The presentation on Numerical Methods covered the following topics : 1. Introduction 2. Bisection Method with proof 3. False Position method with proof 4. Successive Approximation method 5. Newton Raphson (N-R)Method 6. Iterative Formulae for finding qth root, square root and reciprocal of positive number N, Using N-R method 7. Secant Method 8. Power Method and this is useful for engineering and B,Sc students.

1. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
1
Numerical Techniques
Unit- 1: Roots of Equations
1. Introduction
2. Bisection Method with proof
3. False Position method with proof
4. Successive Approximation method
5. Newton Raphson (N-R)Method
6. Iterative Formulae for finding qth root, square
root and reciprocal of positive number N, Using N-R
method
7. Secant Method
8. Power Method
Table of contents

2. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
2
Numerical Techniques
1. Introduction
Types of
equations
Algebraic
equation
Transcendental equation
Def: An equation of the form
Is called an algebraic equation
2
0 1 2
( ) ... 0
n
n
f x a a x a x a x     
Def: An equation which is not an algebraic
equation is called Transcendental equation.
i.e. : It contains some functions like
trigonometric, Logarithmic, hyperbolic etc.
3
: ( ) 2 3 0E g f x x x   
: ( ) sin 0
x
E g f x x x e  

3. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
3
Numerical Techniques
1. Introduction
Def. : Roots of equation
2
T h e v a lu e s o f " " w h ic h sa tisfie s e q u a tio n ( ) 0
a re c a lle d ro o ts o f a n e q u a tio n ( ) 0 .
E g . : if ( ) - 1 0 th e n 1 a n d - 1 a re
c a lle d ro o ts o f g iv e n e q u a tio n .
x f x
f x
f x x x x


   
Iterative Methods to find roots of equation
S uppose w e w ant to find the root " " of the equation ( ) 0.f x 
0
L e t b e a n a p p ro x im a te v a lu e o f ro o t " "x 
0
1 2 3
U sin g , w e g en erate a seq u en ce o f n u m b e rs m ean s
iterates , , , ... u n d er certain co n d itio n s.
x
x x x


4. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
4
Numerical Techniques
1. Introduction
Iterative Methods to find roots of equation
T h is seq u en ce co n verg es to ro o t " "
T h e m e th o d o f fin d in g th e se a p p ro x im a tio n fro m
a n in itia l g u e sse s is c a lle d a n ite ra tiv e m e th o d .

W e w ill d iscu ss so m e very fam ilier iter ative m eth o d s
to fin d th e so lu tio n o f alg eb raic an d tran scen d en tal
eq u atio n s.


5. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
5
Numerical Techniques
1. Introduction
Intermediate Value Theorem (IVT)
If an y co n tin u o u s fu n ctio n ( ) assu m es v alu es
o f o p p o site sig n at th e en d p o in ts o f an in terv al [ , ]
i.e.: ( ) ( ) 0 th en th e in terv al w ill co n tain at
least o n e ro o t o f th e eq u atio n (
f x
a b
f a f b
f

 
) 0 .
i.e. : T h ere ex ist ( , ) su ch th at ( ) 0 .
x
a b f 

  

6. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
6
Numerical Techniques
2. Bisection Method (Bolzano Method)
L e t ( ) b e a n y c o n tin u o u s fu n c tio n b e tw e e n a n y tw o
p o in ts "a " a n d "b ".
L e t ( ) 0 a n d ( ) 0 .
f x
f a f b

  
Y
Xa b
2
x 3
x 4
x 1
x
Exact Root
0
( )y f x

7. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
7
Numerical Techniques
2. Bisection Method (Bolzano Method)
1
T h en th e first ap p ro x im atio n is .
2
a b
x

 
1 1
1 1
1
If ( ) 0 , th en is th e ro o t o f th e eq u atio n ( ) 0
o th erw ise th e ro o t lies b etw een " " an d " " o r " an d " "
d ep en d in g u p o n ( ) is p o sitiv e o r n eg a tiv e.
f x x f x
a x x b
f x
  
T h en w e b isect th e in terval as b efo re a n d co n tin u e th e p ro cess
u n till th e ro o ts are fo u n d are fo u n d to d esired accu racy.


8. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
8
Numerical Techniques
Ex-1: Find the smallest positive root of the equation
using bisection method correct to two decimal places
3
9 1 0x x  
3
H e re ( ) 9 1f x x x  
0 1 2 3
( ) 1 7 9 1
x
f x  
H ere w e w ant to obtain sm allest root
th en w e tak e an in terval (0,1)
b ecau se (0 ) 1 0 an d (1) 7 0
T h erefo re ro o t lies b etw een 0 an d 1
f f    
Solu. 1

9. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
9
Numerical Techniques
Solu. 1
1 0 1 0.5 -3.3750 <0
2 0 0.5 0.25 -1.2344 <0
3 0 0.25 0.125 -0.1230 <0
4 0 0.125 0.0625 0.4377 >0
5 0.0625 0.0125 0.09338 0.1571 >0
6 0.09338 0.125 0.1094 0.0167 >0
7 0.1094 0.125 0.1172 -0.0532 <0
8 0.1094 0.1172 0.1133

10. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
10
Numerical Techniques
Solu. 1
H en ce th e sm allest p o sitive ro o t co rrect
to tw o d ecim al p laces is 0 .1 1

11. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
11
Numerical Techniques
Ex-2: Using bisection method to find a negative root of
correct to three decimal places3
1 1 0x x  
3
H e re ( ) 1 1f x x x  
0 1 2 3
( ) 1 1 9 2 1 7
x
f x
  

T h erefo re th e n eg ative ro o t o f g iven eq u atio n lies b etw een
( 3, 2 ) b ecau se ( 2 ) 2 0 an d ( 3) 1 7 0
T h erefo re ro o t lies b etw een -3 an d -2
f f        
Solu. 1

12. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
12
Numerical Techniques
Solu. 1
1 -3 -2 -2.5 -2.125 <0
2 -2.54 -2 -2.25 1.8594 >0
3 -2.5 -2.25 -2.375 -0.0215 <0
4 -2.375 -2.25 -2.3125 0.946 >0
5 -2.375 -2.3125 -2.3438 0.4684 >0
6 -2.375 -2.3438 -2.3594 0.2252 >0
7 -2.375 -2.3594 -2.3672 0.1023 >0
8 -2.375 -2.3672 -2.3711 0.0405 >0
9 -2.375 -2.3711 -2.3731 0.0087 >0
10 -2.375 -2.3731 -2.37405 -0.0072 <0

13. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
13
Numerical Techniques
Solu. 1
11 -2.3741 -2.3731 -2.3736 0.0008 >0
12 -2.3741 -2.3736 -2.3739
H en ce th e n eg ative ro o t o f g iven eq u atio n co rrect u p to th ree
d ecim al p laces is -2 .3 7 3


14. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
14
Numerical Techniques
Ex-3: Using bisection method to find a positive root of
correct to three decimal placeslog 1.2 0x x  
H ere ( ) lo g 1 .2f x x x 
0 1 2 3
( ) 1 .2 1 .2 0 .5 9 0 .2 3 1 9
x
f x   
T h erefo re th e p o sitive ro o t o f g iven eq u atio n lies b etw een
(2, 3) b ecau se (2 ) 0 .5 9 0 an d (3) 0 .2 3 1 9 0
T h erefo re ro o t lies b etw een 2 an d 3
f f    
Solu. 3

15. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
15
Numerical Techniques
Solu. 3
1 2 3 2.5 -0.205 <0
2 2.5 3 2.75
0.0081
6
>0
3 2.5 2.625 -0.099 <0
4 2.625 2.75 2.6875 -0.0461 <0
5 2.6875 2.75 2.7188 -0.019 <0
6 2.7188 2.75 2.7344 -0.0054 <0
7 2.7344 2.7422
0.0013
5
>0
8 2.7383 -0.0020 <0
9 2.7402 -0.0034 <0
10 2.7412
0.0004
8
>0
2.75
2.75
2.7344 2.7422
2.7383 2.7422
2.7402 2.7422

16. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
16
Numerical Techniques
Solu. 1
11 2.7402 2.7412 2.7407
0.0004
7
>0
12 2.7402 2.7407 2.7405
H ence the positive root of given equation correct up to three
decim al places is 2.740


17. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
17
Numerical Techniques
3. False Position / Regula Falsi Method
T h is m e th o d g iv e s ro o t o f a n e q u a tio n ( ) 0
a n d c lo s e ly (s im ila r) re s e m b le s th e B is e c tio n M e th o d .
f x 
( )y f x

18. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
18
Numerical Techniques
3. False Position / Regula Falsi Method
C h o o se tw o p o in ts "a" an d "b " su ch th at ( ) 0 an d ( ) 0 .
i.e. : T h e g rap h o f ( ) b etw een p o in ts " a" an d "b " cu t X -ax is.
T h u s ro o t o f th e eq u atio n lies b etw een "a" an d "b ".
E q u atio n o f th e li
f a f b
y f x
  
 

 n e jo in in g th e p o in ts ( , ( )) an d ( , ( )) is
( ) ( )
( ) ( )
A a f a B b f b
f b f a
y f a x a
b a

   

 
( ) ( )
0 ( ) ( )
f b f a
f a x a
b a

    

   ( ) ( ) ( ) ( )f a b a f b f a x a      

19. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
19
Numerical Techniques
3. False Position / Regula Falsi Method
   ( ) ( ) ( ) ( ) ( ) ( )b f a a f a f b x a f a x a      
( ) ( ) ( ) ( ) ( ) ( )a f a b f a x f b a f b x f a a f a     
( ) ( ) ( ) ( ) ( ) ( )a f a b f a a f a a f b x f b x f a     
 ( ) ( ) ( ) ( ) ( ) ( )a f a b f a a f a a f b x f b f a     
 ( ) ( ) ( ) ( )a f b b f a x f b f a   
( ) ( )
(1)
( ) ( )
a f b b f a
x
f b f a

       


20. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
20
Numerical Techniques
3. False Position / Regula Falsi Method
1
1 1
E q ------(1 ) is th e fo rm u la to fin d th e ro o ts o f g iven
eq u atio n .
N o w fin d u sin g fo rm u la -----(1 )
If ( ) 0 th en is th e ro o t o f th e eq u atio n ( ) 0
O th erw ise th e ro o t lies b etw een " " an d "
n
x
f x x f x
a


  
 1
1 1
" o r
" " an d " " d ep en d in g u p o n ( ) is p o sitive o r n eg ative.
R ep eat ab o ve step s u n till w e g et ap p ro x im ate ro o t o f
( ) 0 very clo se to real (actu al) ro o t.
x
x b f x
f x



21. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
21
Numerical Techniques
3
E x -1 : F in d a real ro o t o f th e eq u atio n 2 5 0 b y th e
m eth o d o f false p o sitio n co rrect to th ree d ecim al p laces.
x x  
3
H e re ( ) 2 5f x x x  
0 1 2 3
( ) 5 6 1 1 6
x
f x   
Solu. 1
T h e re fo re th e ro o t lie s b e tw e e n 2 a n d 3
L e t 2 & 3
( ) ( 2 ) 1 & ( ) (3) 1 6
a b
f a f f b f

  
     

22. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
22
Numerical Techniques
Solu. 1
1 ite ra tio n :
s t
1
( ) ( )
( ) ( )
a f b b f a
x
f b f a



1
( 2 ) (1 6 ) (3) ( 1)
(1 6 ) ( 1)
x
 
 
 
