topic 3 numerical methods

7/30/2019 Topic 3 Numerical Methods

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Topic 3 Numerical Methods

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3.1 Solutions of Non-Linear Equations

Solve the following equations :

1. Solve 0542 xx

2. Solve 083 x

3. Solve 03 xex

So, in this subtopic we are to going discuss numerical methods in

finding the approximate numerical solutions of such equations.

Introduction

Many equations cannot be solved exactly, but various methods of

finding approximate numerical solutions exist.

The most commonly used methods have two main parts:

(a) finding an initial approximate value(b) improving this value by an iterative process

Initial Values:

The initial value of the roots of f(x) = 0 can be located

approximately by either a graphical or an algebraic method.

GRAPHICAL METHOD:

Either

(a) Plot ( or sketch ) the graph ofy =f(x).

The real roots are the points where the curve cuts the x axis.


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Or

(b) Rewritef(x) = 0 in the form F(x) = G(x).

Plot ( or sketch ) y =F(x) andy = G(x). The real roots are at

the points where these graphs intersect.

Example 1

Find the approximate value of the equation 04ln xx by

using the graphical method.


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Example 2

On the same axes, sketch 1 xe y and .1 xy Show that the

equation01)1ln(

xx has only one root .The sketch shows that there is only one point of intersection and so

the equation 01)1ln( xx has only one root .


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Example 3

By sketching the graph of 3xy and 13 xy on the same

coordinate axes, show that the equation 0133

xx has threereal roots.


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Algebraic Method

Find two values a and b such that f(a) and f(b) have

different signs.

At least one root must lie between a and b if f(x) is

continuous.

If more than one root is suspected between a and b, sketch

y =f(x).

Example 4

Show that the equation 043 xx has a root between 3.1x

and 4.1x . Approximate the root to two decimal places.


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3.2 Iterative Method

All iterative methods follow the same basic pattern. A sequence of

approximations ,......,,, 4321 xxxx is found, each one closer to the

root of 0xf . Each approximation is found from the one

before it using a specified method. The process continue until the

required accuracy is reached.

Iteration Method

Rewrite the equation 0xf in the form xgx

If the initial approximation is ,1x then calculate

12 xgx

23 xgx

34 xgx

..

This method fails if 1' 1 xg near the root.

So we are looking for 1' 1 xg .

Example 5

Using the iteration method, find the solution of xexxf near

5.0x correct to three decimal places.


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Example 6

Show that the equation 0412

xx has a root between 0.2 and

0.3. Taking 0.2 as the first approximation, use the iteration method

to find the root of the equation. Give your answer correct to three

significant figures.


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Example 7

Show that the equation 05.0sin xx has a root lying in the

interval 1,2 . Using the iteration formula 5.0sin1 nn xx ,

find the root correct to three decimal places.


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Example 8

Given the equation 02ln xx

a) Show that there exist a real root between 1.5 and 1.6

b) Find the appropriate iterative function for the equation.

c) Hence, find the real root of the equation correct to three decimal

places by using iterative method.


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EXERCISE

1. Show that the equation 03 xex has a root between 0and 1. By taking the first approximation x1 as 0.8, use

iteration method to find the root correct to 3 decimal places.

(Ans: 0.792)

2. Find an approximate root of the equation .043 xx Find this root correct to 3 decimal places using iteration

method. (Ans: 1.379)

3. Sketch the graph of )2(ln xy where x


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3.3 Newton-Raphson method

Newton-Raphson Method

If 1x is the first approximation to the root of the equation

0xf , then second, third,. approximations are written as

,........,, 432 xxx and are given by the formula

11

12' xf

xfxx

Repeat this process as required

nn

nnxf

xfxx

'1


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Example 9

Using the Newton-Raphson Method, find the solution of xexxf

near 5.0x correct to three decimal places.


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Example 10

Show that the equation 063 xx has a root between 1 and 2.

Using the Newton-Raphson method with the starting point 1.6,

determine an approximation to this root, giving your answer to

three significant figures.


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Example 11

Use Newton-Raphson method to find an approximate value of 315

. Give your answer correct to three decimal places.


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Example 12

Sketch the graph of xey and y = 2x where x < 2, on the

same axes. Get the first approximation, 0x for the equation

ex

= 2 x where 0 < 0x < 1. Hence, by using Newton-Raphson

method , solve the equation ofx

ex

2

1for x < 2 to three

decimal places.


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Example 13

Show that the equation 0sin2 xx has a root between 1x

(radian) and 2x (radian). Find the root of equation by using

a)Iteration methodb)Newton Raphson method

giving your answer correct to two decimal places.

topic 3 numerical methods

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