Maximum Marks: 20 Due Date: 24 -11-2019

DON’T MISS THESE: Important instructions before attempting the solution of this assignment:

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Question #1: Find the root of the equation, Perform three iteration of the equation,

ln (x −1) + sinx = 0 by using Newton Raphson method.

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

x + y + z = 6; 2x − y + z = 3; x + z = 4

Maximum Marks: 20 Due Date: 24 -11-2019

DON’T MISS THESE: Important instructions before attempting the solution of this assignment:

• To solve this assignment, you should have good command over 01 - 12 lectures.

• Try to get the concepts, consolidate your concepts and ideas from these questions which you learn in the 01 to 12 lectures.

• Upload assignments properly through LMS, No Assignment will be accepted through email.

• Write your ID on the top of your solution file.

• Don’t use colourful back grounds in your solution files.

• Use Math Type or Equation Editor Etc. for mathematical symbols.

• You should remember that if we found the solution files of some students are same then we will reward zero marks to all those students.

• Try to make solution by yourself and protect your work from other students, otherwise you and the student who send same solution file as you will be given zero mark.

• Also remember that you are supposed to submit your assignment in Word format any other like scan images etc. will not be accepted and we will give zero mark corresponding to these assignments.

Question #1: Find the root of the equation, Perform three iteration of the equation,

ln (x −1) + sinx = 0 by using Newton Raphson method.

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

x + y + z = 6; 2x − y + z = 3; x + z = 4

Question #1: Find the root of the equation, Perform three iteration of the equation,

ln (x −1) + sinx =0 by using Newton Raphson method.

Ans: Let f(x) = ln(x+1) + sinx = 0 and f(x) = 1/(x-1) + cosx

F (1.5) = ln(0.5) + (1.5) = - 0.0667

F(2) = ln(1) + sin(2) = 0.035

Since f (1.5) f (2) < 0 so roots lies in interval [1.5, 2]

Let x0 = 1.75 . x0 can be taken in the interval any real number [ 1.5 , 2 ], we let mid point

of this interval .

As we know Newton Raphson method is

Xn+1 = xn – f ( xn ) / f(xn)

First iteration

X1 = x0 –f(x0) / f(x0) = 1.75 - f(1.75) / f(1.75)

= 1.75 – (-0.2571 / 2.3329) = 1.8602

Second iteration:

X2 = x1 - f(x) / f(x) = 1.8602 –[ f(1.8602) / f(1.8602)]

= 1.8602 - ( -0.1181 / 2.1620 ) = 1.9148

Third iteration:

X3 = x2- f(x2) / f(x2) = 1.9148 –f(1.9148) / f(1.9148)

= 1.9148 – [-0.0556/2.0926]

= 1.9414

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

x + y + z = 6;2x − y + z = 3;x + z = 4

ANS: In Gaussian elimination method we convert the augmented matrix into reduce

Echelon form therefore,

Augmented matrix is

R2- 2R1 , R3 – R1

-1R2 , -1R3

R23

R3-3R2

X + Y+ Z = 6 ;………………….(1)

Y = 2,

Z = 3

Put into eq (1),

we get X = 1 ,