1
3 5
1 7
x 
1
2 .0 5 8 8x 
1
( ) ( 2 .0 5 8 8) 8 .7 2 6 5 4 .1 1 7 6 5 0 .3 9 1 1 0f x f       

23. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
23
Numerical Techniques
Solu. 1
T herefore the next root lies betw een 2.0588 and 3.
2
( ) ( )
( ) ( )
a f b b f a
x
f b f a



2
( 2 .0 5 3 3) (1 6 ) (3) ( 0 .3 9 1 1)
(1 6 ) ( 0 .3 9 1 1)
x
 
 
 
2
3 2 .9 4 0 8 1 .1 7 3 3
1 6 .3 9 1 1
x

 
2
2 .0 8 1 2x 
2
( ) ( 2 .0 8 1 2 ) 9 .0 1 4 5 4 .1 6 2 4 5 0 .1 4 7 9 0f x f       
2 ite ra tio n :
n d
Let a = 2.0588 and b = 3
Now, f(a) = f(2.0588) = -0.3911 <
0
And f(b) = f(3) = 16 > 0

24. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
24
Numerical Techniques
Solu. 1
T herefore the next root lies betw een 2.0812 and 3.
3
( ) ( )
( ) ( )
a f b b f a
x
f b f a



3
( 2 .0 8 1 2 ) (1 6 ) (3) ( 0 .1 4 7 9 )
(1 6 ) ( 0 .1 4 7 9 )
x
 
 
 
3
3 3 .2 9 9 2 0 .4 4 3 7
1 6 .1 4 7 9
x

 
3
2 .0 8 9 6x 
3
( ) ( 2 .0 8 9 6 ) 9 .1 2 4 1 4 .1 7 9 2 5 0 .0 5 5 1 0f x f       
3 ite ra tio n :
r d
Let a = 2.0812 and b = 3
Now, f(a) = f(2.0812) = -0.1479 <
0
And f(b) = f(3) = 16 > 0

25. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
25
Numerical Techniques
Solu. 1
T herefore the next root lies betw een 2.0896 and 3.
4
( ) ( )
( ) ( )
a f b b f a
x
f b f a



4
( 2 .0 8 9 6 ) (1 6 ) (3) ( 0 .0 5 5 1)
(1 6 ) ( 0 .0 5 5 1)
x
 
 
 
4
3 3 .4 3 3 6 0 .1 6 5 3
1 6 .0 5 5 1
x

 
4
2 .0 9 2 7x 
4
( ) ( 2 .0 9 2 7 ) 9 .1 6 4 8 4 .1 8 5 4 5 0 .0 2 0 6 0f x f       
4 ite ra tio n :
th
Let a = 2.0896 and b = 3
Now, f(a) = f(2.0896) = -0.0551 <
0
And f(b) = f(3) = 16 > 0

26. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
26
Numerical Techniques
Solu. 1
T herefore the next root lies betw een 2.0927 and 3.
5
( ) ( )
( ) ( )
a f b b f a
x
f b f a



5
( 2 .0 9 2 7 ) (1 6 ) (3) ( 0 .0 2 0 6 )
(1 6 ) ( 0 .0 2 0 6 )
x
 
 
 
5
3 3 .4 8 3 2 0 .0 6 1 8
1 6 .0 2 0 6
x

 
5
2 .0 9 3 9x 
5
( ) ( 2 .0 9 3 9 ) 9 .1 8 0 5 4 .1 8 7 8 5 0 .0 0 7 3 0f x f       
5 ite ra tio n :
th
Let a = 2.0927 and b = 3
Now, f(a) = f(2.0927) = -0.0206 <
0
And f(b) = f(3) = 16 > 0

27. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
27
Numerical Techniques
Solu. 1
T herefore the next root lies betw een 2.0939 and 3.
6
( ) ( )
( ) ( )
a f b b f a
x
f b f a



6
( 2 .0 9 3 9 ) (1 6 ) (3) ( 0 .0 0 7 3)
(1 6 ) ( 0 .0 0 7 3)
x
 
 
 
6
3 3 .5 0 2 4 0 .0 2 1 9
1 6 .0 0 7 3
x

 
6
2 .0 9 4 3x 
6
( ) ( 2 .0 9 4 3) 9 .1 8 5 8 4 .1 8 8 6 5 0 .0 0 2 8 0f x f       
6 ite ra tio n :
th
Let a = 2.0939 and b = 3
Now, f(a) = f(2.0939) = -0.0073 <
0
And f(b) = f(3) = 16 > 0

28. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
28
Numerical Techniques
Solu. 1
T herefore the next root lies betw een 2.0943 and 3.
7
( ) ( )
( ) ( )
a f b b f a
x
f b f a



7
( 2 .0 9 4 3) (1 6 ) (3) ( 0 .0 0 2 8 )
(1 6 ) ( 0 .0 0 2 8 )
x
 
 
 
7
3 3 .5 0 8 8 0 .0 0 8 4
1 6 .0 0 2 8
x

 
7
2 .0 9 4 4x 
6 7
N o w th e ro o ts 2 .0 9 4 3 an d 2 .0 9 4 4 h avin g
3 -eq u al d ig it after d ecim al p o in t th erefo re th e
req u ired ro o t co rrect u p to th ree d ecim al p laces is 2 .0 9 4 .
x x 
7 ite ra tio n :
th
Let a = 2.0943 and b = 3
Now, f(a) = f(2.0943) = -0.0028 <
0
And f(b) = f(3) = 16 > 0
Answe
r

29. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
29
Numerical Techniques
1 0
E x -2 : F in d a real ro o t o f th e eq u atio n lo g 1 .2 b y th e
m eth o d o f false p o sitio n co rrect to th ree d ecim al p laces.
x x 
1 0
H e re ( ) lo g 1 .2f x x x 
0 1 2 3
( ) 1 .2 1 .2 0 .5 9 7 9 0 .2 3 1 4
x
f x   
Solu. 2
T h e re fo re th e ro o t lie s b e tw e e n 2 a n d 3
L e t 2 & 3
( ) ( 2 ) 0 .5 9 7 9 & ( ) (3) 0 .2 3 1 4
a b
f a f f b f

  
     

30. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
30
Numerical Techniques
Solu. 2
1 ite ra tio n :
s t
1
( ) ( )
( ) ( )
a f b b f a
x
f b f a



1
( 2 ) (0 .2 3 1 4 ) (3) ( 0 .5 9 7 9 )
(0 .2 3 1 4 ) ( 0 .5 9 7 9 )
x
 
 
 
1
0 .4 6 2 8 1 .7 9 3 7
0 .8 2 9 3
x

 
1
2 .7 2 1 0x 
1 1 0
( ) ( 2 .7 2 1 0 ) ( 2 .7 2 1 0 ) lo g ( 2 .7 2 1 0 ) 1 .2 0 .0 1 7 1 0f x f      

31. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
31
Numerical Techniques
Solu. 2
T herefore the next root lies betw een 2.7210 and 3.
2
( ) ( )
( ) ( )
a f b b f a
x
f b f a



2
( 2 .7 2 1 0 ) (0 .2 3 1 4 ) (3) ( 0 .0 1 7 1)
(0 .2 3 1 4 ) ( 0 .0 1 7 1)
x
 
 
 
2
0 .6 2 9 6 0 .0 5 1 3
0 .2 4 8 5
x

 
2
2 .7 4 0 0x 
2
( ) ( 2 .7 4 0 0 ) 0 .0 0 0 5 6 0f x f    
2 ite ra tio n :
n d
Let a = 2.7210 and b = 3
Now, f(a) = f(2.7210) = -0.0171 <
0
And f(b) = f(3) = 0.2314 > 0

32. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
32
Numerical Techniques
Solu. 2
T herefore the next root lies betw een 2.7400 and 3.
3
( ) ( )
( ) ( )
a f b b f a
x
f b f a



3
( 2 .7 4 0 0 ) (0 .2 3 1 4 ) (3) ( 0 .0 0 0 5 6 )
(0 .2 3 1 4 ) ( 0 .0 0 0 5 6 )
x
 
 
 
3
0 .6 3 4 0 0 .0 0 1 6 8
0 .2 3 2 0
x

 
3
2 .7 4x 
3
( ) (2 .7 4 ) 0 .0 0 0 6 0f x f    
3 ite ra tio n :
r d
Let a = 2.7400 and b = 3
Now, f(a) = f(2.7400) = -0.00056 <
0
And f(b) = f(3) = 0.2314 > 0

33. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
33
Numerical Techniques
Solu. 2
T herefore the next root lies betw een 2.74 and 3.
4
( ) ( )
( ) ( )
a f b b f a
x
f b f a



4
( 2 .7 4 ) (0 .2 3 1 4 ) (3) ( 0 .0 0 0 6 )
(0 .2 3 1 4 ) ( 0 .0 0 0 6 )
x
 
 
 
4
0 .6 3 4 0 0 .0 0 1 8
0 .2 3 2
x

 
4
2 .7 4 0 5x 
4
( ) ( 2 .7 4 0 5 ) 0 .0 0 0 1 0f x f    
4 ite ra tio n :
th
Let a = 2.74 and b = 3
Now, f(a) = f(2.74) = -0.0006 < 0
And f(b) = f(3) = 0.2314 > 0

34. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
34
Numerical Techniques
Solu. 2
T herefore the next root lies betw een 2.7405 and 3.
5
( ) ( )
( ) ( )
a f b b f a
x
f b f a



5
( 2 .7 4 0 5 ) (0 .2 3 1 4 ) (3) ( 0 .0 0 0 1)
(0 .2 3 1 4 ) ( 0 .0 0 0 1)
x
 
 
 
5
0 .6 3 4 2 0 .0 0 0 3
0 .2 3 1 5
x

 
5
2 .7 4 0 8x 
5 ite ra tio n :
th
Let a = 2.7405 and b = 3
Now, f(a) = f(2.7405) = -0.0001 <
0
And f(b) = f(3) = 0.2314 > 0
4 5
N o w th e ro o ts 2 .7 4 0 5 an d 2 .7 4 0 8 h avin g
3 -eq u al d ig it after d ecim al p o in t th erefo re th e
req u ired ro o t co rrect u p to th ree d ecim al p laces is 2 .7 4 0 .
x x 
Answe
r

35. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
35
Numerical Techniques
E x -3 : F in d a sm allest p o sitive ro o t o f th e eq u atio n 0 b y
the m ethod of false position c o rrect to th ree d ecim al p laces.
x
x e

 
H e re ( )
x
f x x e

 
0 1
( ) 1 0 .6 3 2 1
x
f x 
Solu. 3
T h e re fo re th e ro o t lie s b e tw e e n 0 a n d 1
L e t 0 & 1
( ) (0 ) 1 & ( ) (1) 0 .6 3 2 1
a b
f a f f b f

  
     

36. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
36
Numerical Techniques
Solu. 3
1 ite ra tio n :
s t
1
( ) ( )
( ) ( )
a f b b f a
x
f b f a



1
(0 ) (0 .6 3 2 1) (1) ( 1)
(0 .6 3 2 1) ( 1)
x
 
 
 
1
1
1 .6 3 2 1
x 
1
0 .6 1 2 7x 
1
( ) (0 .6 1 2 7 ) 0 .6 1 2 7 0 .5 4 1 9 0 .0 7 0 8 0f x f     

37. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
37
Numerical Techniques
Solu. 3
T herefore the next root lies betw een 0 and 0.6127.
2
( ) ( )
( ) ( )
a f b b f a
x
f b f a



