5327 × 10, 0.07004, 0.005272
2. 392.92

Short Answer Questions

1. The value of π is 3.1416 correct to four decimal places, find the error.
2. Round off the numbers 34789 and 3.7256 to three significant figures.
3. Find the relative error if the number 852.43 is correct to five significant figure.
4. If the number 0.0700 is correct to 3 significant figures, find the relative error.
5. Find the absolute error if the number 0.0033543 is truncated to three ­decimal places.
6. If the number 25.34217 is rounded off to four significant figures then find E_a, E_r.
7. Find the absolute error in correct to 4 significant digits.