2
(0 ) (0 .0 7 0 8) (0 .6 1 2 7 ) ( 1)
(0 .0 7 0 8) ( 1)
x
 
 
 
2
0 .6 1 2 7
1 .0 7 0 8
x 
2
0 .5 7 2 2x 
2
( ) (0 .5 7 2 2 ) 0 .5 7 2 2 0 .5 6 4 3 0 .0 0 7 9 0f x f     
2 ite ra tio n :
n d
Let a = 0 and b = 0.6127
Now, f(a) = f(0) = -1 < 0
And f(b) = f(0.6127) = 0.0708 >
0

38. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
38
Numerical Techniques
Solu. 3
T herefore the next root lies betw een 0 and 0.5722.
3
( ) ( )
( ) ( )
a f b b f a
x
f b f a



3
(0 ) (0 .0 0 7 9 ) (0 .5 7 2 2 ) ( 1)
(0 .0 0 7 9 ) ( 1)
x
 
 
 
3
0 .5 7 2 2
1 .0 0 7 9
x 
3
0 .5 6 7 7x 
3
( ) (0 .5 6 7 7 ) 0 .5 6 7 7 0 .5 6 6 8 0 .0 0 0 9 0f x f     
3 ite ra tio n :
r d
Let a = 0 and b = 0.5722
Now, f(a) = f(0) = -1 < 0
And f(b) = f(0.5722) = 0.0079 >
0

39. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : False
Position Method
39
Numerical Techniques
Solu. 3
T herefore the next root lies betw een 0 and 0.5677.
4
( ) ( )
( ) ( )
a f b b f a
x
f b f a



4
(0 ) (0 .0 0 0 9 ) (0 .5 6 7 7 ) ( 1)
(0 .0 0 0 9 ) ( 1)
x
 
 
 
4
0 .6 3 4 0 0 .0 0 1 8
0 .2 3 2
x

 
4
0 .5 6 7 2x 
4 ite ra tio n :
th
Let a = 0 and b = 0.5677
Now, f(a) = f(0) = -1 < 0
And f(b) = f(0.5677) = 0.0009 > 0
3 4
N o w th e ro o ts 0 .5 6 7 7 an d 0 .5 6 7 2 h avin g
3 -eq u al d ig it after d ecim al p o in t th erefo re th e
req u ired ro o t co rrect u p to th ree d ecim al p laces is 0 .5 6 7 .
x x 
Answe
r

40. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
40
Numerical Techniques : Tutorial - 1
SR.No Questions Answers
Ex-1 Find a real root of the equation 𝑥3
− 2𝑥 − 5 = 0 correct up to three
decimal places by using Bisection Method.
2.094
Ex-2 Find a positive root of the equation 𝑥𝑒 𝑥
= 1, lying (0, 1) using Bisection
Method.
0.567
Ex-3 Find a root of the equation 𝑥3
− 𝑥 − 11 = 0, using Bisection Method up
to forth approximation.
2.3125
Ex-4 Find a positive root of the equation 2𝑠𝑖𝑛𝑥 − 𝑥 = 0, using Bisection
Method up to fifth approximation.
1.90625
Ex-5 Find a real root of the equation 𝑥 − 𝑐𝑜𝑠𝑥 = 0, correct up to three
decimal places using Bisection Method.
1.813
Ex-6 Apply Regula – Falsi Method to solve the equation 𝑙𝑛𝑥 − 𝑥 + 3 = 0,
correct up to two decimal places.
4.45
Ex-7 Apply Regula – Falsi Method to solve the equation 𝑥2
− 𝑙𝑜𝑔𝑥 − 12 = 0,
correct up to three decimal places.
3.542
Ex-8 Apply Regula – Falsi Method to solve the equation 𝑥 𝑒 𝑥
= 2, correct up
to four decimal places.
0.8526
Ex-9 Apply Regula – Falsi Method to solve the equation 𝑥3
− 2𝑥 − 5 = 0,
correct up to three decimal places.
2.094
Ex-10 Apply Regula – Falsi Method to solve the equation 𝑥 𝑒 𝑥
= 𝑐𝑜𝑠𝑥, correct
up to three decimal places.
0.517

41. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
41
Numerical Techniques _Tutorial – I_ Solution
0 1 2 3
( ) 5 6 1 1 6
x
f x   
T h erefo re th e ro o t o f g iven eq u atio n lie s b etw een
(2, 3) b ecau se (2 ) 1 0 an d (3) 1 6 0
T h erefo re ro o t lies b etw een 2 an d 3
f f    
Solu. 1
3
H e re ( ) 2 5f x x x  
3
E x -1 : F in d real ro o t o f th e eq u atio n 2 5 0 , co rrect u p to
th ree d ecim al p laces u sin g B is ectio n M eth o d .
x x  

42. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
42
Numerical Techniques_ Tutorial – I_ Solution
Solu. 1
1 2 3 2.5 5.625 >0
2 2 2.5 2.25 1.8906 >0
3 2 2.25 2.125 0.3437 >0
4 2 2.125 2.0625 -0.3513 <0
5 2.0625 2.125 2.09375 -0.0089 <0
6 2.09375 2.125 2.10938 0.1668 >0
7 2.09375 2.10938 2.10156
0.0785
6
>0
8 2.09375 2.10156 2.09766
0.0347
1
>0
9 2.09375 2.09766 2.09570
0.0128
6
>0
10 2.09375 2.09570 2.09473
0.0019
5
>0

43. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
43
Numerical Techniques_Tutorial-1_Solution
Solu. 1
11 2.09375 2.09473 2.09424 -0.0035 <0
12 2.09434 2.09473 2.094485
H en ce th e n eg ative ro o t o f g iven eq u atio n co rrect u p to th ree
d ecim al p laces is 2 .0 9 4


44. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
44
Numerical Techniques _Tutorial – I_ Solution
0 1
( ) 1 1 .7 1 8
x
f x 
T h erefo re th e ro o t o f g iven eq u atio n lie s b etw een
(0,1) b ecau se (0 ) 1 0 an d (1) 1 .7 1 8 0
T h erefo re ro o t lies b etw een 0 an d 1
f f    
Solu. 2
H e re ( ) 1
x
f x x e 
E x -2 : F in d p o sitive ro o t o f th e eq u atio n 1, co rrect u p to
th ree d ecim al p laces u sin g B is ectio n M eth o d .
x
x e 

45. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
45
Numerical Techniques_ Tutorial – I_ Solution
Solu. 2
1 0 1 0.5 -0.1756 <0
2 0.5 1 0.75 0.5876 >0
3 0.5 0.75 0.625 0.1677 >0
4 0.5 0.625 0.5625 -0.0128 <0
5 0.5625 0.625 0.5938 0.0753 >0
6 0.5625 0.5938 0.5781 0.0305 >0
7 0.5625 0.5781 0.5703 0.0087 >0
8 0.5625 0.5703 0.5664 -0.0021 <0
9 0.5664 0.5703 0.5684 0.0035 >0
10 0.5664 0.5684 0.5674 0.0007 >0

46. Semester :III
Mr. Tushar J Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Tutorial-1
46
Numerical Techniques_Tutorial-1_Solution
Solu. 2
11 0.5664 0.5674 0.5669 -0.0006 <0
12 0.5669 0.5674 0.5672
H en ce th e n eg ative ro o t o f g iven eq u atio n co rrect u p to th ree
d ecim al p laces is 0 .5 6 7

0.0001 >0
13 0.5669 0.5672 0.5671

47. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
1
Numerical Techniques
Unit- 1: Roots of Equations
1. Introduction
2. Bisection Method with proof
3. False Position method with proof
4. Successive Approximation method
5. Newton Raphson (N-R)Method
6. Iterative Formulae for finding qth root, square
root and reciprocal of positive number N, Using N-R
method
7. Secant Method
8. Power Method
Table of contents

48. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
2
Numerical Techniques
5. Newton - Raphson Method:
 
0
L et x
0 .
b e th e in itia l a p p ro xim a tio n
o f th e eu q tio n f x


  0 0
,A x f x
0
x1
x2
x

49. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
3
Numerical Techniques
5. Newton - Raphson Method
  0 0
0
L et A , b e th e p o in t o n cu rve
acco rd in g to p o in t.
x f x
x

 
 
 
0
0 0
d f x
y f x x x
d x
   
 m = Slope of curve
N ow , T angent line of curve at A is
     0 0 0
'y f x f x x x   
   0 0
y f x m x x   

50. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
4
Numerical Techniques
5. Newton - Raphson Method
 1
, , 0 .N ow A D cuts X axis at D x
     0 0 1 0
0 'f x f x x x   
     0 1 0 0
'f x x x f x   
 
 
0
1 0
0
'
f x
x x
f x
   
 
 
0
1 0
0
(1)
'
f x
x x
f x
        

51. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
5
Numerical Techniques
5. Newton - Raphson Method
1 2 3
. _ _ _ _ (1) is th e 1 ap p ro x im atio n .
N o w , o b tain , , , .... u sin g eq .(1).
S o o n u n til th e ro o t is o b tain ed .
n st
n
E q
x x x
 
 
1
In g en eral,
'
n
n n
n
f x
x x
f x

 

52. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
6
Numerical Techniques
Ex-1: Find the root of the equation using N- R
method correct to four decimal places.
Sol.:
3
1 0x x  
3
H ere ( ) 1f x x x  
w e take an interval (1, 2)
b ecau se (1) 1 0 an d (2 ) 5 0
T h erefo re ro o t lies b etw een 1 an d 2 .
f f    
x 0 1 2
f(x) -1 -1 5

53. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
7
Numerical Techniques
Sol.:
0
an d w e tak e 1 .5x 
 
3
Let 1f x x x   
 
2
' 3 1f x x  
   0 0
0.875 & ' 5.75f x f x  
1 ap p ro x im atio n :
st

 
 
0
1 0
0
'
f x
x x
f x
 
0 .8 7 5
1 .5
5 .7 5
  1 .3 4 7 8 3

54. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
8
Numerical Techniques
Sol.:
   1 1
0.100699 & ' 4.44994f x f x  
2 ap p ro x im atio n :
n d

 
 
1
2 1
1
'
f x
x x
f x
 
0 .1 0 0 6 9 9
1 .3 4 7 8 3
4 .4 4 9 9 4
 
   2 2
0.00206 & ' 4.26846f x f x  
1 .3 2 5 2 0

55. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
9
Numerical Techniques
Sol.:
3 ap p ro x im atio n :
rd

 
 
2
3 2
2
'
f x
x x
f x
 
0 .0 0 2 0 6
1 .3 2 5 2 0
4 .2 6 8 4 6
 
   3 3
0.00009 & ' 4.26465f x f x  
1 .3 2 4 7 2

56. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
10
Numerical Techniques
Sol.:
4 ap p ro x im atio n :
th

 
 
3
4 3
3
'
f x
x x
f x
 
0 .0 0 0 0 9
1 .3 2 4 7 2
4 .2 6 4 6 5
 
1 .3 2 4 7 0
H ence the root of the equation correct
to four decim al places is 1.3247.

57. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
11
Numerical Techniques
Ex-II: Find the real positive root of the equation
which is near using N- R method correct to four
decimal places.
Sol.:
sin cos 0x x x 
x 
H ere ( ) sin cosf x x x x 
 ' sin cos sinf x x x x x  
co sx x
0
L et 3 .1 4 1 6x  
   0 0
1 & ' 3.1416f x f x    

58. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
12
Numerical Techniques
Sol.:
1 ap p ro x im atio n :
st

 
 
0
1 0
0
'
f x
x x
f x
 
( 1)
3 .1 4 1 6
( 3 .1 4 1 6 )

 

   1 1
0.06620 & ' 2.68147f x f x    
2 .8 2 3 2 9

59. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
13
Numerical Techniques
Sol.:
2 ap p ro x im atio n :
n d

 
 
1
2 1
1
'
f x
x x
f x
 
( 0 .0 6 6 2 0 )
2 .8 2 3 2 9
( 2 .6 8 1 4 7 )

 

2 .7 9 8 6 1
   2 2
0.00059 & ' 2.63561f x f x    

60. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
14
Numerical Techniques
Sol.:
3 ap p ro x im atio n :
rd

 
 
2
3 2
2
'
f x
x x
f x
 
( 0 .0 0 0 5 9 )
2 .7 9 8 6 1
( 2 .6 3 5 6 1)

 

2 .7 9 8 3 9
   3 3
0.00001 & ' 2.63519f x f x    

61. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
15
Numerical Techniques
Sol.:
4 ap p ro x im atio n :
th

 
 
3
4 3
3
'
f x
x x
f x
 
( 0 .0 0 0 0 1)
2 .7 9 8 3 9
( 2 .6 3 5 1 9 )

 

2 .7 9 8 3 9
H ence the root of the equation correct
to three decim al places is 2.7983.

62. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
16
Numerical Techniques
Ex-III: Solve correct to three decimal
places if root lies between -2 and -1 using N-R method.
Sol.:
 
4 3
10 7 0f x x x x    
4 3
H ere ( ) 1 0 7f x x x x   
 
3 2
' 4 3 10f x x x  
H ere th e ro o t o f eq u atio n lies
b etw een -2 an d -1 .
0
T h u s 1 .5x  
   0 0
0.4375 & ' 10.25f x f x   

63. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
17
Numerical Techniques
Sol.:
1 ap p ro x im atio n :
st

 
 
0
1 0
0
'
f x
x x
f x
 
(0 .4 3 7 5)
1 .5
( 1 0 .2 5)
  

1.5 0.0427  
1 .4 5 7 3 
   1 1
0.0321 & ' 8.7505f x f x   

64. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
18
Numerical Techniques
Sol.:
2 ap p ro x im atio n :
n d

 
 
1
2 1
1
'
f x
x x
f x
 
0 .0 3 2 1
1 .4 5 7 3
( 8 .7 5 0 8)
  

1.4573 0.00367  
1 .4 5 6 3 
   2 2
0.0002 & ' 8.6251f x f x   

65. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
19
Numerical Techniques
Sol.:
3 ap p ro x im atio n :
rd

 
 
2
3 2
2
'
f x
x x
f x
 
(0 .0 0 0 2 )
1 .4 5 3 6
( 8 .6 2 5 1)
  

1.4536 0.000023  
1 .4 5 3 6 
H ence the root of the equation correct
to three decim al places is -1.453.

66. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
20
Numerical Techniques
6. Iterative Formulae for finding qth root and reciprocal of
positive number N, Using N-R method
 
th
i q root of positive num ber N :
1
L et
q
x N
q
x N 
0
q
x N  
 Let 0
q
f x x N  
 
1
'
q
f x qx

 

67. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
21
Numerical Techniques
6. Iterative Formulae for finding qth root and reciprocal of
positive number N, Using N-R method
N ow from N ew ton-R aphson m ethod,
 
 
1
'
n
n n
n
f x
x x
f x

 
 
1
q
n
n q
n
x N
x
q x


 
 1
1
1
q q
n n n
q
n
q x x x N
q x


  
 
  
 1 1
1
1n n q
n
N
x q x
q x
 
 
   
 

68. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
22
Numerical Techniques
6. Iterative Formulae for finding qth root and reciprocal of
positive number N, Using N-R method
 ii R eciprocal of positive num ber N :
1
L et x
N

1
N
x
 
1
0N
x
  

69. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
23
Numerical Techniques
6. Iterative Formulae for finding qth root and reciprocal of
positive number N, Using N-R method
 
1
f x N
x
 
  2
1
'f x
x
 
N ow from N ew ton-R aphson form ula,
 
 
1
'
n
n n
n
f x
x x
f x

 
2
1
1
n
n
n
N
x
x
x
 
 
 
 
 
 
 

70. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
24
Numerical Techniques
6. Iterative Formulae for finding qth root and reciprocal of
positive number N, Using N-R method
1
2
1
1
n
n
n n
n
N x
x
x x
x

 
 
 
 
 
 
 
 1n n n
x N x x  
2
n n n
x x N x  
 1
2n n n
x x N x
 

71. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
25
Numerical Techniques
Ex-I: Find the value of correct to 4 decimal places using N-
R method.
Sol.:
8
L et N = 8 and q= 2
0
N ow 3 because 9 3x  
W h ich is n ear to 8 .
1 ap p ro x im atio n :
st

 1 0 1
0
1
1 q
N
x q x
q x

 
   
 
   
 
2 1
1 8
1 3
2 3

 
  
  
 
1
3 2 .6 6 6 6 7
2
 

72. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
26
Numerical Techniques
Sol.:
1
2 .8 3 3 3 3x 
2 ap p ro x im atio n :
n d

 2 1 1
1
1
1 q
N
x q x
q x

 
   
 
   
 
2 1
1 8
1 2 .8 3 3 3 3
2 2 .8 3 3 3 3

 
  
  
 
1
2 .8 3 3 3 3 2 .8 2 3 5 3
2
 
2
2 .8 2 8 4 3x 

73. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
27
Numerical Techniques
Sol.:
3 ap p ro x im atio n :
rd

 3 2 1
2
1
1 q
N
x q x
q x

 
   
 
   
 
2 1
1 8
1 2 .8 2 8 4 3
2 2 .8 2 8 4 3

 
  
  
 
1
2 .8 2 8 4 3 2 .8 2 8 4 2
2
 
2.82843
T h u s, th e valu e o f 8 co rrect to th e
fo u r d ecim al p laces is 2 .8 2 8 4.

74. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
28
Numerical Techniques
Ex-II: Find the value of 1/69 by using N- R method correct up
to 3 decimal places.
Sol.:
1 ap p ro x im atio n :
st

0
L et 6 9 an d
0 .0 1 (B ecau se 1 / 6 9 0 .0 1 ap p ro x im ately )
N
x

 
N ow by N -R form ula,
 1 0 0
2x x N x 
     0.01 2 69 0.01   
   0.01 1.31
0 .0 1 3 1

75. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
29
Numerical Techniques
Sol.:
2 ap p ro x im atio n :
n d

 2 1 1
2x x N x 
     0.0131 2 69 0.0131   
   0.0131 1.0961
0 .0 1 4 4
3 ap p ro x im atio n :
rd

 3 2 2
2x x N x 
     0.0144 2 69 0.0144   
   0.0144 1.0064

76. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
30
Numerical Techniques
Sol.:
3
0.0145x 
T h u s, th e valu e o f 1 / 6 9 co rrect to th e
th ree d ecim al p laces is 0 .0 1 4.

77. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
31
Numerical Techniques
Ex-III: Find the value of correct up to 4 decimal places
by using N- R method.
Sol.:
1 ap p ro x im atio n :
st

N ow by N -R form ula,
 
1
33 0

L et 30 and 3N q  
0
0 .3 3x 
3
1 1
B ecau se 0 .3 3
32 7
 
 1 0 1
0
1
1 q
N
x q x
q x

 
   
 

78. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
32
Numerical Techniques
Sol.:
   
 
1 4
1 3 0
4 0 .3 3
3 0 .3 3
x 
 
   
   
 
1
1 .3 2 0 .3 5 5 7 8
3
  

0 .3 2 1 4 1
2 ap p ro x im atio n :
n d

 2 1 1
1
1
1 q
N
x q x
q x

 
   
 

79. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
33
Numerical Techniques
Sol.:
3 ap p ro x im atio n :
rd

   
 
2 4
1 3 0
4 0 .3 2 1 4 1
3 0 .3 2 1 4 1
x 
 
   
   
 
1
1 .2 8 5 6 4 0 .3 2 0 1 5
3
  

0 .3 2 1 8 3
 3 2 1
2
1
1 q
N
x q x
q x

 
   
 

80. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
34
Numerical Techniques
Sol.:
   
 
3 4
1 3 0
4 0 .3 2 1 8 3
3 0 .3 2 1 8 3
x 
 
   
   
 
1
1 .2 8 7 3 2 0 .3 2 1 8 3
3
  

0 .3 2 1 8 3
 
1
3T hus, the value of 30 correct to the
four decim al places is 0.3218.


81. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
35
Numerical Techniques_Tutorial_Solution
Tutorial
Ex-I: Find the root of correct up to 3 decimal
places if by using N- R method.
 
3
cosf x x x 
0
1x 
Sol.:
 
3
0
Let cos and 1f x x x x  
 
2
' 3 sinf x x x  
   0 0
0.4597 & ' 3.8415f x f x  

82. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
36
Numerical Techniques_Tutorial_Solution
Sol.:
1 ap p ro x im atio n :
st

 
 
0
1 0
0
'
f x
x x
f x
 
0 .4 5 9 7
1
3 .8 4 1 5
 
0 .8 8 0 3
   1 1
0.0453 & ' 3.0957f x f x  

83. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
37
Numerical Techniques_Tutorial_Solution
Sol.:
2 ap p ro x im atio n :
n d

 
 
1
2 1
1
'
f x
x x
f x
 
0 .0 4 5 3
0 .8 8 0 3
3 .0 9 5 7
 
0 .8 6 5 7
   2 1
0.0007 & ' 3.00098f x f x  

84. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
38
Numerical Techniques_Tutorial_Solution
Sol.:
3 ap p ro x im atio n :
rd

 
 
2
3 2
2
'
f x
x x
f x
 
0 .0 0 0 7
0 .8 6 5 7
3 .0 0 9 8
 
0 .8 6 5 5
H en ce, th e ro o t o f g iven eq u atio n is 0 .8 6 5
co rrect u p to 3 d ecim al p laces.

85. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
39
Numerical Techniques_Tutorial_Solution
Tutorial
Ex-II: Find the root of the equation between
1.5 and 2 correct up to 4 decimal places by using N- R method.
Sol.:
  0
Let 1.2 sin 0.5 and 1.5f x x x x   
  1.2 sin 0.5f x x x  
 ' 1 1.2 cosf x x  
   0 0
0.19699 & ' 0.91511f x f x   

86. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
40
Numerical Techniques_Tutorial_Solution
Sol.:
1 ap p ro x im atio n :
st

 
 
0
1 0
0
'
f x
x x
f x
 
 0 .1 9 6 9 9
1 .5
0 .9 1 5 1 1

 
1 .7 1 5 2 7
   1 1
0.0277 & ' 1.17276f x f x  

87. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
41
Numerical Techniques_Tutorial_Solution
Sol.:
2 ap p ro x im atio n :
n d

 
 
1
2 1
1
'
f x
x x
f x
 
0 .0 2 7 7 7
1 .7 1 5 2 7
1 .1 7 2 7 6
 
1 .6 9 1 5 9
   2 1
0.00033 & ' 1.14460f x f x  

88. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
42
Numerical Techniques_Tutorial_Solution
Sol.:
3 ap p ro x im atio n :
rd

 
 
2
3 2
2
'
f x
x x
f x
 
0 .0 0 0 3 3
1 .6 9 1 5 9
1 .1 4 4 6 0
 
1 .6 9 1 3 0
   3 3
0.0000021 & ' 1.14425f x f x  

89. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
43
Numerical Techniques_Tutorial_Solution
Sol.:
4 ap p ro x im atio n :
th

 
 
3
4 3
3
'
f x
x x
f x
 
0 .0 0 0 0 0 2 1
1 .6 9 1 3 0
1 .1 4 4 2 5
 
1 .6 9 1 3 0
H en ce, th e ro o t o f g iven eq u atio n is 1 .6 9 1 3
co rrect u p to 4 d ecim al p laces.

90. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
44
Numerical Techniques_Tutorial_Solution
Tutorial
Sr. No. Question Answer
3 Find the positive root of
correct to 3 decimal places using N-R
method.
1.855
4 Solve correct up to
four decimal places if by N-R method.
0.8655
5 Solve correct up to 4
decimal places if =0.6 by N-R method.
0.5885
4
10x x 
  sin
x
f x x e

 
 
3
cosf x x x 

91. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
45
Numerical Techniques_Tutorial_Solution
Tutorial
Ex-VI: Derive Newton – Raphson’s formula for finding the cube
root of a positive number N. Hence find correct up to 4
decimal places.
Sol.:
3
1 2
1
3
L et x N
3
x N 
3
0x N  
 
3
f x x N  
 
2
' 3f x x 

92. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
46
Numerical Techniques_Tutorial_Solution
Sol.:
N ow , by N - R form ula,
 
 
1
'
n
n n
n
f x
x x
f x

 
 3
2
3
n
n
n
x N
x
x

 
3 3
2
3
3
n n
n
x x N
x
 

3
2
2
3
n
n
x N
x



93. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
47
Numerical Techniques_Tutorial_Solution
Sol.:
1 2
1
2
3
n n
n
N
x x
x

 
  
 
0
N o w , 1 2 & 2N x 
3
B ecau se 8 2 w h ich is n ear to 1 2
1 ap p ro x im atio n :
st

1 0 2
0
1 1 2
2
3
x x
x
 
  
 
 
1 1 2
2 2
3 4
 
 
 
 
1
2.33333x 

94. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
48
Numerical Techniques_Tutorial_Solution
Sol.:
2 ap p ro x im atio n :
n d

2 1 2
1
1 1 2
2
3
x x
x
 
  
 
1 1 2
4 .6 6 6 6 7
3 5 .4 4 4 4 3
 
 
 
 
2 .2 9 0 2 5
3 2 2
2
1 1 2
2
3
x x
x
 
  
 
3 ap p ro x im atio n :
rd


95. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
49
Numerical Techniques_Tutorial_Solution
Sol.:
3
1 1 2
4 .5 8 0 5
3 5 .2 4 5 2 4
x
 
 
 
 
2 .2 8 9 4 3
4 ap p ro x im atio n :
th

4 3 2
3
1 1 2
2
3
x x
x
 
  
 
1 1 2
4 .5 7 8 8 6
3 5 .2 4 1 4 9
 
 
 
 
4
2.28943x 
3
, 1 2 2 .2 8 9 4T h u s 

96. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
50
Numerical Techniques_Tutorial_Solution
Tutorial
Ex-VII: Find the values of correct up to 3 decimal places.
Sol.:
 
1
54 0

L et N = 40 and 5q  
 
0 1/ 5
1 1
0.5
232
x   
1 ap p ro x im atio n :
st

 1 0 1
0
1
1 q
N
x q x
q x

 
   
 

97. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
51
Numerical Techniques_Tutorial_Solution
Sol.:
   
 
1 6
1 4 0
6 0 .5
5 0 .5
x 
 
    
  
0 .4 7 5
2 ap p ro x im atio n :
n d

   
 
2 1 6
1
1 4 0
6
5
x x
x

 
    
  
 
1
2 .8 5 0 .4 5 9 4
5
   
0 .4 7 8 1

98. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
52
Numerical Techniques_Tutorial_Solution
Sol.:
3 ap p ro x im atio n :
rd

 3 2 6
2
1
6
5
N
x x
x

 
    
 
 
1
2 .8 6 8 6 0 .4 7 7 7
5
   
0 .4 7 8 2
 
1
5T h u s, 4 0 0 .4 7 8 co rrect u p to
3 d ecim al p laces.



99. Semester :III
Mr. Tushar j. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
53
Numerical Techniques_Tutorial_Solution
Tutorial
Sr. No. Question Answer
8 Find the value of correct up to
4 decimal places by using N – R
method.
3.8709
9 Find the value of correct up to
4 decimal places by using N – R
method.
4.1231
10 Find the value of 1/42 correct up to 3
decimal places by using N – R
method.
0.023
3
5 8
1 7

100. Semester :III
Mr. Tushar J. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
1
Numerical Techniques
Unit- 1: Roots of Equations
1. Introduction
2. Bisection Method with proof
3. False Position method with proof
4. Successive Approximation method
5. Newton Raphson (N-R)Method
6. Iterative Formulae for finding qth root, square
root and reciprocal of positive number N, Using N-R
method
7. Secant Method
8. Power Method
Table of contents

101. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
2
Numerical Techniques
1. Introduction
Def. : Eigen Values and Eigen Vectors:
Mr. Tushar J. Bhatt
1
2
L e t A b e a n y sq u a re m a trix o f o rd e r ,
th e n a n y n o n z e ro v e c to r
is sa id to b e a n e ig e n v e c to r o f a m a trix ,
if th e re e x ists a n u m b e r su c h th a t .
H e re is sa id to b e c h a ra c t
n
n n
x
x
X
x
A
A X X 


 
 
 
 
 
 

e ristic ro o t o r
e ig e n v a lu e o f th e m a trix .A

102. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
3
Numerical Techniques
1. Introduction
Note:
Mr. Tushar J. Bhatt
1) T h e su m o f th e eig en valu es o f a m atrix is eq u al to th e
su m o f its p rin cip al d iag o n al elem en t s.
2 ) T h e p ro d u ct o f all th e eig en valu es o f a m atrix is eq u al
to th e d eterm in an t o f th e m atrix .
Power Method:
T h e p o w er m eth o d is an iterative m eth o d to fin d
th e n u m erically larg est(d o m in an t) eig en v alu e an d
th e co rresp o n d in g eig en vecto r o f th e m atrix A .

103. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
4
Numerical Techniques
Working Rules:
L e t A b e a n y s q u a re m a trix .
Mr. Tushar J. Bhatt
0
0
1
i) T a k e fo r m a trix o f o rd e r 2 2
0
1
a n d 0 fo r m a trix o f o rd e r 3 3 ,
0
a s in itia l e ig e n v e c to r.
X
X
 
  
 
 
 
 
 
  
0
ii) F in d .A X
iii) T a k e n u m e ric a lly la rg e st n u m b e r c o m m o n
fro m th e re su lta n t v e c to r.

104. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
5
Numerical Techniques
Mr. Tushar J. Bhatt
1 2 2
iv) F in d an d d o th e ab o ve p ro cess to fin d .A X X
0 1 1
. . 1
if is la rg e s t n u m b e r a m o n g a ll , a n d .
a
a b
i e A X b b X
c c
b
b a b c

 
  
  
    
   
 
 
v) C o n tin u e th is p ro cess u n til tw o co n se cu tive sam e
eig en vecto rs are o b tain ed .
1
vi) If th en is th e larg est eig en valu e
an d is th e co rresp o n d in g eig en ve cto r.
i i i
i
X X
X



105. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
6
Numerical Techniques
0
1
L et 0
0
X
 
 

 
  
Sol.
Mr. Tushar J. Bhatt
0
A X 
1 1
X
1
A X  2 2
X
2 1 0
1 2 1
0 1 2
 
 
 
 
  
1
0
0
 
 
 
  
2
1
0
 
 
 
 
  
1
2 0 .5
0
 
 
 
 
  
2 1 0
1 2 1
0 1 2
 
 
 
 
  
1
0 .5
0
 
 

 
  
2 .5
2
0 .5
 
 
 
 
  
1
2 .5 0 .8
0 .2
 
 
 
 
  

106. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
7
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
2
A X  3 3
X
4
A X  5 5
X
2 1 0
1 2 1
0 1 2
 
 
 
 
  
1
0 .8
0 .2
 
 

 
  
2 .8
2 .8
1 .2
 
 
 
 
  
1
2 .8 1
0 .4 3
 
 
 
 
  
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .8 7
1
0 .5 4
 
 

 
  
2 .7 4
3 .4 1
2 .0 8
 
 
 
 
  
0 .8
3 .4 1 1
0 .6 1
 
 
 
 
  
3
A X 
2 1 0
1 2 1
0 1 2
 
 
 
 
  
1
1
0 .4 3
 
 

 
  
3
3 .4 3
1 .8 6
 
 
 
 
  
0 .8 7
3 .4 3 1
0 .5 4
 
 
 
 
  
4 4
X

107. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
8
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
5
A X 
6 6
X
7
A X  8 8
X
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .8
1
0 .6 1
 
 

 
  
2 .6
3 .4 1
2 .2 2
 
 
 
 
  
0 .7 6
3 .4 1 1
0 .6 5
 
 
 
 
  
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .7 4
1
0 .6 7
 
 

 
  
2 .4 8
3 .4 1
2 .3 4
 
 
 
 
  
0 .7 3
3 .4 1 1
0 .6 9
 
 
 
 
  
6
A X 
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .7 6
1
0 .6 5
 
 

 
  
2 .5 2
3 .4 1
2 .3
 
 
 
 
  
0 .7 4
3 .4 1 1
0 .6 7
 
 
 
 
  
7 7
X

108. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
9
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
8
A X 
9 9
X
1 0
A X  1 1 1 1
X
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .7 3
1
0 .6 9
 
 

 
  
2 .4 6
3 .4 2
2 .3 8
 
 
 
 
  
0 .7 2
3 .4 2 1
0 .7
 
 
 
 
  
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .7 1
1
0 .7
 
 

 
  
2 .4 2
3 .4 1
2 .4
 
 
 
 
  
0 .7 1
3 .4 1 1
0 .7
 
 
 
 
  
9
A X 
2 1 0
1 2 1
0 1 2
 
 
 
 
  
0 .7 2
1
0 .7
 
 

 
  
2 .4 4
3 .4 2
2 .4
 
 
 
 
  
0 .7 1
3 .4 2 1
0 .7
 
 
 
 
  
1 0 1 0
X

109. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
10
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
1 0 1 1
0 .7 1
H ere, 1
0 .7
X X
 
 
  
 
  
1 1
S o , 3 .4 1 is th e larg est eig en valu e o f th e m atrix A
0 .7 1
an d 1 is th e co rresp o n d in g eig en vecto r.
0 .7
 
 
 

 
  

110. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
11
Numerical Techniques
0
1
H ere 1
1
X
 
 

 
  
Sol.
Mr. Tushar J. Bhatt
0
A X 
1 1
X
1
A X 
2 2
X
2 1 1
1 3 2
1 2 3
 
 

 
  
1
1
1
 
 
 
  
2
4
6
 
 

 
  
0 .3 3
6 0 .6 7
1
 
 

 
  
0 .3 3
0 .6 7
1
 
 
 
  
0 .9 9
3 .6 8
4 .6 7
 
 

 
  
0 .2 1
4 .6 7 0 .7 9
1
 
 

 
  
2 1 1
1 3 2
1 2 3
 
 

 
  

111. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
12
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
2
A X  3 3
X
4
A X  5 5
X
0 .2 1
0 .7 9
1
 
 
 
  
0 .6 3
4 .1 6
4 .7 9
 
 

 
  
0 .1 3
4 .7 9 0 .8 7
1
 
 

 
  
0 .0 8
0 .9 2
1
 
 
 
  
0 .2 4
4 .6 8
4 .9 2
 
 

 
  
0 .0 5
4 .9 2 0 .9 5
1
 
 

 
  
3
A X 
0 .1 3
0 .8 7
1
 
 
 
  
0 .3 9
4 .4 8
4 .8 7
 
 

 
  
0 .0 8
4 .8 7 0 .9 2
1
 
 

 
  
4 4
X
2 1 1
1 3 2
1 2 3
 
 

 
  
2 1 1
1 3 2
1 2 3
 
 

 
  
2 1 1
1 3 2
1 2 3
 
 

 
  

112. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
13
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
5
A X 
6 6
X
7
A X  8 8
X
0 .0 5
0 .9 5
1
 
 
 
  
0 .1 5
4 .8
4 .9 5
 
 

 
  
0 .0 3
4 .9 5 0 .9 7
1
 
 

 
  
0 .0 2
0 .9 8
1
 
 
 
  
0 .0 6
4 .9 2
4 .9 8
 
 

 
  
0 .0 1
4 .9 8 0 .9 9
1
 
 

 
  
6
A X 
0 .0 3
0 .9 7
1
 
 
 
  
0 .0 9
4 .8 8
4 .9 7
 
 

 
  
0 .0 2
4 .9 7 0 .9 8
1
 
 

 
  
7 7
X
2 1 1
1 3 2
1 2 3
 
 

 
  
2 1 1
1 3 2
1 2 3
 
 

 
  
2 1 1
1 3 2
1 2 3
 
 

 
  

113. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
14
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
8
A X 
9 9
X
0 .0 1
0 .9 9
1
 
 
 
  
0 .0 3
4 .9 6
4 .9 9
 
 

 
  
0 .0 1
4 .9 9 0 .9 9
1
 
 

 
  
2 1 1
1 3 2
1 2 3
 
 

 
  
8 9
0 .0 1
H ere, 0 .9 9
1
X X
 
 
 
 
  
9
S o , 4 .9 9 5 is th e d o m in an t eig en valu e o f th e A
0 .0 1
an d 0 .9 9 is th e co rresp o n d in g eig en vec to r.
1
  
 
 
 
  

114. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
15
Numerical Techniques
0
1
L et
0
X
 
  
 
Sol.
Mr. Tushar J. Bhatt
0
A X 
4 2
1 3
 
 
 
1
0
 
 
 
4
1
 
  
 
1
4
0 .2 5
 
  
 
1 1
X
1
A X 
4 2
1 3
 
 
 
1
0 .2 5
 
 
 
4 .5
1 .7 5
 
  
 
1
4 .5
0 .3 9
 
  
 
2 2
X
2
A X 
4 2
1 3
 
 
 
1
0 .3 9
 
 
 
4 .7 8
2 .1 7
 
  
 
1
4 .7 8
0 .4 5
 
  
 
3 3
X

115. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
16
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
3
A X 
4 2
1 3
 
 
 
1
0 .4 5
 
 
 
4 .9
2 .3 5
 
  
 
1
4 .9
0 .4 8
 
  
 
4 4
X
4
A X 
4 2
1 3
 
 
 
1
0 .4 8
 
 
 
4 .9 6
2 .4 4
 
  
 
1
4 .9 6
0 .4 9
 
  
 
5 5
X
5
A X 
6
A X 
4 2
1 3
 
 
 
4 2
1 3
 
 
 
1
0 .4 9
 
 
 
4 .9 8
2 .4 7
 
  
 
1
4 .9 8
0 .5 0
 
  
 
6 6
X
1
0 .5 0
 
 
 
5
2 .5
 
  
 
1
5
0 .5 0
 
  
 
7 7
X

116. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Bisection
Method
17
Numerical Techniques
Sol.
Mr. Tushar J. Bhatt
6 7
1
H ere,
0 .5 0
X X
 
   
 
7
S o , 5 is th e larg est eig en valu e o f th e m atrix A
1
an d is th e co rresp o n d in g eig en vecto r.
0 .5 0
 
 
 
 
N ow , suppose second eigen value of A is ,x
S um of all diagonal elem ents = S um of eigen values
4 3 5 x   
2x 
S o, 2 is the another eigen value of A .

117. Semester :III
Mr. Tushar J. Bhatt
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Table of
content
1
Numerical Techniques
Unit- 1: Roots of Equations
1. Introduction
2. Bisection Method with proof
3. False Position method with proof
4. Newton Raphson (N-R)Method
5. Iterative Formulae for finding qth root, square
root and reciprocal of positive number N, Using N-R
method
6. Secant Method
7. Successive Approximation method
8. Power Method
Table of content:

118. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
2
Numerical Techniques
Secant Method
In N -R M e th o d , tw o fu n c tio n s f(x ) a n d f '(x ) a re re q u ire d
to b e e v a lu a te d p e r s te p .

Mr. Tushar J. Bhatt
If th e y a re n o t c o m p lic a te d e x p re s s io n s th e n N -R M e th o d
is d e s ira b le .

T h e S e c a n t m e th o d w ill re q u ire o n ly o n e v a lu e f(x )
a n d is a lm o s t a s fa s t a s N -R M e th o d .

   
   
1
1
1
T h e G e n e ra l F o rm u la fo r s e c a n t m e th o d i s
n n n
n n
n n
f x x x
x x
f x f x





 


119. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
3
Numerical Techniques
Mr. Tushar J. Bhatt
Secant Method
W O R K IN G R U L E S :
0 1
S tep -1 : F in d th e in itial valu es x an d x (if n o t g iven ).
   
   
1 1 0
2 1
1 0
S tep -2 : O b tain th e first ap p ro x im atio n u sin g th e secan t fo rm u la:
f x x x
x x
f x f x

 

   
   
1
1 3 4 5
1
S tep -3 : U se th e fo rm u la to fin d x , x , x ,....
an d co n tin u e th e p ro cess u n till th e ro o ts are fo u n d u p to d esired accu racy.
n n n
n n
n n
f x x x
x x
f x f x




  

S u p p o se w e w a n t to fin d th e ro o t o f th e e q u a tio n f(x )= 0 .

120. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
4
Numerical Techniques
Ex-1: Find a real root of the equation by secant
method starting from and correct upto three
decimal places
3
2 5 0x x  
3
L e t ( ) 2 5f x x x  
   
3
0
2 2 2 2 5 1f x f      
Sol
Mr. Tushar J. Bhatt
0
2x  1
3x 
0 1
H e re 2 a n d 3x x 
   
3
1
3 3 2 3 5 1 6f x f     

121. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
5
Numerical Techniques
Mr. Tushar J. Bhatt
   
   
1 1 0
2 1
1 0
B y S e c a n t M e th o d ,
f x x x
x x
f x f x

 

   
   
1 6 3 2
3
1 6 1

 
 
2
2 .0 5 8 8x 
   
 
2
3
2 .0 5 8 8
2 .0 5 8 8 2 2 .0 5 8 8 5 0 .3 9 1 1
f x f
    
3
2 5A ns A ns  

122. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
6
Numerical Techniques
Mr. Tushar J. Bhatt
3
2 .0 8 1 3x 
   
   
0 .3 9 1 1 2 .0 5 8 8 3
2 .0 5 8 8
0 .3 9 1 1 1 6
 
 
 
   
   
2 2 1
3 2
2 1
f x x x
x x
f x f x

 

   
 
3
3
2 .0 8 1 3
2 .0 8 1 3 2 2 .0 8 1 3 5 0 .1 4 7 3
f x f
    

123. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
7
Numerical Techniques
Mr. Tushar J. Bhatt
4
2 .0 9 4 9x 
   
   
0 .1 4 7 3 2 .0 8 1 3 2 .0 5 8 8
2 .0 8 1 3
0 .1 4 7 3 0 .3 9 1 1
 
 
  
   
 
4
3
2 .0 9 4 9
2 .0 9 4 9 2 2 .0 9 4 9 5 0 .0 0 3 8
f x f
   
   
   
3 3 2
4 3
3 2
f x x x
x x
f x f x

 


124. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
8
Numerical Techniques
Mr. Tushar J. Bhatt
4
2 .0 9 4 6x 
   
   
0 .0 0 3 8 2 .0 9 4 9 2 .0 8 1 3
2 .0 9 4 9
0 .0 0 3 8 0 .1 4 7 3

 
 
H en ce th e real ro o t co rrect
u p to th ree d ecim al p laces is 2 .0 9 4
   
   
4 4 3
5 4
4 3
f x x x
x x
f x f x

 


125. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
9
Numerical Techniques
Ex-2 : Find the positive root of the equation by
secant method correct upto two decimal places.
0
x
e x

 
L e t ( )
x
f x e x

 
   
0
0
0 0 1f x f e   
Sol.
Mr. Tushar J. Bhatt
A tleast one root lies betw een 0 and 1.
   
1
1
1 1 0 .6 3 2f x f e

    
 
0 1
1 0 .6 3 2
x
f x 
0 1
L e t x 0 1a n d x 

126. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
10
Numerical Techniques
Mr. Tushar J. Bhatt
   
   
1 1 0
2 1
1 0
B y S e c a n t M e th o d ,
f x x x
x x
f x f x

 

   
   
0 .6 3 2 1 0
1
0 .6 3 2 1
 
 
 
2
0 .6 1 3x 
   2
0 .6 1 3
0 .6 1 3
0 .6 1 3
0 .0 7 1
f x f
e


 
 
A ns
e A ns

 

127. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
11
Numerical Techniques
Mr. Tushar J. Bhatt
3
0 .5 6 4x 
   
   
0 .0 7 1 0 .6 1 3 1
0 .6 1 3
0 .0 7 1 0 .6 3 2
 
 
  
   
   
2 2 1
3 2
2 1
f x x x
x x
f x f x

 

   3
0 .5 6 4
0 .5 6 4
0 .5 6 4
0 .0 0 5
f x f
e


 


128. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
12
Numerical Techniques
Mr. Tushar J. Bhatt
H ence the root correct upto tw o decim al places is 0.56.
4
0 .5 6 7x 
   
   
0 .0 0 5 0 .5 6 4 0 .6 1 3
0 .5 6 4
0 .0 0 5 0 .0 7 1

 
 
   
   
3 3 2
4 3
3 2
f x x x
x x
f x f x

 


129. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
13
Numerical Techniques
Ex-3 : Find the positive root of the equation by
secant method starting from and correct upto six
decimal places
2 sin 0x x 
L et ( ) 2 sinf x x x 
   0
2 2 2 sin 2 0 .1 8 1 4 0 5 1f x f   
Sol.
Mr. Tushar J. Bhatt
0
2x  1
1 .9x 
0 1
H e re 2 a n d 1 .9x x 
   1
1 .9 1 .9 2 sin 1 .9 0 .0 0 7 3 9 9 8f x f   

130. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
14
Numerical Techniques
Mr. Tushar J. Bhatt
   
   
1 1 0
2 1
1 0
B y S e c a n t M e th o d ,
f x x x
x x
f x f x

 

   
   
0 .0 0 7 3 9 9 8 1 .9 2
1 .9
0 .0 0 7 3 9 9 8 0 .1 8 1 4 0 5 1

 

   
 
0 .0 0 7 4 0 .1
1 .9
0 .1 7 4

 

2
1 .8 9 5 7 4 7 4x 
   2
1 .8 9 5 7 4 7 4
1 .8 9 5 7 4 7 4 2 s in 1 .8 9 5 7 4 7 4
0 .0 0 0 4 1 4 7
f x f
 

2 sinAns Ans 

131. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
15
Numerical Techniques
Mr. Tushar J. Bhatt
3
1 .8 9 5 4 9 4 9x 
   
   
0 .0 0 0 4 1 4 7 1 .8 9 5 7 4 7 4 1 .9
1 .8 9 5 7 4 7 4
0 .0 0 0 4 1 4 7 0 .0 0 7 3 9 9 8

 

   
   
2 2 1
3 2
2 1
f x x x
x x
f x f x

 

   3
1 .8 9 5 4 9 4 9
1 .8 9 5 4 9 4 9 2 s in 1 .8 9 5 4 9 4 9
0 .0 0 0 0 0 1 1
f x f
 


132. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
16
Numerical Techniques
Mr. Tushar J. Bhatt
H en ce th e p o sitive ro o t co rrect
u p to th ree d ecim al p laces is 1.8 9 5 4 9 4
4
1 .8 9 5 4 9 4 2x 
   
   
0 .0 0 0 0 0 1 1 1 .8 9 5 4 9 4 9 1 .8 9 5 7 4 7 4
1 .8 9 5 4 9 4 9
0 .0 0 0 0 0 1 1 0 .0 0 0 4 1 4 7

 

   
   
3 3 2
4 3
3 2
f x x x
x x
f x f x

 


133. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
17
Numerical Techniques
Mr. Tushar J. Bhatt
Successive Approximation Method (Iteration Method)
S u p p o se w e w a n t to fin d th e ro o t o f th e e q u a tio n f(x )= 0 .
 R ew rite th e g iven eq u atio n as .x g x 
0
L e t x b e a n in itia l a p p ro x im a tio n .
 1 0
T h e firs t a p p ro x im a tio n is
x g x


 2 1
T h e s e c o n d a p p ro x im a tio n is
x g x


 3 2
T h e th ird a p p ro x im a tio n is
x g x


 1
In g e n e ra l,
, 0 ,1, 2 , 3, .....n n
x g x n

 
1 2 3 4
O b ta in , , , , ...... u n till w e g e t th e ro o t
u p to d e s ire d a c c u ra c y.
x x x x

134. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
18
Numerical Techniques
Mr. Tushar J. Bhatt
Successive Approximation Method (Iteration Method)
 
 
0
L e t I b e th e in te rv a l c o n ta in in g th e e x a c t ro o t
o f th e e q u a tio n f(x )= 0 ,
i.e .,
If ' 1; fo r a ll x in I,
th e n o n ly th is m e th o d is a p p lic a b le .
x x
x g x
g x



Condition for Convergence:

135. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
19
Numerical Techniques
Ex-1 : Solve the equation by successive
approximation method correct upto 3 decimal places.
 
1
cos 1
3
x x 
 
1
G iv e n th a t c o s 1
3
x x 
Sol.
Mr. Tushar J. Bhatt
0
L e t 0 .5x 
   
1
L e t c o s 1
3
f x x x  
 
0 1
0 .6 6 6 7 0 .4 8 6 6
x
f x 
A tleast one root lies betw een 0 and 1.
   
1
L e t g c o s 1
3
x x 
   
1
g ' sin
3
x x 

136. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
20
Numerical Techniques
Mr. Tushar J. Bhatt
   
1
g ' 0 .1 s in 0 .1 0 .0 3 3 3 1
3
   
   
1
g ' 0 .5 s in 0 .5 0 .1 5 9 8 1
3
   
   
1
g ' 0 .9 s in 0 .9 0 .2 6 1 1 1
3
   
   H ence,g' 1; for all x in 0,1 .x 
S u ccessive A p p ro x im atio n m eth o d is ap p licab le.

137. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
21
Numerical Techniques
T h e ro o t is 0 .6 0 7 co rrect u p to 3 d ecim al p laces.
Mr. Tushar J. Bhatt
     1 0
B y S u c c e s s iv e A p p ro x im a tio n M e th o d ,
1
0 .5 c o s 0 .5 1 0 .6 2 5 9
3
x g x g    
     2 1
1
0 .6 2 5 9 c o s 0 .6 2 5 9 1 0 .6 0 3 5
3
x g x g    
     3 2
1
0 .6 0 3 5 c o s 0 .6 0 3 5 1 0 .6 0 7 8
3
x g x g    
     4 3
1
0 .6 0 7 8 c o s 0 .6 0 7 8 1 0 .6 0 7 0
3
x g x g    
 
1
c o s 1
3
A n s  

138. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
22
Numerical Techniques
Ex-2 : Solve the equation by successive
approximation method correct upto 4 decimal places.
3 2
1 0x x  
3 2
N o w 1 0 c a n b e re -w ritte n in fo llo w in g w a y s :x x  
Sol.
Mr. Tushar J. Bhatt
0
L e t 0 .5x 
 
3 2
L et 1f x x x  
 
0 1
1 1
x
f x 
A tleast one root lies betw een 0 and 1.
 
3 2
2
2
(1 ) 1 0
1 1 0
1
1
1
1
x x
x x
x
x
x
x
  
   
 

 


139. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
23
Numerical Techniques
 
3 2
3 2
1
2 3
(2 ) 1 0
1
1
x x
x x
x x
  
  
  
Mr. Tushar J. Bhatt
 
3 2
2 3
1
3 2
(3 ) 1 0
1
1
x x
x x
x x
  
  
  
 
1
F o r (1 ), L e t g
1
x
x


   
3
2
1
g ' 1
2
x x

  
   
3
2
1
g ' 0 .1 0 .1 1 0 .4 3 3 3 9 1
2

    
S u ccessive A p p ro x im atio n m eth o d is ap p li cab le fo r (1 ).
   
3
2
1
g ' 0 .5 0 .5 1 0 .2 7 2 1 7 1
2

    
   
3
2
1
g ' 0 .9 0 .9 1 0 .1 9 0 9 1 1
2

    
   H ence,g' 1; for all x in 0,1 .x 

140. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
24
Numerical Techniques
T h e ro o t is 0 .7 5 4 8 co rrect u p to 4 d ecim a l p laces.
Mr. Tushar J. Bhatt
   1 0
B y S u c c e s s iv e A p p ro x im a tio n M e th o d ,
1
0 .5 0 .8 1 6 5 0
0 .5 1
x g x g   

   2 1
1
0 .8 1 6 5 0 0 .7 4 1 9 6
0 .8 1 6 5 0 1
x g x g   

   3 2
1
0 .7 4 1 9 6 0 .7 5 7 6 7
0 .7 4 1 9 6 1
x g x g   

   4 3
1
0 .7 5 7 6 7 0 .7 5 4 2 8
0 .7 5 7 6 7 1
x g x g   

 
1
F ro m (1 ), L e t g
1
x
x


   5 4
1
0 .7 5 4 2 8 0 .7 5 5 0 1
0 .7 5 4 2 8 1
x g x g   

   6 5
1
0 .7 5 5 0 1 0 .7 5 4 8 5
0 .7 5 5 0 1 1
x g x g   

   7 6
1
0 .7 5 4 8 5 0 .7 5 4 8 8
0 .7 5 4 8 5 1
x g x g   

1
1A n s



141. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
25
Numerical Techniques
Ex-3 : Solve the equation by iteration method
correct upto 4 decimal places.
10
x
e x


G iv e n th a t 1 0
1 0
x
x e
e x x


  
Sol.
Mr. Tushar J. Bhatt
0
L e t 0 .5x 
 L et 1 0
x
f x e x

 
 
0 1
1 9 .6 3 2 1 2
x
f x 
A tleast one root lies betw een 0 and 1.
 L e t g
1 0
x
e
x


   
1
g '
1 0
x
x e

 

142. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
26
Numerical Techniques
Mr. Tushar J. Bhatt
   0 .11
g ' 0 .1 0 .0 9 0 4 8 1
1 0
e

   
   0 .51
g ' 0 .5 0 .0 6 0 6 5 1
1 0
e

   
   0 .91
g ' 0 .9 0 .0 4 0 6 6 1
1 0
e

   
   H ence,g' 1; for all x in 0,1 .x 
S u ccessive A p p ro x im atio n m eth o d is ap p licab le.

143. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
27
Numerical Techniques
T h e ro o t is 0 .0 9 1 2 co rrect u p to 4 d ecim a l p laces.
Mr. Tushar J. Bhatt
   
0 .5
1 0
B y S u c c e s s iv e A p p ro x im a tio n M e th o d ,
0 .5 0 .0 6 0 6 5
1 0
e
x g x g

   
   
0 .0 6 0 6 5
2 1
0 .0 6 0 6 5 0 .0 9 4 1 1
1 0
e
x g x g

   
   
0 .0 9 4 1 1
3 2
0 .0 9 4 1 1 0 .0 9 1 0 2
1 0
e
x g x g

   
   
0 .0 9 1 0 2
4 3
0 .0 9 1 0 2 0 .0 9 1 3 0
1 0
e
x g x g

   
   
0 .0 9 1 3 0
5 4
0 .0 9 1 3 0 0 .0 9 1 2 7
1 0
e
x g x g

   
   
0 .0 9 1 2 7
6 5
0 .0 9 1 2 7 0 .0 9 1 2 8
1 0
e
x g x g

   
1 0
A n s
e



144. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
28
Numerical Techniques
Mr. Tushar J. Bhatt
Rate of Convergence of Iterative Methods :
In N u m e ric a l A n a lys is , th e s p e e d a t w h ic h th e
a p p ro x im a tio n s a p p ro a c h e s th e e x a c t ro o t o f th e
e q u a tio n is c a lle d ra te o f c o n v e rg e n c e .
Sr. No. Iterative Method Rate of Convergence
1. Bisection Method 1
2. False Position Method 1
3. Newton – Raphson Method 2
4. Secant Method 1.618
5. Successive Approximation
Method
1
T h e fo llo w in g ta b le g iv e s th e ra te o f c o n v e rg e n c e
o f a ll ite ra tiv e m e th o d s w e h a v e s tu d ie d :

145. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
29
Ex-1 Solve by taking and correct
upto 3 decimal places using Secant Method
3 2
L e t ( ) 2 3 1f x x x x   
   
3 2
0
2 2 2 2 3 2 1 9f x f       
Sol
Mr. Tushar J. Bhatt
0 1
H e re 2 a n d 1x x 
3 2
2 3 1 0x x x    0
2x  1
1x 
   
3 2
1
1 1 2 1 3 1 1 1f x f        
Tutorial -4
Numerical Techniques_Tutorial_Solution

146. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
30
Mr. Tushar J. Bhatt
   
   
1 1 0
2 1
1 0
B y S e c a n t M e th o d ,
f x x x
x x
f x f x

 

   
   
1 1 2
1
1 9
 
 
 
2
1 .1x 
   
     
2
3 2
1 .1
1 .1 2 1 .1 3 1 .1 1
0 .5 4 9 0
f x f
   
 
Numerical Techniques_Tutorial_Solution
3 2
2 3 1A n s A n s A n s   

147. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
31
Mr. Tushar J. Bhatt
3
1 .2 2 1 7x 
   
   
0 .5 4 9 0 1 .1 1
1 .1
0 .5 4 9 0 1
 
 
  
   
   
2 2 1
3 2
2 1
f x x x
x x
f x f x

 

   
     
3
3
1 .2 2 1 7
1 .2 2 1 7 2 1 .2 2 1 7 3 1 .2 2 1 7 1
0 .1 4 3 6
f x f
   

Numerical Techniques_Tutorial_Solution

148. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
32
Mr. Tushar J. Bhatt
4
1 .1 9 6 5x 
   
   
0 .1 4 3 6 1 .2 2 1 7 1 .1
1 .2 2 1 7
0 .1 4 3 6 0 .5 4 9 0

 
 
   
     
4
3 2
1 .1 9 6 5
1 .1 9 6 5 2 1 .1 9 6 5 3 1 .1 9 6 5 1
0 .0 1 3 6
f x f
   
 
   
   
3 3 2
4 3
3 2
f x x x
x x
f x f x

 

Numerical Techniques_Tutorial_Solution

149. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
33
Mr. Tushar J. Bhatt
5
1 .1 9 8 7x 
   
   
0 .0 1 3 6 1 .1 9 6 5 1 .2 2 1 7
1 .1 9 6 5
0 .0 1 3 6 0 .1 4 3 6
 
 
 
   
   
4 4 3
5 4
4 3
f x x x
x x
f x f x

 

   
     
5
3 2
1 .1 9 8 7
1 .1 9 8 7 2 1 .1 9 8 7 3 1 .1 9 8 7 1
0 .0 0 0 1
f x f
   
 
Numerical Techniques_Tutorial_Solution

150. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
34
Mr. Tushar J. Bhatt
6
1 .1 9 8 7x 
   
   
0 .0 0 0 1 1 .1 9 8 7 1 .1 9 6 5
1 .1 9 8 7
0 .0 0 0 1 0 .0 1 3 6
 
 
  
H en ce th e ro o t co rrect
u p to 3 d ecim al p laces is 1 .1 9 8 .
   
   
5 5 4
6 5
5 4
f x x x
x x
f x f x

 

Numerical Techniques_Tutorial_Solution

151. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
35
Ex-2 Solve by taking and correct upto
4 decimal places using Secant Method.
L e t ( ) 3 s in
x
f x e x x  
   
0
0
0 3 0 sin 0 1f x f e     
Sol
Mr. Tushar J. Bhatt
0 1
H e re 0 a n d 1x x 
3 sin 0
x
e x x   0
0x  1
1x 
   
1
1
1 3 1 sin 1 1 .1 2 3 1 9f x f e      
Numerical Techniques_Tutorial_Solution

152. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
36
Mr. Tushar J. Bhatt
   
   
1 1 0
2 1
1 0
B y S e c a n t M e th o d ,
f x x x
x x
f x f x

 

   
   
1 .1 2 3 1 9 1 0
1
1 .1 2 3 1 9 1
 
 
 
2
0 .4 7 0 9 9x 
   
   
2
0 .4 7 0 9 9
0 .4 7 0 9 9
3 0 .4 7 0 9 9 s in 0 .4 7 0 9 9
0 .2 6 5 1 6
f x f
e

  
 
Numerical Techniques_Tutorial_Solution

153. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
37
Mr. Tushar J. Bhatt
3
0 .3 0 7 5 1x 
   
   
0 .2 6 5 1 6 0 .4 7 0 9 9 1
0 .4 7 0 9 9
0 .2 6 5 1 6 1 .1 2 3 1 9
 
 
  
   
   
2 2 1
3 2
2 1
f x x x
x x
f x f x

 

   
   
3
0 .3 0 7 5 1
0 .3 0 7 5 1
3 0 .3 0 7 5 1 s in 0 .3 0 7 5 1
0 .1 3 4 8 2
f x f
e

  

Numerical Techniques_Tutorial_Solution
3 sin
A n s
e A n s A n s  

154. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
38
Mr. Tushar J. Bhatt
4
0 .3 6 2 6 1x 
   
   
0 .1 3 4 8 2 0 .3 0 7 5 1 0 .4 7 0 9 9
0 .3 0 7 5 1
0 .1 3 4 8 2 0 .2 6 5 1 6

 
 
   
   
3 3 2
4 3
3 2
f x x x
x x
f x f x

 

   
   
4
0 .3 6 2 6 1
0 .3 6 2 6 1
3 0 .3 6 2 6 1 s in 0 .3 6 2 6 1
0 .0 0 5 4 8
f x f
e

  
 
Numerical Techniques_Tutorial_Solution

155. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
39
Mr. Tushar J. Bhatt
5
0 .3 6 0 4 6x 
   
   
0 .0 0 5 4 8 0 .3 6 2 6 1 0 .3 0 7 5 1
0 .3 6 2 6 1
0 .0 0 5 4 8 0 .1 3 4 8 2
 
 
 
   
   
4 4 3
5 4
4 3
f x x x
x x
f x f x

 

   
   
5
0 .3 6 0 4 6
0 .3 6 0 4 6
3 0 .3 6 0 4 6 s in 0 .3 6 0 4 6
0 .0 0 0 0 9
f x f
e

  
 
Numerical Techniques_Tutorial_Solution

156. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Secant
Method
40
Mr. Tushar J. Bhatt
6
0 .3 6 0 4 2x 
   
   
0 .0 0 0 0 9 0 .3 6 0 4 6 0 .3 6 2 6 1
0 .3 6 0 4 6
0 .0 0 0 0 9 0 .0 0 5 4 8
 
 
  
H en ce th e ro o t co rrect
u p to 4 d ecim al p laces is 0 .3 6 0 4 .
   
   
5 5 4
6 5
5 4
f x x x
x x
f x f x

 

Numerical Techniques_Tutorial_Solution

157. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
41
Numerical Techniques_Tutorial_Solution
Sr. No. Question Answer
3 Find the root of
correct upto 5 decimal places by taking
and using Secant method.
2.94282
4 Find the root of
correct upto 4 decimal places by taking
and using Secant method.
0.9045
5 Find the root of
correct upto 4 decimal places by taking
and using Secant method.
0.5177
3
9 1 0x x  
Mr. Tushar J. Bhatt
0
2x 
1
3x 
sin cotx x
cos 0
x
xe x 
0
1x 
0
0x 
1
0 .5x 
1
1x 

158. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
42
Numerical Techniques
Ex-6 : Solve the equation by successive
approximation method correct upto 4 decimal places.
1 0
2 7 logx x 
 1 0
1
G iv e n th a t 7 lo g
2
x x 
Sol.
Mr. Tushar J. Bhatt
0
L e t 3 .5x 
  10
L et 2 7 lo gf x x x  
 
1 2 3 4
5 3 .3 0 1 0 3 1 .4 7 7 1 2 0 .3 9 7 9 4
x
f x   
A tleast one root lies betw een 3 and 4.
   1 0
1
L e t g 7 lo g
2
x x 
  1 0
lo g1 1 1 1
g ' 0 lo g
2 lo g 1 0 2 lo g 1 0 lo g 1 0
e
e e e
x
x x
x x
   
       
   

159. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
43
Numerical Techniques
Mr. Tushar J. Bhatt
 
 
1
g ' 3 .1 0 .0 7 0 0 5 1
2 3 .1 lo g 1 0e
  
   H ence,g' 1; for all x in 3, 4 .x 
S u ccessive A p p ro x im atio n m eth o d is ap p licab le.
 
 
1
g ' 3 .5 0 .0 6 2 0 4 1
2 3 .5 lo g 1 0e
  
 
 
1
g ' 3 .9 0 .0 5 5 6 8 1
2 3 .9 lo g 1 0e
  

160. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
44
Numerical Techniques
T h e ro o t is 3 .7 8 9 2 co rrect u p to 4 d ecim a l p laces.
Mr. Tushar J. Bhatt
     1 0 1 0
B y S u c c e s s iv e A p p ro x im a tio n M e th o d ,
1
3 .5 7 lo g 3 .5 3 .7 7 2 0 3
2
x g x g    
     2 1 1 0
1
3 .7 7 2 0 3 7 lo g 3 .7 7 2 0 3 3 .7 8 8 2 9
2
x g x g    
 1 0
1
7 lo g
2
A n s 
     3 2 1 0
1
3 .7 8 8 2 9 7 lo g 3 .7 8 8 2 9 3 .7 8 9 2 2
2
x g x g    
     4 3 1 0
1
3 .7 8 9 2 2 7 lo g 3 .7 8 9 2 2 3 .7 8 9 2 7
2
x g x g    

161. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
45
Numerical Techniques
Ex-7 : Solve the equation by successive
approximation method correct upto 3 decimal places.
4
1 0x x  
4
N o w 1 0 c a n b e re -w ritte n in fo llo w in g w a y s :x x  
Sol.
Mr. Tushar J. Bhatt
0
L e t 1 .5x 
 
3 2
L et 1f x x x  
 
0 1 2
1 1 1 3
x
f x  
A tleast one root lies betw een 1 and 2.
 
4
3
3
(1 ) 1 0
1 1 0
1
1
x x
x x
x
x
  
   
 


162. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
46
Numerical Techniques
 
4
4
1
4
(2 ) 1 0
1
1
x x
x x
x x
  
  
  
Mr. Tushar J. Bhatt
4
4
(3 ) 1 0
1
x x
x x
  
  
  3
1
F o r (1 ), L e t g
1
x
x


 
 
2
2
3
3
g '
1
x
x
x
 

 
 
 
2
2
3
3 1 .1
g ' 1 .1 3 3 .1 3 2 2 1
1 .1 1
   

S u ccessive A p p ro x im atio n m eth o d is n o t a p p licab le fo r (1 ).
   H en ce,g ' is n o t 1; fo r all x in 1, 2 .x 

163. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
47
Numerical Techniques
Mr. Tushar J. Bhatt
   
1
4F o r (2 ), L e t g 1x x 
   
3
4
1
g ' 1
4
x x

 
   
3
4
1
g ' 1 .1 1 .1 1 0 .1 4 3 3 1
4

   
S u ccessive A p p ro x im atio n m eth o d is ap p li cab le fo r (2 ).
   H ence,g' 1; for all x in 1, 2 .x 
   
3
4
1
g ' 1 .5 1 .5 1 0 .1 2 5 7 1
4

   
   
3
4
1
g ' 1 .9 1 .9 1 0 .1 1 2 5 1
4

   

164. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : Successive
Approximation
Method
48
Numerical Techniques
T h e ro o t is 1 .2 2 0 co rrect u p to 3 d ecim al p laces.
Mr. Tushar J. Bhatt
     
1
4
1 0
B y S u ccessiv e A p p ro x im atio n M eth o d ,
1 .5 1 .5 1 1 .2 5 7 4x g x g    
   
1
4F ro m (2 ), L e t g 1x x   
1
41A n s  
     
1
4
2 1
1 .2 5 7 4 1 .2 5 7 4 1 1 .2 2 5 8x g x g    
     
1
4
3 2
1 .2 2 5 8 1 .2 2 5 8 1 1 .2 2 1 4x g x g    
     
1
4
4 3
1 .2 2 1 4 1 .2 2 1 4 1 1 .2 2 0 8x g x g    
     
1
4
5 4
1 .2 2 0 8 1 .2 2 0 8 1 1 .2 2 0 8x g x g    

165. Semester :III
Subject : NT
Code :18SAHMT301
Unit No. :1
Topic : N-R Method
49
Numerical Techniques_Tutorial_Solution
Sr. No. Question Answer
8 Solve the equation by
successive approximation method
correct upto 4 decimal places.
1.3247
9 Solve the equation by
successive approximation method
correct upto 4 decimal places.
0.2607
10 Solve the equation by
successive approximation method
correct upto 4 decimal places.
2.1322
Mr. Tushar J. Bhatt
3
1 0x x  
3 2 sin
x
x x e 
1
1 tanx x

 